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<title>Forcing (mathematics)</title>

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Forcing (mathematics)

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</style><div role="note" class="hatnote navigation-not-searchable">For the use of forcing in <a href="Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>, see <a href="Forcing_(computability)" title="Forcing (computability)">Forcing (computability)</a>.</div>

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In the mathematical discipline of <a href="Set_theory" title="Set theory">set theory</a>, forcing is a technique for proving <a href="Consistency" title="Consistency">consistency</a> and <a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">independence</a> results. Intuitively, forcing can be thought of as a technique to expand the set theoretical <a href="Universe_(mathematics)" title="Universe (mathematics)">universe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> to a larger universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

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<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

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</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy"> by introducing a new "generic" object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

Forcing was first used by <a href="Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> in 1963, to prove the independence of the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> and the <a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> from <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>. It has been considerably reworked and simplified in the following years, and has since served as a powerful technique, both in set theory and in areas of <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a> such as <a href="Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>. <a href="Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a> uses the notions of forcing from both recursion theory and set theory. Forcing has also been used in <a href="Model_theory" title="Model theory">model theory</a>, but it is common in model theory to define <a href="Generic_filter" title="Generic filter">genericity</a> directly without mention of forcing.

<div class="mw-heading mw-heading2"><h2 id="Intuition">Intuition</h2></div>

Forcing is usually used to construct an expanded universe that satisfies some desired property. For example, the expanded universe might contain many new real numbers (at least <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}}</annotation>

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</math><img src="./765bb468708b2eec66bf2dc6505a4c92959d697d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{2}}" loading="lazy"> of them), identified with <a href="Subset" title="Subset">subsets</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> of natural numbers, that were not there in the old universe, and thereby violate the <a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>.

In order to intuitively justify such an expansion, it is best to think of the "old universe" as a <a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">model</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> of the set theory, which is itself a set in the "real universe" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. By the <a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem theorem</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> can be chosen to be a "bare bones" model that is <a href="Skolem's_paradox" title="Skolem's paradox">externally countable</a>, which guarantees that there will be many subsets (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">) of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> that are not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. Specifically, there is an <a href="Ordinal_number" title="Ordinal number">ordinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}^{M}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}^{M}}</annotation>

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</math><img src="./016561651925a1b33a26c52afbbfc43542f9ce73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.379ex; height:3.176ex;" alt="{\displaystyle \aleph _{2}^{M}}" loading="lazy"> that "plays the role of the <a href="Cardinal_number" title="Cardinal number">cardinal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}}">

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</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}}</annotation>

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</math><img src="./765bb468708b2eec66bf2dc6505a4c92959d697d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{2}}" loading="lazy">" in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, but is actually countable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Working in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, it should be easy to find one distinct subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

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</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> per each element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}^{M}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}^{M}}</annotation>

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</math><img src="./016561651925a1b33a26c52afbbfc43542f9ce73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.379ex; height:3.176ex;" alt="{\displaystyle \aleph _{2}^{M}}" loading="lazy">. (For simplicity, this family of subsets can be characterized with a single subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq \aleph _{2}^{M}\times \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle X\subseteq \aleph _{2}^{M}\times \mathbb {N} }</annotation>

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</math><img src="./6fa9b1401f7fd09179cba4ee888357331dc0a617.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.976ex; height:3.176ex;" alt="{\displaystyle X\subseteq \aleph _{2}^{M}\times \mathbb {N} }" loading="lazy">.)

However, in some sense, it may be desirable to "construct the expanded model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

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<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

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</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">". This would help ensure that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

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</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> "resembles" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> in certain aspects, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}^{M[X]}}">

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<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}^{M[X]}}</annotation>

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</math><img src="./9d812327e5878140c408b573a09e6d0f911dc099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.694ex; height:3.676ex;" alt="{\displaystyle \aleph _{2}^{M[X]}}" loading="lazy"> being the same as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph _{2}^{M}}">

</mrow>

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</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \aleph _{2}^{M}}</annotation>

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</math><img src="./016561651925a1b33a26c52afbbfc43542f9ce73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.379ex; height:3.176ex;" alt="{\displaystyle \aleph _{2}^{M}}" loading="lazy"> (more generally, that cardinal collapse does not occur), and allow fine control over the properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

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</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy">. More precisely, every member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

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</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> should be given a (non-unique) name in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. The name can be thought as an expression in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, just like in a <a href="Simple_field_extension" class="mw-redirect" title="Simple field extension">simple field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=K(\theta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=K(\theta )}</annotation>

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</math><img src="./c22f60c3373ec02531ad1e87a68f805f982a2d48.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.647ex; height:2.843ex;" alt="{\displaystyle L=K(\theta )}" loading="lazy"> every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> can be expressed in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">. A major component of forcing is manipulating those names within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, so sometimes it may help to directly think of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> as "the universe", knowing that the theory of forcing guarantees that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

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<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

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</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> will correspond to an actual model.

A subtle point of forcing is that, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is taken to be an arbitrary "missing subset" of some set in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

</semantics>

</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> constructed "within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">" may not even be a model. This is because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> may encode "special" information about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> that is invisible within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> (e.g. the <a href="Countability" class="mw-redirect" title="Countability">countability</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">), and thus prove the existence of sets that are "too complex for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> to describe".<a href="#cite_note-FOOTNOTECohen2008111-1">[1]</a>

Forcing avoids such problems by requiring the newly introduced set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to be a generic set relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.<a href="#cite_note-FOOTNOTECohen2008111-1">[1]</a> Some statements are "forced" to hold for any generic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">: For example, a generic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is "forced" to be infinite. Furthermore, any property (describable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">) of a generic set is "forced" to hold under some forcing condition. The concept of "forcing" can be defined within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, and it gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> enough reasoning power to prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[X]}</annotation>

</semantics>

</math><img src="./87b00856eb008c4ea9bc42894bb2bfa0b8605ac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.716ex; height:2.843ex;" alt="{\displaystyle M[X]}" loading="lazy"> is indeed a model that satisfies the desired properties.

Cohen's original technique, now called <a href="Ramified_forcing" title="Ramified forcing">ramified forcing</a>, is slightly different from the unramified forcing expounded here. Forcing is also equivalent to the method of <a href="Boolean-valued_model" title="Boolean-valued model">Boolean-valued models</a>, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.<a href="#cite_note-FOOTNOTEShoenfield1971-3">[3]</a>

<div class="mw-heading mw-heading3"><h3 id="The_role_of_the_model">The role of the model</h3></div>

In order for the above approach to work smoothly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> must in fact be a <a href="Standard_model_(set_theory)" title="Standard model (set theory)">standard transitive model</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, so that membership and other elementary notions can be handled intuitively in both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. A standard transitive model can be obtained from any standard model through the <a href="Mostowski_collapse_lemma" title="Mostowski collapse lemma">Mostowski collapse lemma</a>, but the existence of any standard model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> (or any variant thereof) is in itself a stronger assumption than the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">.

To get around this issue, a standard technique is to let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> be a standard transitive model of an arbitrary finite subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> (any axiomatization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> has at least one <a href="Axiom_schema" title="Axiom schema">axiom schema</a>, and thus an infinite number of axioms), the existence of which is guaranteed by the <a href="Reflection_principle#In_ZFC" title="Reflection principle">reflection principle</a>. As the goal of a forcing argument is to prove <a href="Consistency" title="Consistency">consistency</a> results, this is enough since any inconsistency in a theory must manifest with a derivation of a finite length, and thus involve only a finite number of axioms.

<div class="mw-heading mw-heading2"><h2 id="Forcing_conditions_and_forcing_posets">Forcing conditions and forcing posets</h2></div>

Each forcing condition can be regarded as a finite piece of information regarding the object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> adjoined to the model. There are many different ways of providing information about an object, which give rise to different forcing notions. A general approach to formalizing forcing notions is to regard forcing conditions as abstract objects with a <a href="Poset" class="mw-redirect" title="Poset">poset</a> structure.

A forcing poset is an ordered triple, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {P} ,\leq ,\mathbf {1} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\mathbb {P} ,\leq ,\mathbf {1} )}</annotation>

</semantics>

</math><img src="./b4973a936c0bfc67b18ca94f029b911da02a4ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.442ex; height:2.843ex;" alt="{\displaystyle (\mathbb {P} ,\leq ,\mathbf {1} )}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq }</annotation>

</semantics>

</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> is a <a href="Preorder" title="Preorder">preorder</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation>

</semantics>

</math><img src="./235ffc0f1788b720aef5caa7b97246a84421fd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }" loading="lazy"> is the largest element. Members of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> are the forcing conditions (or just conditions). The order relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\leq q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\leq q}</annotation>

</semantics>

</math><img src="./9e46e733ff7a6d62f7685d4c20e4a467e984dfdf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.343ex;" alt="{\displaystyle p\leq q}" loading="lazy"> means "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is stronger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy">". (Intuitively, the "smaller" condition provides "more" information, just as the smaller interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3.1415926,3.1415927]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [3.1415926,3.1415927]}</annotation>

</semantics>

</math><img src="./c908181ccc7898fdf7a0fc77a6c915ae063e1432.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.221ex; height:2.843ex;" alt="{\displaystyle [3.1415926,3.1415927]}" loading="lazy"> provides more information about the number <a href="Pi" title="Pi">π</a> than the interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [3.1,3.2]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [3.1,3.2]}</annotation>

</semantics>

</math><img src="./257e754c74e41fcca2cc6c6a91a6ed0d57d18e39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.271ex; height:2.843ex;" alt="{\displaystyle [3.1,3.2]}" loading="lazy"> does.) Furthermore, the preorder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq }</annotation>

</semantics>

</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> must be <a href="Atom_(order_theory)" title="Atom (order theory)">atomless</a>, meaning that it must satisfy the splitting condition:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy">, there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q,r\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q,r\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./d689c66ded20fc4bba019a2a529d4b904ce9839f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.413ex; height:2.509ex;" alt="{\displaystyle q,r\in \mathbb {P} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q,r\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q,r\leq p}</annotation>

</semantics>

</math><img src="./2302d3b672217c37102be87120a5868d436f4599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.42ex; height:2.343ex;" alt="{\displaystyle q,r\leq p}" loading="lazy">, with no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./b060224d0bd21c5604eaebb6b71cdc5dab6c531f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s\in \mathbb {P} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\leq q,r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\leq q,r}</annotation>

</semantics>

</math><img src="./f141724b2ec9e273b3954dc799250556fd072bdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.341ex; height:2.343ex;" alt="{\displaystyle s\leq q,r}" loading="lazy">.</li></ul>

In other words, it must be possible to strengthen any forcing condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> in at least two incompatible directions. Intuitively, this is because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is only a finite piece of information, whereas an infinite piece of information is needed to determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

There are various conventions in use. Some authors require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq }</annotation>

</semantics>

</math><img src="./440568a09c3bfdf0e1278bfa79eb137c04e94035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \leq }" loading="lazy"> to also be <a href="Antisymmetric_relation" title="Antisymmetric relation">antisymmetric</a>, so that the relation is a <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a>. Some use the term <a href="Partial_order" class="mw-redirect" title="Partial order">partial order</a> anyway, conflicting with standard terminology, while some use the term <a href="Preorder" title="Preorder">preorder</a>. The largest element can be dispensed with. The reverse ordering is also used, most notably by <a href="Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a> and his co-authors.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> be any infinite set (such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy">), and let the generic object in question be a new subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\subseteq S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\subseteq S}</annotation>

</semantics>

</math><img src="./34c97c538e8c00a7487681e1be483d6fd83c1eeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.578ex; height:2.343ex;" alt="{\displaystyle X\subseteq S}" loading="lazy">. In Cohen's original formulation of forcing, each forcing condition is a finite set of sentences, either of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in X}</annotation>

</semantics>

</math><img src="./f6201478b1190a333ea731849684429a697638dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.05ex; height:2.176ex;" alt="{\displaystyle a\in X}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\notin X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\notin X}</annotation>

</semantics>

</math><img src="./aac0c56aff8bfe7589cbc7357d17e7513b9f3519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.676ex;" alt="{\displaystyle a\notin X}" loading="lazy">, that are self-consistent (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in X}</annotation>

</semantics>

</math><img src="./f6201478b1190a333ea731849684429a697638dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.05ex; height:2.176ex;" alt="{\displaystyle a\in X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\notin X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\notin X}</annotation>

</semantics>

</math><img src="./aac0c56aff8bfe7589cbc7357d17e7513b9f3519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.676ex;" alt="{\displaystyle a\notin X}" loading="lazy"> for the same value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> do not appear in the same condition). This forcing notion is usually called Cohen forcing.

The forcing poset for Cohen forcing can be formally written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Fin} (S,2),\supseteq ,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {Fin} (S,2),\supseteq ,0)}</annotation>

</semantics>

</math><img src="./e2b3dbe7b6e3d15accf8effa10a6df3cf720fec4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.81ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Fin} (S,2),\supseteq ,0)}" loading="lazy">, the finite partial functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}</annotation>

</semantics>

</math><img src="./4815c09d4b9a557e4497ffe956e7bdd585b9d740.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.816ex; height:3.843ex;" alt="{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}" loading="lazy"> under reverse inclusion. Cohen forcing satisfies the splitting condition because given any condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, one can always find an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in S}</annotation>

</semantics>

</math><img src="./fe43ae205afd3f10432bddabc3bdae5ecfa6b412.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.57ex; height:2.176ex;" alt="{\displaystyle a\in S}" loading="lazy"> not mentioned in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, and add either the sentence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in X}</annotation>

</semantics>

</math><img src="./f6201478b1190a333ea731849684429a697638dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.05ex; height:2.176ex;" alt="{\displaystyle a\in X}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\notin X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\notin X}</annotation>

</semantics>

</math><img src="./aac0c56aff8bfe7589cbc7357d17e7513b9f3519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.05ex; height:2.676ex;" alt="{\displaystyle a\notin X}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> to get two new forcing conditions, incompatible with each other.

Another instructive example of a forcing poset is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Bor} (I),\subseteq ,I)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {Bor} (I),\subseteq ,I)}</annotation>

</semantics>

</math><img src="./adb6929781b65e9ed3286c8103a14cbb3d0e03d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.558ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Bor} (I),\subseteq ,I)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation>

</semantics>

</math><img src="./87ec65159c44769434523e46928bc1b82681f842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bor} (I)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bor} (I)}</annotation>

</semantics>

</math><img src="./bc3d90059470cfe7329fb766b466365eba6ee296.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.701ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bor} (I)}" loading="lazy"> is the collection of <a href="Borel_subset" class="mw-redirect" title="Borel subset">Borel subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> having non-zero <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>. The generic object associated with this forcing poset is a random real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in [0,1]}</annotation>

</semantics>

</math><img src="./6d4a8acea9f5c4e59d8f5fd0ea3c695efa7252fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.542ex; height:2.843ex;" alt="{\displaystyle r\in [0,1]}" loading="lazy">. It can be shown that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> falls in every Borel subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"> with measure 1, provided that the Borel subset is "described" in the original unexpanded universe (this can be formalized with the concept of Borel codes). Each forcing condition can be regarded as a random event with probability equal to its measure. Due to the ready intuition this example can provide, probabilistic language is sometimes used with other divergent forcing posets.

<div class="mw-heading mw-heading2"><h2 id="Generic_filters">Generic filters</h2></div>

Even though each individual forcing condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> cannot fully determine the generic object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subseteq \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\subseteq \mathbb {P} }</annotation>

</semantics>

</math><img src="./7e461c9fca9b2726f217e178cb560c1a9fb681dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.345ex; height:2.343ex;" alt="{\displaystyle G\subseteq \mathbb {P} }" loading="lazy"> of all true forcing conditions does determine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. In fact, without loss of generality, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is commonly considered to be the generic object adjoined to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, so the expanded model is called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy">. It is usually easy enough to show that the originally desired object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is indeed in the model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy">.

Under this convention, the concept of "generic object" can be described in a general way. Specifically, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> should be a generic filter on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. The "<a href="Filter_(mathematics)" title="Filter (mathematics)">filter</a>" condition means that it makes sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a set of all true forcing conditions:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\subseteq \mathbb {P} ;}</annotation>

</semantics>

</math><img src="./a49803b565d6afa21577224854051134a258a3f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.992ex; height:2.509ex;" alt="{\displaystyle G\subseteq \mathbb {P} ;}" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} \in G;}</annotation>

</semantics>

</math><img src="./17b00e2e0e5dfd40e68bff6492760d5d117df5d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.651ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} \in G;}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\geq q\in G}</annotation>

</semantics>

</math><img src="./7b08f0ba0ef208607fa2639fa33ab7a6205f7528.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.094ex; height:2.509ex;" alt="{\displaystyle p\geq q\in G}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G;}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G;}</annotation>

</semantics>

</math><img src="./eff7fe604149215f018209df3002d3f75925b25e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.573ex; height:2.509ex;" alt="{\displaystyle p\in G;}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q\in G}</annotation>

</semantics>

</math><img src="./bb0ea25d481b2bb5f5a2d1b4bf850ee296d859a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.03ex; height:2.509ex;" alt="{\displaystyle p,q\in G}" loading="lazy">, then there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in G}</annotation>

</semantics>

</math><img src="./e2ba6195fb254bc1e1fb197ccd9fc060e5303d5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.716ex; height:2.176ex;" alt="{\displaystyle r\in G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq p,q.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\leq p,q.}</annotation>

</semantics>

</math><img src="./3fe8458cc81a91293f5757671ac6742c13bb0a9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.067ex; height:2.343ex;" alt="{\displaystyle r\leq p,q.}" loading="lazy"></li></ul>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to be "generic relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">" means:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\in M}</annotation>

</semantics>

</math><img src="./fcc61a0a33c4cfc06613957d8ccc5ad551be546c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.207ex; height:2.176ex;" alt="{\displaystyle D\in M}" loading="lazy"> is a "dense" subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> (that is, for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy">, there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\in D}</annotation>

</semantics>

</math><img src="./58a6900217ea448bda38fd342598a296f946c5e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.834ex; height:2.509ex;" alt="{\displaystyle q\in D}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\leq p}</annotation>

</semantics>

</math><img src="./c473ed13155b8cedb2f8f87bd8377581e4df2b30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.337ex; height:2.343ex;" alt="{\displaystyle q\leq p}" loading="lazy">), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap D\neq \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cap D\neq \varnothing }</annotation>

</semantics>

</math><img src="./7506b0db35c810365693ef8ea1b08d0202dfe434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:2.676ex;" alt="{\displaystyle G\cap D\neq \varnothing }" loading="lazy">.</li></ul>

Given that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a countable model, the existence of a generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> follows from the <a href="Rasiowa%E2%80%93Sikorski_lemma" title="Rasiowa–Sikorski lemma">Rasiowa–Sikorski lemma</a>. In fact, slightly more is true: Given a condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy">, one can find a generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G}</annotation>

</semantics>

</math><img src="./646b90327c05c7a10155bb0fe34e46195b9447de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.926ex; height:2.509ex;" alt="{\displaystyle p\in G}" loading="lazy">. Due to the splitting condition on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a filter, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} \setminus G}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} \setminus G}</annotation>

</semantics>

</math><img src="./fd8f8d572a5bcff715c068a0a2681480c2192a14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.442ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} \setminus G}" loading="lazy"> is dense. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\in M}</annotation>

</semantics>

</math><img src="./258641106d43be188995d03111a607742bc4b927.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.11ex; height:2.176ex;" alt="{\displaystyle G\in M}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} \setminus G\in M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} \setminus G\in M}</annotation>

</semantics>

</math><img src="./314c51d50245c8ed7765368a24976540cedbe2d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.725ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} \setminus G\in M}" loading="lazy"> because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">. For this reason, a generic filter is never in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="P-names_and_interpretations">P-names and interpretations</h2></div>

Associated with a forcing poset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> is the <a href="Class_(set_theory)" title="Class (set theory)">class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{(\mathbb {P} )}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V^{(\mathbb {P} )}}</annotation>

</semantics>

</math><img src="./33261995e8acb84d31dd529ab30cb80092ded312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.433ex; height:2.843ex;" alt="{\displaystyle V^{(\mathbb {P} )}}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names. A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-name is a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of the form

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq \{(u,p)\mid u~{\text{is a}}~\mathbb {P} {\text{-name and}}~p\in \mathbb {P} \}.}</annotation>

</semantics>

</math><img src="./1747b0a60bb292cdd4281ba46a0a16d07295614d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.847ex; height:2.843ex;" alt="{\displaystyle A\subseteq \{(u,p)\mid u~{\text{is a}}~\mathbb {P} {\text{-name and}}~p\in \mathbb {P} \}.}" loading="lazy"></dd></dl>

Given any filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">, the interpretation or valuation map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names is given by

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {val} (u,G)=\{\operatorname {val} (v,G)\mid \exists p\in G:~(v,p)\in u\}.}</annotation>

</semantics>

</math><img src="./8464df93cc3d1986cc67912e05ac75a6f7072862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.836ex; height:2.843ex;" alt="{\displaystyle \operatorname {val} (u,G)=\{\operatorname {val} (v,G)\mid \exists p\in G:~(v,p)\in u\}.}" loading="lazy"></dd></dl>

The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names are, in fact, an expansion of the <a href="Von_Neumann_universe" title="Von Neumann universe">universe</a>. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in V}</annotation>

</semantics>

</math><img src="./fa374e20b2db7f6b8caa71ff1865f7f84f215c9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.958ex; height:2.176ex;" alt="{\displaystyle x\in V}" loading="lazy">, one defines <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\check {x}}}</annotation>

</semantics>

</math><img src="./0d779a5add32c7e9eb6a1d6e561232e4bc2c6f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\check {x}}}" loading="lazy"> to be the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-name

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\check {x}}=\{({\check {y}},\mathbf {1} )\mid y\in x\}.}</annotation>

</semantics>

</math><img src="./99194736fe5912d91a59203e8486bd428fb955fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.145ex; height:2.843ex;" alt="{\displaystyle {\check {x}}=\{({\check {y}},\mathbf {1} )\mid y\in x\}.}" loading="lazy"></dd></dl>

Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} \in G}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} \in G}</annotation>

</semantics>

</math><img src="./ecde992fa692ac0297007a2665ad8494deb7eaca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} \in G}" loading="lazy">, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {val} ({\check {x}},G)=x}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {val} ({\check {x}},G)=x}</annotation>

</semantics>

</math><img src="./ec6df37d48eca899879c48fb508bb66c9e3e9091.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.465ex; height:2.843ex;" alt="{\displaystyle \operatorname {val} ({\check {x}},G)=x}" loading="lazy">. In a sense, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\check {x}}}</annotation>

</semantics>

</math><img src="./0d779a5add32c7e9eb6a1d6e561232e4bc2c6f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\check {x}}}" loading="lazy"> is a "name for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">" that does not depend on the specific choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

This also allows defining a "name for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">" without explicitly referring to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">:

</munder>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {G}}=\{({\check {p}},p)\mid p\in \mathbb {P} \}}</annotation>

</semantics>

</math><img src="./a55f571fce662f91216af5fe631e7ef3bae2e0b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:19.992ex; height:3.343ex;" alt="{\displaystyle {\underline {G}}=\{({\check {p}},p)\mid p\in \mathbb {P} \}}" loading="lazy"></dd></dl>

so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {val} ({\underline {G}},G)=\{\operatorname {val} ({\check {p}},G)\mid p\in G\}=G}">

</munder>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {val} ({\underline {G}},G)=\{\operatorname {val} ({\check {p}},G)\mid p\in G\}=G}</annotation>

</semantics>

</math><img src="./ab45bc25f4b5e365533a8d2267ed206f5d809b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:36.724ex; height:3.343ex;" alt="{\displaystyle \operatorname {val} ({\underline {G}},G)=\{\operatorname {val} ({\check {p}},G)\mid p\in G\}=G}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Rigorous_definitions">Rigorous definitions</h3></div>

The concepts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names, interpretations, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\check {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\check {x}}}</annotation>

</semantics>

</math><img src="./0d779a5add32c7e9eb6a1d6e561232e4bc2c6f40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\check {x}}}" loading="lazy"> may be defined by <a href="Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a>. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varnothing }</annotation>

</semantics>

</math><img src="./00595c5e33692e724937fdcc8870496acce1ac74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }" loading="lazy"> the empty set, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha +1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha +1}</annotation>

</semantics>

</math><img src="./c8de281682e16a35de79915ddc9e3582d215e0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.49ex; height:2.343ex;" alt="{\displaystyle \alpha +1}" loading="lazy"> the <a href="Successor_ordinal" title="Successor ordinal">successor ordinal</a> to ordinal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation>

</semantics>

</math><img src="./10d6ec962de5797ba4f161c40e66dca74ae95cc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}" loading="lazy"> the <a href="Power_set" title="Power set">power-set</a> operator, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> a <a href="Limit_ordinal" title="Limit ordinal">limit ordinal</a>, define the following hierarchy:

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\operatorname {Name} (\varnothing )&=\varnothing ,\\\operatorname {Name} (\alpha +1)&={\mathcal {P}}(\operatorname {Name} (\alpha )\times \mathbb {P} ),\\\operatorname {Name} (\lambda )&=\bigcup \{\operatorname {Name} (\alpha )\mid \alpha <\lambda \}.\end{aligned}}}</annotation>

</semantics>

</math><img src="./2459866353c29314d5ff26c04efd774222e9b4a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:39.626ex; height:10.176ex;" alt="{\displaystyle {\begin{aligned}\operatorname {Name} (\varnothing )&=\varnothing ,\\\operatorname {Name} (\alpha +1)&={\mathcal {P}}(\operatorname {Name} (\alpha )\times \mathbb {P} ),\\\operatorname {Name} (\lambda )&=\bigcup \{\operatorname {Name} (\alpha )\mid \alpha <\lambda \}.\end{aligned}}}" loading="lazy"></dd></dl>

Then the class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names is defined as

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mtext>is an ordinal</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V^{(\mathbb {P} )}=\bigcup \{\operatorname {Name} (\alpha )~|~\alpha ~{\text{is an ordinal}}\}.}</annotation>

</semantics>

</math><img src="./b92bc50b56405b6b87df9ab58745126b1572a7d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:38.426ex; height:3.843ex;" alt="{\displaystyle V^{(\mathbb {P} )}=\bigcup \{\operatorname {Name} (\alpha )~|~\alpha ~{\text{is an ordinal}}\}.}" loading="lazy"></dd></dl>

The interpretation map and the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto {\check {x}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto {\check {x}}}</annotation>

</semantics>

</math><img src="./b5aac5c81e5f88176e0033e750b01a757395d5bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.273ex; height:2.176ex;" alt="{\displaystyle x\mapsto {\check {x}}}" loading="lazy"> can similarly be defined with a hierarchical construction.

<div class="mw-heading mw-heading2"><h2 id="Forcing">Forcing</h2></div>

Given a generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\subseteq \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\subseteq \mathbb {P} }</annotation>

</semantics>

</math><img src="./7e461c9fca9b2726f217e178cb560c1a9fb681dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.345ex; height:2.343ex;" alt="{\displaystyle G\subseteq \mathbb {P} }" loading="lazy">, one proceeds as follows. The subclass of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{(\mathbb {P} )}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M^{(\mathbb {P} )}}</annotation>

</semantics>

</math><img src="./209137168db061d81452b49f11f155929a6cf83a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.015ex; height:2.843ex;" alt="{\displaystyle M^{(\mathbb {P} )}}" loading="lazy">. Let

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]=\left\{\operatorname {val} (u,G)~{\Big |}~u\in M^{(\mathbb {P} )}\right\}.}</annotation>

</semantics>

</math><img src="./3687b1838c0b860cb136a00988ee422c4221f1a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.825ex; height:4.843ex;" alt="{\displaystyle M[G]=\left\{\operatorname {val} (u,G)~{\Big |}~u\in M^{(\mathbb {P} )}\right\}.}" loading="lazy"></dd></dl>

To reduce the study of the set theory of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> to that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, one works with the "forcing language", which is built up like ordinary <a href="First-order_logic" title="First-order logic">first-order logic</a>, with membership as the binary relation and all the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names as constants.

Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}</annotation>

</semantics>

</math><img src="./df738ff08e99da4fd34b2f84fa4ea5a0589f6ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:21.088ex; height:3.009ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}" loading="lazy"> (to be read as "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> forces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> in the model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> with poset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">"), where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is a condition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> is a formula in the forcing language, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation>

</semantics>

</math><img src="./14f13cb025ff2e136dcbd2fc81ddf965b728e3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}" loading="lazy">'s are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names, to mean that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a generic filter containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}</annotation>

</semantics>

</math><img src="./3607c60acc0cf8c8f6b00ed773865061f67f4531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.721ex; height:2.843ex;" alt="{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}" loading="lazy">. The special case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} \Vdash _{M,\mathbb {P} }\varphi }">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} \Vdash _{M,\mathbb {P} }\varphi }</annotation>

</semantics>

</math><img src="./679ea92cb87a925736021f649d404c4926540309.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.246ex; height:2.843ex;" alt="{\displaystyle \mathbf {1} \Vdash _{M,\mathbb {P} }\varphi }" loading="lazy"> is often written as "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} \Vdash _{M,\mathbb {P} }\varphi }">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} \Vdash _{M,\mathbb {P} }\varphi }</annotation>

</semantics>

</math><img src="./6b6ba0d88fca536ad09d1984587f850419f39c2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.33ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} \Vdash _{M,\mathbb {P} }\varphi }" loading="lazy">" or simply "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }\varphi }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }\varphi }</annotation>

</semantics>

</math><img src="./5195f567c47e64561560519fcdf0f2034b6fa43a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.264ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }\varphi }" loading="lazy">". Such statements are true in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy">, no matter what <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is.

What is important is that this external definition of the forcing relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi }</annotation>

</semantics>

</math><img src="./fb8df333075d66c2320f7edc1258fbdd9d535926.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:9.168ex; height:2.843ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi }" loading="lazy"> is equivalent to an internal definition within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, defined by <a href="Transfinite_induction" title="Transfinite induction">transfinite induction</a> (specifically <a href="Epsilon-induction" title="Epsilon-induction"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \in }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \in }</annotation>

</semantics>

</math><img src="./6fe4d5b0a594c1da89b5e78e7dfbeed90bdcc32f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:1.843ex;" alt="{\displaystyle \in }" loading="lazy">-induction</a>) over the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names on instances of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in v}</annotation>

</semantics>

</math><img src="./1a7c1525beef5c7493c3ccf17f852ea791a26e2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.298ex; height:1.843ex;" alt="{\displaystyle u\in v}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=v}</annotation>

</semantics>

</math><img src="./2c14384063d79ad3532fd37aecc8616441da4631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.556ex; height:1.676ex;" alt="{\displaystyle u=v}" loading="lazy">, and then by ordinary induction over the complexity of formulae. This has the effect that all the properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> are really properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, and the verification of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> becomes straightforward. This is usually summarized as the following three key properties:

<ul><li>Truth: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}</annotation>

</semantics>

</math><img src="./3607c60acc0cf8c8f6b00ed773865061f67f4531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.721ex; height:2.843ex;" alt="{\displaystyle M[G]\models \varphi (\operatorname {val} (u_{1},G),\ldots ,\operatorname {val} (u_{n},G))}" loading="lazy"> <a href="If_and_only_if" title="If and only if">if and only if</a> it is forced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, that is, for some condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G}</annotation>

</semantics>

</math><img src="./646b90327c05c7a10155bb0fe34e46195b9447de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.926ex; height:2.509ex;" alt="{\displaystyle p\in G}" loading="lazy">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}</annotation>

</semantics>

</math><img src="./df738ff08e99da4fd34b2f84fa4ea5a0589f6ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:21.088ex; height:3.009ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}" loading="lazy">.</li>

<li>Definability: The statement "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}</annotation>

</semantics>

</math><img src="./df738ff08e99da4fd34b2f84fa4ea5a0589f6ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:21.088ex; height:3.009ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}" loading="lazy">" is definable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.</li>

<li>Coherence: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})\land q\leq p\implies q\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})\land q\leq p\implies q\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}</annotation>

</semantics>

</math><img src="./16327108cd4f161e6836b9d4f39828b1ffa5125c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:56.293ex; height:3.009ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})\land q\leq p\implies q\Vdash _{M,\mathbb {P} }\varphi (u_{1},\ldots ,u_{n})}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Internal_definition">Internal definition</h3></div>

There are many different but equivalent ways to define the forcing relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.<a href="#cite_note-FOOTNOTEKunen1980-4">[4]</a> One way to simplify the definition is to first define a modified forcing relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> that is strictly stronger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy">. The modified relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> still satisfies the three key properties of forcing, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi }">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi }</annotation>

</semantics>

</math><img src="./cc904d16629eee41d03fa4c80710c3492aec5326.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:9.168ex; height:3.176ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi '}">

</mrow>

</mrow>

</msubsup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi '}</annotation>

</semantics>

</math><img src="./a0fd77ebf684f72122e5b60029b6a5c03f465d67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:9.853ex; height:3.509ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\varphi '}" loading="lazy"> are not necessarily equivalent even if the first-order formulae <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi '}</annotation>

</semantics>

</math><img src="./41699a838e24ad24212d37ff18eabb15a7765cd4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.205ex; height:3.009ex;" alt="{\displaystyle \varphi '}" loading="lazy"> are equivalent. The unmodified forcing relation can then be defined as

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }\varphi \iff p\Vdash _{M,\mathbb {P} }^{*}\neg \neg \varphi .}</annotation>

</semantics>

</math>

In fact, Cohen's original concept of forcing is essentially <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> rather than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy">.<a href="#cite_note-FOOTNOTEShoenfield1971-3">[3]</a>

The modified forcing relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> can be defined recursively as follows:

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}u\in v}</annotation>

</semantics>

</math><img src="./949f2bcea079926fc5075727e0abe15e4976a153.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:12.946ex; height:3.176ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}u\in v}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w=u).}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w=u).}</annotation>

</semantics>

</math><img src="./354099702daa1bafe5e2d3658a5261e7d044f1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:36.674ex; height:3.343ex;" alt="{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w=u).}" loading="lazy"></li>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}u\neq v}</annotation>

</semantics>

</math><img src="./f70dc06f5fd4a279b9cc6e6aadac195d02b5f566.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:13.204ex; height:3.176ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}u\neq v}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin u)\vee (\exists (w,q)\in u)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin v).}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin u)\vee (\exists (w,q)\in u)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin v).}</annotation>

</semantics>

</math><img src="./52b4a0ecd530634ab207e765049e9c5fd9df6784.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:74.768ex; height:3.343ex;" alt="{\displaystyle (\exists (w,q)\in v)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin u)\vee (\exists (w,q)\in u)(q\geq p\wedge p\Vdash _{M,\mathbb {P} }^{*}w\notin v).}" loading="lazy"></li>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\neg \varphi }</annotation>

</semantics>

</math><img src="./ad0258fc30b502485da347ab4b5dc6082cc32c2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:10.719ex; height:3.176ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\neg \varphi }" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (\exists q\leq p)(q\Vdash _{M,\mathbb {P} }^{*}\varphi ).}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (\exists q\leq p)(q\Vdash _{M,\mathbb {P} }^{*}\varphi ).}</annotation>

</semantics>

</math><img src="./1cbecd73d9edf81fb6d102d12cbb03902fa67809.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.425ex; height:3.343ex;" alt="{\displaystyle \neg (\exists q\leq p)(q\Vdash _{M,\mathbb {P} }^{*}\varphi ).}" loading="lazy"></li>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}(\varphi \vee \psi )}</annotation>

</semantics>

</math><img src="./bc1c582d0a7f4d8747ce626c16d204ddd8e95ea0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:15.073ex; height:3.343ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}(\varphi \vee \psi )}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p\Vdash _{M,\mathbb {P} }^{*}\varphi )\vee (p\Vdash _{M,\mathbb {P} }^{*}\psi ).}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (p\Vdash _{M,\mathbb {P} }^{*}\varphi )\vee (p\Vdash _{M,\mathbb {P} }^{*}\psi ).}</annotation>

</semantics>

</math><img src="./e3bd8e7542007c9ee62ebe501a6da619782d35f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:24.999ex; height:3.343ex;" alt="{\displaystyle (p\Vdash _{M,\mathbb {P} }^{*}\varphi )\vee (p\Vdash _{M,\mathbb {P} }^{*}\psi ).}" loading="lazy"></li>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\exists x\,\varphi (x)}</annotation>

</semantics>

</math><img src="./956f95b8eefe593047d94aec1807332d4ca23727.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; margin-left: -0.089ex; width:15.316ex; height:3.343ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }^{*}\exists x\,\varphi (x)}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\exists u\in M^{(\mathbb {P} )})(p\Vdash _{M,\mathbb {P} }^{*}\varphi (u)).}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\exists u\in M^{(\mathbb {P} )})(p\Vdash _{M,\mathbb {P} }^{*}\varphi (u)).}</annotation>

</semantics>

</math><img src="./176cab83986f0d41ef6841f33d877d84e0836f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:26.961ex; height:3.843ex;" alt="{\displaystyle (\exists u\in M^{(\mathbb {P} )})(p\Vdash _{M,\mathbb {P} }^{*}\varphi (u)).}" loading="lazy"></li></ol>

Other symbols of the forcing language can be defined in terms of these symbols: For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u=v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u=v}</annotation>

</semantics>

</math><img src="./2c14384063d79ad3532fd37aecc8616441da4631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.556ex; height:1.676ex;" alt="{\displaystyle u=v}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg (u\neq v)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg (u\neq v)}</annotation>

</semantics>

</math><img src="./eb3081d10c9add923de3edbdf92f4a5d749db29a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.915ex; height:2.843ex;" alt="{\displaystyle \neg (u\neq v)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\,\varphi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall x\,\varphi (x)}</annotation>

</semantics>

</math><img src="./ac47f8ee4cd467c41bf6344a208d91e180167a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.668ex; height:2.843ex;" alt="{\displaystyle \forall x\,\varphi (x)}" loading="lazy"> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \neg \exists x\,\neg \varphi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \neg \exists x\,\neg \varphi (x)}</annotation>

</semantics>

</math><img src="./b70e40c0e7471bca045a9a933a71b2ad0812222c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.769ex; height:2.843ex;" alt="{\displaystyle \neg \exists x\,\neg \varphi (x)}" loading="lazy">, etc. Cases 1 and 2 depend on each other and on case 3, but the recursion always refers to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-names with lesser <a href="Rank_(set_theory)" class="mw-redirect" title="Rank (set theory)">ranks</a>, so transfinite induction allows the definition to go through.

By construction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> (and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy">) automatically satisfies Definability. The proof that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy"> also satisfies Truth and Coherence is by inductively inspecting each of the five cases above. Cases 4 and 5 are trivial (thanks to the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vee }</annotation>

</semantics>

</math><img src="./7b76220c6805c9b465d6efbc7686c624f49f3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists }</annotation>

</semantics>

</math><img src="./77ed842b6b90b2fdd825320cf8e5265fa937b583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }" loading="lazy"> as the elementary symbols<a href="#cite_note-5">[5]</a>), cases 1 and 2 relies only on the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a filter, and only case 3 requires <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to be a generic filter.<a href="#cite_note-FOOTNOTEShoenfield1971-3">[3]</a>

Formally, an internal definition of the forcing relation (such as the one presented above) is actually a transformation of an arbitrary formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x_{1},\dots ,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (x_{1},\dots ,x_{n})}</annotation>

</semantics>

</math><img src="./e4110f1cb84cf29c6c3e3be18146d1d2c63aea84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.44ex; height:2.843ex;" alt="{\displaystyle \varphi (x_{1},\dots ,x_{n})}" loading="lazy"> to another formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\Vdash _{\mathbb {P} }\varphi (u_{1},\dots ,u_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{\mathbb {P} }\varphi (u_{1},\dots ,u_{n})}</annotation>

</semantics>

</math><img src="./b6b52e41e29412d358e860832189e20a9ac2572d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.903ex; height:2.843ex;" alt="{\displaystyle p\Vdash _{\mathbb {P} }\varphi (u_{1},\dots ,u_{n})}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> are additional variables. The model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> does not explicitly appear in the transformation (note that within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in M^{(\mathbb {P} )}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u\in M^{(\mathbb {P} )}}</annotation>

</semantics>

</math><img src="./b1291d64bcd5317d90630c80fa7bb61eec6a2a10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.185ex; height:2.843ex;" alt="{\displaystyle u\in M^{(\mathbb {P} )}}" loading="lazy"> just means "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-name"), and indeed one may take this transformation as a "syntactic" definition of the forcing relation in the universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> of all sets regardless of any countable transitive model. However, if one wants to force over some countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, then the latter formula should be interpreted under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> (i.e. with all quantifiers ranging only over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">), in which case it is equivalent to the external "semantic" definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy"> described at the top of this section:

<dl><dd>For any formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x_{1},\dots ,x_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (x_{1},\dots ,x_{n})}</annotation>

</semantics>

</math><img src="./e4110f1cb84cf29c6c3e3be18146d1d2c63aea84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.44ex; height:2.843ex;" alt="{\displaystyle \varphi (x_{1},\dots ,x_{n})}" loading="lazy"> there is a theorem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> of the theory <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> (for example conjunction of finite number of axioms) such that for any countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models T}</annotation>

</semantics>

</math><img src="./ef8f6025f56d1b7fe6a9cf76678edcba0e8723d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.384ex; height:2.843ex;" alt="{\displaystyle M\models T}" loading="lazy"> and any atomless partial order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} \in M}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} \in M}</annotation>

</semantics>

</math><img src="./e456f3d022acfad71c9fc9f9bf89f26742d1517c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} \in M}" loading="lazy"> and any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">-generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall a_{1},\ldots ,a_{n}\in M^{\mathbb {P} })(\forall p\in \mathbb {P} )(p\Vdash _{M,\mathbb {P} }\varphi (a_{1},\dots ,a_{n})\,\Leftrightarrow \,M\models p\Vdash _{\mathbb {P} }\varphi (a_{1},\dots ,a_{n})).}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\forall a_{1},\ldots ,a_{n}\in M^{\mathbb {P} })(\forall p\in \mathbb {P} )(p\Vdash _{M,\mathbb {P} }\varphi (a_{1},\dots ,a_{n})\,\Leftrightarrow \,M\models p\Vdash _{\mathbb {P} }\varphi (a_{1},\dots ,a_{n})).}</annotation>

</semantics>

</math></dd></dl>

This the sense under which the forcing relation is indeed "definable in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">".

<div class="mw-heading mw-heading2"><h2 id="Consistency">Consistency</h2></div>

The discussion above can be summarized by the fundamental consistency result that, given a forcing poset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy">, we may assume the existence of a generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, not belonging to the universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy"> is again a set-theoretic universe that models <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">. Furthermore, all truths in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy"> may be reduced to truths in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> involving the forcing relation.

Both styles, adjoining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> to either a countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> or the whole universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.

<div class="mw-heading mw-heading2"><h2 id="Cohen_forcing">Cohen forcing</h2></div>

The simplest nontrivial forcing poset is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {Fin} (\omega ,2),\supseteq ,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\operatorname {Fin} (\omega ,2),\supseteq ,0)}</annotation>

</semantics>

</math><img src="./d9500b0b72f8d4728077cd72c3613c08d21163ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.756ex; height:2.843ex;" alt="{\displaystyle (\operatorname {Fin} (\omega ,2),\supseteq ,0)}" loading="lazy">, the finite partial functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}">

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}</annotation>

</semantics>

</math><img src="./4815c09d4b9a557e4497ffe956e7bdd585b9d740.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.816ex; height:3.843ex;" alt="{\displaystyle 2~{\stackrel {\text{df}}{=}}~\{0,1\}}" loading="lazy"> under reverse inclusion. That is, a condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is essentially two disjoint finite subsets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {p^{-1}}[1]}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {p^{-1}}[1]}</annotation>

</semantics>

</math><img src="./b9c2238dc0ad883c7914f61da0486e68fb29d081.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.048ex; height:3.176ex;" alt="{\displaystyle {p^{-1}}[1]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {p^{-1}}[0]}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {p^{-1}}[0]}</annotation>

</semantics>

</math><img src="./ecd683ab5bd9e7723f40eaf1ef8a7fa6d1079199.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:6.048ex; height:3.176ex;" alt="{\displaystyle {p^{-1}}[0]}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, to be thought of as the "yes" and "no" parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, with no information provided on values outside the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> is stronger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">" means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\supseteq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\supseteq p}</annotation>

</semantics>

</math><img src="./de855639626e49fcad8945ccf816d6c67dda6d7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.337ex; height:2.343ex;" alt="{\displaystyle q\supseteq p}" loading="lazy">, in other words, the "yes" and "no" parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are supersets of the "yes" and "no" parts of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, and in that sense, provide more information.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> be a generic filter for this poset. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are both in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cup q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\cup q}</annotation>

</semantics>

</math><img src="./4e6404ce8f5dd18d1f16710b4df4c04cb06d0d10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\cup q}" loading="lazy"> is a condition because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a filter. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=\bigcup G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=\bigcup G}</annotation>

</semantics>

</math><img src="./6f5585aa3f94a1c09b0386b617a63145ffb0a1b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.01ex; height:3.843ex;" alt="{\displaystyle g=\bigcup G}" loading="lazy"> is a well-defined partial function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2}</annotation>

</semantics>

</math><img src="./901fc910c19990d0dbaaefe4726ceb1a4e217a0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" loading="lazy"> because any two conditions in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> agree on their common domain.

In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> is a total function. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in \omega }</annotation>

</semantics>

</math><img src="./9ffa46e74c9bc8008d95dab90fcc53a676861391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.681ex; height:1.843ex;" alt="{\displaystyle n\in \omega }" loading="lazy">, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}=\{p\mid p(n)~{\text{is defined}}\}}">

<msub>

</mrow>

</msub>

<mtext>is defined</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{n}=\{p\mid p(n)~{\text{is defined}}\}}</annotation>

</semantics>

</math><img src="./70bbf04e743a94069192ab3cd2ca856d69021a88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.072ex; height:2.843ex;" alt="{\displaystyle D_{n}=\{p\mid p(n)~{\text{is defined}}\}}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation>

</semantics>

</math><img src="./0fe03857347bf517e7fbda4085b0dafd6018cf18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}" loading="lazy"> is dense. (Given any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">'s domain, adjoin a value for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">—the result is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation>

</semantics>

</math><img src="./0fe03857347bf517e7fbda4085b0dafd6018cf18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}" loading="lazy">.) A condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G\cap D_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G\cap D_{n}}</annotation>

</semantics>

</math><img src="./0e712111c9892090c4c06c9d8e2bd9ec36cba19f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.652ex; height:2.509ex;" alt="{\displaystyle p\in G\cap D_{n}}" loading="lazy"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> in its domain, and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\subseteq g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\subseteq g}</annotation>

</semantics>

</math><img src="./322af083305878c8c7d9aa639d831678c94e5cf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.473ex; height:2.343ex;" alt="{\displaystyle p\subseteq g}" loading="lazy">, we find that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation>

</semantics>

</math><img src="./d4ad18070e494503403daf39398e711c1378348e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}" loading="lazy"> is defined.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={g^{-1}}[1]}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X={g^{-1}}[1]}</annotation>

</semantics>

</math><img src="./da1e37555972a1c217dc7fa4c3e659c66bfefbc7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.986ex; height:3.176ex;" alt="{\displaystyle X={g^{-1}}[1]}" loading="lazy">, the set of all "yes" members of the generic conditions. It is possible to give a name for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> directly. Let

</munder>

</mrow>

<mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {X}}=\left\{\left({\check {n}},p\right)\mid p(n)=1\right\}.}</annotation>

</semantics>

</math><img src="./34a535317121fe5a6350246e1c5f30083c39d843.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.036ex; margin-bottom: -0.755ex; width:24.454ex; height:3.343ex;" alt="{\displaystyle {\underline {X}}=\left\{\left({\check {n}},p\right)\mid p(n)=1\right\}.}" loading="lazy"></dd></dl>

Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {val} ({\underline {X}},G)=X.}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {val} ({\underline {X}},G)=X.}</annotation>

</semantics>

</math><img src="./de1961fa90bca888edd04a338ba4823602eba49d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:15.413ex; height:3.343ex;" alt="{\displaystyle \operatorname {val} ({\underline {X}},G)=X.}" loading="lazy"> Now suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq \omega }</annotation>

</semantics>

</math><img src="./f8bd32eb0e1642c33ce7011598be0282d8e70179.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.287ex; height:2.343ex;" alt="{\displaystyle A\subseteq \omega }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. We claim that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\neq A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\neq A}</annotation>

</semantics>

</math><img src="./aafe71d73f4e2121e4826aa596ed947f05fffdfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.822ex; height:2.676ex;" alt="{\displaystyle X\neq A}" loading="lazy">. Let

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}=\{p\mid (\exists n)(n\in \operatorname {Dom} (p)\land (p(n)=1\iff n\notin A))\}.}</annotation>

</semantics>

</math><img src="./770434a2e202137684b2dde8552afb18d7fc8d09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.862ex; height:2.843ex;" alt="{\displaystyle D_{A}=\{p\mid (\exists n)(n\in \operatorname {Dom} (p)\land (p(n)=1\iff n\notin A))\}.}" loading="lazy"></dd></dl>

Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{A}}</annotation>

</semantics>

</math><img src="./b78af4a9d2d8aaec85ea4e6b0f997a31def04311.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle D_{A}}" loading="lazy"> is dense. (Given any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, find <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> that is not in its domain, and adjoin a value for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> contrary to the status of "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in A}</annotation>

</semantics>

</math><img src="./c5dbbde2da8fd6a9b5cca520041518e9ed305e3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.978ex; height:2.176ex;" alt="{\displaystyle n\in A}" loading="lazy">".) Then any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G\cap D_{A}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G\cap D_{A}}</annotation>

</semantics>

</math><img src="./c25fc6e5f968f1646e701231f0f4a73a832acb85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.898ex; height:2.509ex;" alt="{\displaystyle p\in G\cap D_{A}}" loading="lazy"> witnesses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\neq A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\neq A}</annotation>

</semantics>

</math><img src="./aafe71d73f4e2121e4826aa596ed947f05fffdfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.822ex; height:2.676ex;" alt="{\displaystyle X\neq A}" loading="lazy">. To summarize, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a "new" subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, necessarily infinite.

Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega \times \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \times \omega _{2}}</annotation>

</semantics>

</math><img src="./047081e70709ce535ed9cf7f84066cff27abda23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.786ex; height:2.009ex;" alt="{\displaystyle \omega \times \omega _{2}}" loading="lazy">, that is, consider instead finite partial functions whose inputs are of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,\alpha )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (n,\alpha )}</annotation>

</semantics>

</math><img src="./4f80ca801edd0dccc0522a36ea806588b32c385e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.726ex; height:2.843ex;" alt="{\displaystyle (n,\alpha )}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<\omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n<\omega }</annotation>

</semantics>

</math><img src="./4d449b5187ade54e76099c8fff438b0a6d2ca135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle n<\omega }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <\omega _{2}}</annotation>

</semantics>

</math><img src="./4c15ff06014d265e13a0dae6e5bf62684d1a02f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.086ex; height:2.176ex;" alt="{\displaystyle \alpha <\omega _{2}}" loading="lazy">, and whose outputs are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, one gets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

</semantics>

</math><img src="./7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy"> new subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">. They are all distinct, by a density argument: Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha <\beta <\omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha <\beta <\omega _{2}}</annotation>

</semantics>

</math><img src="./e5bb8047357f5541be5172d48ddc858513e64727.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.517ex; height:2.509ex;" alt="{\displaystyle \alpha <\beta <\omega _{2}}" loading="lazy">, let

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{\alpha ,\beta }=\{p\mid (\exists n)(p(n,\alpha )\neq p(n,\beta ))\},}</annotation>

</semantics>

</math><img src="./c9530a50d033b0d7f805b29d5b239c4f3788fa38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.823ex; height:3.009ex;" alt="{\displaystyle D_{\alpha ,\beta }=\{p\mid (\exists n)(p(n,\alpha )\neq p(n,\beta ))\},}" loading="lazy"></dd></dl>

then each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{\alpha ,\beta }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{\alpha ,\beta }}</annotation>

</semantics>

</math><img src="./e708218a51b846742e44bfab6e83059cce788a86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.608ex; height:2.843ex;" alt="{\displaystyle D_{\alpha ,\beta }}" loading="lazy"> is dense, and a generic condition in it proves that the αth new set disagrees somewhere with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy">th new set.

This is not yet the falsification of the continuum hypothesis. One must prove that no new maps have been introduced which map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> onto <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{2}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{2}}</annotation>

</semantics>

</math><img src="./7b914a8bfef5d1b9b106048afa0aab4a99251f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{2}}" loading="lazy">. For example, if one considers instead <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (\omega ,\omega _{1})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (\omega ,\omega _{1})}</annotation>

</semantics>

</math><img src="./6692f4d54cd271e745a259cfa343f988698e064d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.246ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (\omega ,\omega _{1})}" loading="lazy">, finite partial functions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy">, the <a href="First_uncountable_ordinal" title="First uncountable ordinal">first uncountable ordinal</a>, one gets in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy"> a bijection from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy">. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{1}}</annotation>

</semantics>

</math><img src="./e20e29ac56d6cc52eaeb2f9c0bf79ef706428ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{1}}" loading="lazy"> has collapsed, and in the forcing extension, is a countable ordinal.

The last step in showing the independence of the continuum hypothesis, then, is to show that Cohen forcing does not collapse cardinals. For this, a sufficient combinatorial property is that all of the <a href="Antichain" title="Antichain">antichains</a> of the forcing poset are countable.

<div class="mw-heading mw-heading2"><h2 id="The_countable_chain_condition">The countable chain condition</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Countable_chain_condition" title="Countable chain condition">Countable chain condition</a></div>

A <a href="Strong_antichain" title="Strong antichain">(strong) antichain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> is a subset such that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q\in A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q\in A}</annotation>

</semantics>

</math><img src="./2edca3bcddbaeca07158fe06487b64cb17fbf94d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.946ex; height:2.509ex;" alt="{\displaystyle p,q\in A}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\neq q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\neq q}</annotation>

</semantics>

</math><img src="./ac1f5fae3451ae6d0b52465cb06703bb11681b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.427ex; height:2.676ex;" alt="{\displaystyle p\neq q}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> are incompatible (written <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\perp q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\perp q}</annotation>

</semantics>

</math><img src="./0bc7ecfc67d94f7c9871a015bc2e98317b699df0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.509ex;" alt="{\displaystyle p\perp q}" loading="lazy">), meaning there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\leq p}</annotation>

</semantics>

</math><img src="./7da9b2cfae9b8972007b5067b742146f05b2e01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.317ex; height:2.343ex;" alt="{\displaystyle r\leq p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\leq q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\leq q}</annotation>

</semantics>

</math><img src="./5078a97beb91363d1dd647145b0e7efaff903556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.217ex; height:2.343ex;" alt="{\displaystyle r\leq q}" loading="lazy">. In the example on Borel sets, incompatibility means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cap q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\cap q}</annotation>

</semantics>

</math><img src="./65b166df66e258c8f6c7234b4863445427410ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\cap q}" loading="lazy"> has zero measure. In the example on finite partial functions, incompatibility means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cup q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\cup q}</annotation>

</semantics>

</math><img src="./4e6404ce8f5dd18d1f16710b4df4c04cb06d0d10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:4.911ex; height:2.343ex;" alt="{\displaystyle p\cup q}" loading="lazy"> is not a function, in other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> assign different values to some domain input.

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> satisfies the <a href="Countable_chain_condition" title="Countable chain condition">countable chain condition</a> (c.c.c.) if and only if every antichain in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> is countable. (The name, which is obviously inappropriate, is a holdover from older terminology. Some mathematicians write "c.a.c." for "countable antichain condition".)

It is easy to see that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bor} (I)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bor} (I)}</annotation>

</semantics>

</math><img src="./bc3d90059470cfe7329fb766b466365eba6ee296.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.701ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bor} (I)}" loading="lazy"> satisfies the c.c.c. because the measures add up to at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">. Also, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (E,2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (E,2)}</annotation>

</semantics>

</math><img src="./97b42a0520b49e3b436a538b51b97f34476d1434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (E,2)}" loading="lazy"> satisfies the c.c.c., but the proof is more difficult.

Given an uncountable subfamily <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subseteq \operatorname {Fin} (E,2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W\subseteq \operatorname {Fin} (E,2)}</annotation>

</semantics>

</math><img src="./cae02b454b8af84269052aee8bda36e7ffe4e307.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.772ex; height:2.843ex;" alt="{\displaystyle W\subseteq \operatorname {Fin} (E,2)}" loading="lazy">, shrink <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> to an uncountable subfamily <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{0}}</annotation>

</semantics>

</math><img src="./7f541f57fd799ba5137a2e50a1a728dde4306c06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{0}}" loading="lazy"> of sets of size at most <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<\omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n<\omega }</annotation>

</semantics>

</math><img src="./4d449b5187ade54e76099c8fff438b0a6d2ca135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.939ex; height:1.843ex;" alt="{\displaystyle n<\omega }" loading="lazy"> (for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> this is uncountable, since otherwise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=\bigcup _{n<\omega }\{w\in W:|w|<n\}}">

</mrow>

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=\bigcup _{n<\omega }\{w\in W:|w|<n\}}</annotation>

</semantics>

</math><img src="./35be3bffb7651d089c7653e1012a999bdaaf8c17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.474ex; height:5.509ex;" alt="{\displaystyle W=\bigcup _{n<\omega }\{w\in W:|w|<n\}}" loading="lazy"> would be a countable union of countable sets, thus countable). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(e_{1})=b_{1}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p(e_{1})=b_{1}}</annotation>

</semantics>

</math><img src="./2c9c6bbdbcab5e281cfedd0bc89fc0efb177af54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.356ex; height:2.843ex;" alt="{\displaystyle p(e_{1})=b_{1}}" loading="lazy"> for uncountably many <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in W_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in W_{0}}</annotation>

</semantics>

</math><img src="./08ce182395d17b896c2115733633a432d30dd5a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.347ex; height:2.509ex;" alt="{\displaystyle p\in W_{0}}" loading="lazy">, shrink this to an uncountable subfamily <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{1}}</annotation>

</semantics>

</math><img src="./74ab879909bd9762251f679bbb2fa738100baa45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.248ex; height:2.509ex;" alt="{\displaystyle W_{1}}" loading="lazy"> and repeat, getting a finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}\in W_{0}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}\in W_{0}}</annotation>

</semantics>

</math><img src="./4abc30d28438b6d6b41020a91187c14ac0cb1291.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.726ex; height:2.843ex;" alt="{\displaystyle \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}\in W_{0}}" loading="lazy"> and an uncountable family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{k}}</annotation>

</semantics>

</math><img src="./aea7ab451421a62d7de329faed9aa86419bebeab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.282ex; height:2.509ex;" alt="{\displaystyle W_{k}}" loading="lazy"> of incompatible conditions of size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n-k}</annotation>

</semantics>

</math><img src="./b98e1d6a69bccd09a4b9b69bdf03a08c1706c8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.446ex; height:2.343ex;" alt="{\displaystyle n-k}" loading="lazy"> such that every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Dom} (p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Dom} (p)}</annotation>

</semantics>

</math><img src="./c813a18b066ec722df5492ebb550a6839ee83132.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.853ex; height:2.843ex;" alt="{\displaystyle \operatorname {Dom} (p)}" loading="lazy"> for at most countable many <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in W_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in W_{k}}</annotation>

</semantics>

</math><img src="./df2bae6c1140bded7e9b322f12875932d513a337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.382ex; height:2.509ex;" alt="{\displaystyle p\in W_{k}}" loading="lazy">. Now, pick an arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in W_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in W_{k}}</annotation>

</semantics>

</math><img src="./df2bae6c1140bded7e9b322f12875932d513a337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.382ex; height:2.509ex;" alt="{\displaystyle p\in W_{k}}" loading="lazy">, and pick from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W_{k}}</annotation>

</semantics>

</math><img src="./aea7ab451421a62d7de329faed9aa86419bebeab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.282ex; height:2.509ex;" alt="{\displaystyle W_{k}}" loading="lazy"> any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> that is not one of the countably many members that have a domain member in common with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}</annotation>

</semantics>

</math><img src="./73c29f82bb16587ac4bee2a478f25bacb64a0277.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:25.479ex; height:2.843ex;" alt="{\displaystyle p\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}</annotation>

</semantics>

</math><img src="./5e8128a69078d7e30b179e138d9282d8d8d0bdb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.29ex; height:2.843ex;" alt="{\displaystyle q\cup \{(e_{1},b_{1}),\ldots ,(e_{k},b_{k})\}}" loading="lazy"> are compatible, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> is not an antichain. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (E,2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (E,2)}</annotation>

</semantics>

</math><img src="./97b42a0520b49e3b436a538b51b97f34476d1434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.238ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (E,2)}" loading="lazy">-antichains are countable.<a href="#cite_note-FOOTNOTECohen2008Section_IV.8,_Lemma_2-6">[6]</a>

The importance of antichains in forcing is that for most purposes, dense sets and maximal antichains are equivalent. A maximal antichain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is one that cannot be extended to a larger antichain. This means that every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in \mathbb {P} }</annotation>

</semantics>

</math><img src="./7891a37dcb8b0ee507f9ef2038a853e245d76657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p\in \mathbb {P} }" loading="lazy"> is compatible with some member of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. The existence of a maximal antichain follows from <a href="Zorn's_lemma" title="Zorn's lemma">Zorn's Lemma</a>. Given a maximal antichain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, let

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=\left\{p\in \mathbb {P} \mid (\exists q\in A)(p\leq q)\right\}.}</annotation>

</semantics>

</math><img src="./bbc4a938abf2ce53c3ccb18d92f37131c6baac78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.651ex; height:2.843ex;" alt="{\displaystyle D=\left\{p\in \mathbb {P} \mid (\exists q\in A)(p\leq q)\right\}.}" loading="lazy"></dd></dl>

Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> is dense, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap D\neq \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cap D\neq \varnothing }</annotation>

</semantics>

</math><img src="./7506b0db35c810365693ef8ea1b08d0202dfe434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:2.676ex;" alt="{\displaystyle G\cap D\neq \varnothing }" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap A\neq \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cap A\neq \varnothing }</annotation>

</semantics>

</math><img src="./2c9426a511a634204e06d856d98b314b813d3f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.059ex; height:2.676ex;" alt="{\displaystyle G\cap A\neq \varnothing }" loading="lazy">. Conversely, given a dense set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">, Zorn's Lemma shows that there exists a maximal antichain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\subseteq D}</annotation>

</semantics>

</math><img src="./7594993cf9bc481feb84c46400216f21a4966f5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.766ex; height:2.343ex;" alt="{\displaystyle A\subseteq D}" loading="lazy">, and then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap D\neq \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cap D\neq \varnothing }</annotation>

</semantics>

</math><img src="./7506b0db35c810365693ef8ea1b08d0202dfe434.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.24ex; height:2.676ex;" alt="{\displaystyle G\cap D\neq \varnothing }" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cap A\neq \varnothing }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\cap A\neq \varnothing }</annotation>

</semantics>

</math><img src="./2c9426a511a634204e06d856d98b314b813d3f7f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.059ex; height:2.676ex;" alt="{\displaystyle G\cap A\neq \varnothing }" loading="lazy">.

Assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} }</annotation>

</semantics>

</math><img src="./1053af9e662ceaf56c4455f90e0f67273422eded.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {P} }" loading="lazy"> satisfies the c.c.c. Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x,y\in V}</annotation>

</semantics>

</math><img src="./89e1c8106b3df898a0338c2ebce32902e4941fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.147ex; height:2.509ex;" alt="{\displaystyle x,y\in V}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:x\to y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:x\to y}</annotation>

</semantics>

</math><img src="./f1e5e985afba6236c1695a35213292b54583fac9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.315ex; height:2.509ex;" alt="{\displaystyle f:x\to y}" loading="lazy"> a function in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy">, one can approximate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> as follows. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> be a name for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> (by the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy">) and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> be a condition that forces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> to be a function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Define a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:x\to {\mathcal {P}}(y)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:x\to {\mathcal {P}}(y)}</annotation>

</semantics>

</math><img src="./19fb94e03e2b2f067fb26caedabad1c3c01a158b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.29ex; height:2.843ex;" alt="{\displaystyle F:x\to {\mathcal {P}}(y)}" loading="lazy">, by

<mover>

</mrow>

</mrow>

</mover>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

</mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mrow>

<mover>

</mover>

</mrow>

</mrow>

<mover>

</mover>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a){\stackrel {\text{df}}{=}}\left\{b\left|(\exists q\in \mathbb {P} )\left[(q\leq p)\land \left(q\Vdash ~u\left({\check {a}}\right)={\check {b}}\right)\right]\right\}.\right.}</annotation>

</semantics>

</math><img src="./59d578f8287a1b5faac8d7d6ffe61c919be16a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.683ex; height:4.843ex;" alt="{\displaystyle F(a){\stackrel {\text{df}}{=}}\left\{b\left|(\exists q\in \mathbb {P} )\left[(q\leq p)\land \left(q\Vdash ~u\left({\check {a}}\right)={\check {b}}\right)\right]\right\}.\right.}" loading="lazy"></dd></dl>

By the definability of forcing, this definition makes sense within <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. By the coherence of forcing, a different <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> must come from an incompatible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">. By c.c.c., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(a)}</annotation>

</semantics>

</math><img src="./fc7c034f57f1a8ef417737a9170547d644b6153b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle F(a)}" loading="lazy"> is countable.

In summary, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is unknown in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> as it depends on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, but it is not wildly unknown for a c.c.c.-forcing. One can identify a countable set of guesses for what the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is at any input, independent of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">.

This has the following very important consequence. If in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\alpha \to \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\alpha \to \beta }</annotation>

</semantics>

</math><img src="./71a95e1bc8374e6acb4cb602d4797868594621a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.649ex; height:2.509ex;" alt="{\displaystyle f:\alpha \to \beta }" loading="lazy"> is a surjection from one infinite ordinal onto another, then there is a surjection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:\omega \times \alpha \to \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:\omega \times \alpha \to \beta }</annotation>

</semantics>

</math><img src="./032930a7aaaff35883d0e920d09757ca564b72db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.773ex; height:2.509ex;" alt="{\displaystyle g:\omega \times \alpha \to \beta }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, and consequently, a surjection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h:\alpha \to \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h:\alpha \to \beta }</annotation>

</semantics>

</math><img src="./f58487cc3d6619d01aafa08db077318e59049503.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.71ex; height:2.509ex;" alt="{\displaystyle h:\alpha \to \beta }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. In particular, cardinals cannot collapse. The conclusion is that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}\geq \aleph _{2}}">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}\geq \aleph _{2}}</annotation>

</semantics>

</math><img src="./e73428a3fd72275ad7b19e54355a78e85e6bad6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.804ex; height:3.009ex;" alt="{\displaystyle 2^{\aleph _{0}}\geq \aleph _{2}}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Easton_forcing">Easton forcing</h2></div>

The exact value of the continuum in the above Cohen model, and variants like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (\omega \times \kappa ,2)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (\omega \times \kappa ,2)}</annotation>

</semantics>

</math><img src="./cb138cd7c75b3631480a447b17fc3747866e8d93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.088ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (\omega \times \kappa ,2)}" loading="lazy"> for cardinals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation>

</semantics>

</math><img src="./54ddec2e922c5caea4e47d04feef86e782dc8e6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }" loading="lazy"> in general, was worked out by <a href="Robert_M._Solovay" title="Robert M. Solovay">Robert M. Solovay</a>, who also worked out how to violate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {GCH}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {GCH}}}</annotation>

</semantics>

</math><img src="./b1c605d515b6f686dd24f1ecc8b1c50da1547905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.681ex; height:2.176ex;" alt="{\displaystyle {\mathsf {GCH}}}" loading="lazy"> (the <a href="Continuum_hypothesis#Generalized_continuum_hypothesis" title="Continuum hypothesis">generalized continuum hypothesis</a>), for <a href="Regular_cardinal" title="Regular cardinal">regular cardinals</a> only, a finite number of times. For example, in the above Cohen model, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {CH}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {CH}}}</annotation>

</semantics>

</math><img src="./c6abd0b5c053128188f7ae544da2f182143cade7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.131ex; height:2.176ex;" alt="{\displaystyle {\mathsf {CH}}}" loading="lazy"> holds in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{\aleph _{0}}=\aleph _{2}}">

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2^{\aleph _{0}}=\aleph _{2}}</annotation>

</semantics>

</math><img src="./90f171d678e3bea0558c3c3074a4c4159b9948aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.804ex; height:3.009ex;" alt="{\displaystyle 2^{\aleph _{0}}=\aleph _{2}}" loading="lazy"> holds in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy">.

<a href="William_Bigelow_Easton" title="William Bigelow Easton">William B. Easton</a> worked out the proper class version of violating the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {GCH}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {GCH}}}</annotation>

</semantics>

</math><img src="./b1c605d515b6f686dd24f1ecc8b1c50da1547905.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.681ex; height:2.176ex;" alt="{\displaystyle {\mathsf {GCH}}}" loading="lazy"> for regular cardinals, basically showing that the known restrictions, (monotonicity, <a href="Cantor's_theorem" title="Cantor's theorem">Cantor's Theorem</a> and <a href="K%C3%B6nig's_theorem_(set_theory)" class="mw-redirect" title="König's theorem (set theory)">König's Theorem</a>), were the only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">-provable restrictions (see <a href="Easton's_theorem" title="Easton's theorem">Easton's Theorem</a>).

Easton's work was notable in that it involved forcing with a proper class of conditions. In general, the method of forcing with a proper class of conditions fails to give a model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">. For example, forcing with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (\omega \times \mathbf {On} ,2)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (\omega \times \mathbf {On} ,2)}</annotation>

</semantics>

</math><img src="./4823ae231c415b1cd02ae6b54f60f925775166e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.242ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (\omega \times \mathbf {On} ,2)}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {On} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {On} }</annotation>

</semantics>

</math><img src="./eebff303626a5ca6b29cf76a9479ad24883abeac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.493ex; height:2.176ex;" alt="{\displaystyle \mathbf {On} }" loading="lazy"> is the proper class of all ordinals, makes the continuum a proper class. On the other hand, forcing with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Fin} (\omega ,\mathbf {On} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Fin} (\omega ,\mathbf {On} )}</annotation>

</semantics>

</math><img src="./6f967c9525e8f54f49cd6dddd11d3d6dac06891a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.239ex; height:2.843ex;" alt="{\displaystyle \operatorname {Fin} (\omega ,\mathbf {On} )}" loading="lazy"> introduces a countable enumeration of the ordinals. In both cases, the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V[G]}</annotation>

</semantics>

</math><img src="./40b9a5385f4657b40731d632f1050b9bd7562b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle V[G]}" loading="lazy"> is visibly not a model of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">.

At one time, it was thought that more sophisticated forcing would also allow an arbitrary variation in the powers of <a href="Regular_cardinal" title="Regular cardinal">singular cardinals</a>. However, this has turned out to be a difficult, subtle and even surprising problem, with several more <a href="PCF_theory" title="PCF theory">restrictions provable</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> and with the forcing models depending on the consistency of various <a href="Large_cardinal" title="Large cardinal">large-cardinal</a> properties. Many open problems remain.

<div class="mw-heading mw-heading2"><h2 id="Random_reals">Random reals</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Random_algebra" title="Random algebra">Random algebra</a></div>

Random forcing can be defined as forcing over the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> of all compact subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"> of positive measure, ordered by the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \subseteq }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \subseteq }</annotation>

</semantics>

</math><img src="./a924f8dcb2847bb8871edfdbf4c6b5cca0669228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \subseteq }" loading="lazy"> (a smaller set in the context of inclusion is a smaller set in the ordering, and represents a condition with more information). There are two types of important dense sets:

<ol><li>For any positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, the set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}=\left\{p\in P:\operatorname {diam} (p)<{\frac {1}{n}}\right\}}">

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{n}=\left\{p\in P:\operatorname {diam} (p)<{\frac {1}{n}}\right\}}</annotation>

</semantics>

</math> is dense, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {diam} (p)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {diam} (p)}</annotation>

</semantics>

</math><img src="./a579eb37484b11387d9369dd031b4e0c47bbef6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.016ex; height:2.843ex;" alt="{\displaystyle \operatorname {diam} (p)}" loading="lazy"> is the <a href="Diameter_of_a_set" title="Diameter of a set">diameter</a> of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">.</li>

<li>For any Borel subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq [0,1]}</annotation>

</semantics>

</math><img src="./25a5404d06244bb945c44295b7c72890f813f8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.515ex; height:2.843ex;" alt="{\displaystyle B\subseteq [0,1]}" loading="lazy"> of measure 1, the set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{B}=\{p\in P:p\subseteq B\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{B}=\{p\in P:p\subseteq B\}}</annotation>

</semantics>

</math> is dense.</li></ol>

For any filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and any pair of elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2}\in G}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2}\in G}</annotation>

</semantics>

</math><img src="./082320b4b911a6070d7bd96180789331d4b95f92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.238ex; height:2.509ex;" alt="{\displaystyle p_{1},p_{2}\in G}" loading="lazy"> there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\in G}</annotation>

</semantics>

</math><img src="./433b902f3d1e6a03a2b45f3df3c824dd91a11133.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.737ex; height:2.509ex;" alt="{\displaystyle q\in G}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\leq p_{1},p_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\leq p_{1},p_{2}}</annotation>

</semantics>

</math><img src="./3ced83df91b3bc4bef82d21de8dbb1abbd0ace11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.649ex; height:2.343ex;" alt="{\displaystyle q\leq p_{1},p_{2}}" loading="lazy">. In this ordering, this means that any filter is closed under finite intersection. Therefore, by <a href="Cantor's_intersection_theorem" title="Cantor's intersection theorem">Cantor's intersection theorem</a>, the intersection of all the elements in any filter is nonempty. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is a filter intersecting the dense set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation>

</semantics>

</math><img src="./0fe03857347bf517e7fbda4085b0dafd6018cf18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}" loading="lazy"> for any positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, then the filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> contains conditions of arbitrarily small positive diameter. Therefore, the intersection of all conditions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> has diameter 0. But the only nonempty sets of diameter 0 are singletons. So there is exactly one real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}\in \bigcap G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}\in \bigcap G}</annotation>

</semantics>

</math><img src="./2f9628bfe1b328a11f84d33635361bf56da8d125.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.209ex; height:3.843ex;" alt="{\displaystyle r_{G}\in \bigcap G}" loading="lazy">.

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq [0,1]}</annotation>

</semantics>

</math><img src="./25a5404d06244bb945c44295b7c72890f813f8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.515ex; height:2.843ex;" alt="{\displaystyle B\subseteq [0,1]}" loading="lazy"> be any Borel set of measure 1. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> intersects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{B}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{B}}</annotation>

</semantics>

</math><img src="./f4c98399752f15ad50d12f7f05417da4869f6d36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.404ex; height:2.509ex;" alt="{\displaystyle D_{B}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}\in B}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}\in B}</annotation>

</semantics>

</math><img src="./d224103f38d73642e5d25861e86793834be517c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.177ex; height:2.509ex;" alt="{\displaystyle r_{G}\in B}" loading="lazy">.

However, a generic filter over a countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. The real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is provably not an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. One issue with this construction is that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in P}</annotation>

</semantics>

</math><img src="./f4dc1da33d78a2bdae06c9edff726feb126dae34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.845ex; height:2.509ex;" alt="{\displaystyle p\in P}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models }</annotation>

</semantics>

</math><img src="./61a22b9bfd453c325ffdf82da69f459e32996864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.102ex; height:2.843ex;" alt="{\displaystyle M\models }" loading="lazy"> "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is compact", but from the viewpoint of some larger universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\supset M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\supset M}</annotation>

</semantics>

</math><img src="./05537234d6c86246ec180c7966f367f211f09f9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.176ex;" alt="{\displaystyle V\supset M}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> can be non-compact and the intersection of all conditions from the generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> can then be empty. To fix this, we consider the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{{\bar {p}}:p\in G\}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C=\{{\bar {p}}:p\in G\}}</annotation>

</semantics>

</math><img src="./dc135e78f49a7f9105b5e5351e3979a88e697c8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.323ex; height:2.843ex;" alt="{\displaystyle C=\{{\bar {p}}:p\in G\}}" loading="lazy"> of topological <a href="Closure_(topology)" title="Closure (topology)">closures</a> of conditions from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. Because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {p}}\supseteq p}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\bar {p}}\supseteq p}</annotation>

</semantics>

</math><img src="./ac175a568a53dfefb8d37fe081258dc1712934b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.717ex; height:2.343ex;" alt="{\displaystyle {\bar {p}}\supseteq p}" loading="lazy">, and because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> is closed under finite intersection, Cantor's intersection theorem applies and the intersection of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> is nonempty. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {diam} ({\bar {p}})=\operatorname {diam} (p)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {diam} ({\bar {p}})=\operatorname {diam} (p)}</annotation>

</semantics>

</math><img src="./0966d77bee1bd20767a247461e20bd3b6d48386a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.321ex; height:2.843ex;" alt="{\displaystyle \operatorname {diam} ({\bar {p}})=\operatorname {diam} (p)}" loading="lazy"> and the ground model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> inherits a metric from the universe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> has elements of arbitrarily small diameter. Finally, there is exactly one real that belongs to all members of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy">. The generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> can be reconstructed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{p\in P:r_{G}\in {\bar {p}}\}}">

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=\{p\in P:r_{G}\in {\bar {p}}\}}</annotation>

</semantics>

</math><img src="./ca45cf47e7bbec2912e2ba9e0dd0a0a279eace30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.715ex; height:2.843ex;" alt="{\displaystyle G=\{p\in P:r_{G}\in {\bar {p}}\}}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in M^{(\mathbb {P} )}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\in M^{(\mathbb {P} )}}</annotation>

</semantics>

</math><img src="./5139a798d7ae7fe8fa2a111f11ed8ea3c3acaa7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.085ex; height:2.843ex;" alt="{\displaystyle a\in M^{(\mathbb {P} )}}" loading="lazy"> is a name for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy"> (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]\models val(a,G)=r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]\models val(a,G)=r_{G}}</annotation>

</semantics>

</math><img src="./440157e8cf175c6d958392f0a9f176b939bb8ba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.489ex; height:2.843ex;" alt="{\displaystyle M[G]\models val(a,G)=r_{G}}" loading="lazy">), and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in M}</annotation>

</semantics>

</math><img src="./44ab7481d485e014d200d3a6c39f1e2c52954b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.047ex; height:2.176ex;" alt="{\displaystyle B\in M}" loading="lazy"> holds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models }</annotation>

</semantics>

</math><img src="./61a22b9bfd453c325ffdf82da69f459e32996864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.102ex; height:2.843ex;" alt="{\displaystyle M\models }" loading="lazy"> "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is a Borel set of measure 1", then by the truth property of forcing

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }a\in {\check {B}}}</annotation>

</semantics>

</math><img src="./be0cad9daddcc3dbe952eef64774df9284d7c378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:13.483ex; height:3.343ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }a\in {\check {B}}}" loading="lazy"></dd></dl>

for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\in G}</annotation>

</semantics>

</math><img src="./646b90327c05c7a10155bb0fe34e46195b9447de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.926ex; height:2.509ex;" alt="{\displaystyle p\in G}" loading="lazy">. There is a name <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> that satisfies

</mrow>

</munder>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {val} (a,G)\in \bigcup _{p\in G}{\bar {p}}}</annotation>

</semantics>

</math><img src="./18d6a3cf15c14afe547f5cb63c5080c98565e1c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.739ex; height:5.843ex;" alt="{\displaystyle \operatorname {val} (a,G)\in \bigcup _{p\in G}{\bar {p}}}" loading="lazy"></dd></dl>

for any generic filter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. For that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">,

<msub>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\Vdash _{M,\mathbb {P} }a\in {\check {B}}}</annotation>

</semantics>

</math><img src="./be0cad9daddcc3dbe952eef64774df9284d7c378.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:13.483ex; height:3.343ex;" alt="{\displaystyle p\Vdash _{M,\mathbb {P} }a\in {\check {B}}}" loading="lazy"></dd></dl>

holds for any condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">.

Every Borel set can be (non-uniquely) built up, starting from intervals with rational endpoints and applying the operations of complement and countable union, a countable number of times. The record of such a construction is called a Borel code. Given a Borel set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">, one recovers a Borel code, and then applies the same construction sequence in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy">, getting a Borel set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B^{*}}</annotation>

</semantics>

</math><img src="./202be1fdef03f14a352508cf0bc32073470d44ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle B^{*}}" loading="lazy">. It can be proven that one gets the same set independent of the code chosen for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, and that basic properties are preserved. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq C}</annotation>

</semantics>

</math><img src="./87dc1864578bb8f696bbfb7ba1c134c0f1f20458.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.629ex; height:2.343ex;" alt="{\displaystyle B\subseteq C}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{*}\subseteq C^{*}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B^{*}\subseteq C^{*}}</annotation>

</semantics>

</math><img src="./ce0368edf666adf97329eb32c530561fe707c942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.769ex; height:2.509ex;" alt="{\displaystyle B^{*}\subseteq C^{*}}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> has measure zero, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B^{*}}</annotation>

</semantics>

</math><img src="./202be1fdef03f14a352508cf0bc32073470d44ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle B^{*}}" loading="lazy"> has measure zero. This mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\mapsto B^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\mapsto B^{*}}</annotation>

</semantics>

</math><img src="./0480b7ba716cd0dfd9196a873c64a66957518d24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.196ex; height:2.343ex;" alt="{\displaystyle B\mapsto B^{*}}" loading="lazy"> is injective.

For any set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\subseteq [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\subseteq [0,1]}</annotation>

</semantics>

</math><img src="./25a5404d06244bb945c44295b7c72890f813f8ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.515ex; height:2.843ex;" alt="{\displaystyle B\subseteq [0,1]}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B\in M}</annotation>

</semantics>

</math><img src="./44ab7481d485e014d200d3a6c39f1e2c52954b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.047ex; height:2.176ex;" alt="{\displaystyle B\in M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\models }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\models }</annotation>

</semantics>

</math><img src="./61a22b9bfd453c325ffdf82da69f459e32996864.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.102ex; height:2.843ex;" alt="{\displaystyle M\models }" loading="lazy"> "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

</semantics>

</math><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> is a Borel set of measure 1" one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}\in B^{*}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}\in B^{*}}</annotation>

</semantics>

</math><img src="./7dcd662fc30c242ba88c3c86ae6e345ff03aed24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.231ex; height:2.676ex;" alt="{\displaystyle r_{G}\in B^{*}}" loading="lazy">.

This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy"> is an "infinite random sequence of 0s and 1s" from the viewpoint of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, which means that it satisfies all statistical tests from the ground model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

So given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{G}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{G}}</annotation>

</semantics>

</math><img src="./31fdac971137ff7d7d706e30722fa28ff0567f07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.573ex; height:2.009ex;" alt="{\displaystyle r_{G}}" loading="lazy">, a random real, one can show that

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G=\left\{B~({\text{in }}M)\mid r\in B^{*}~({\text{in }}M[G])\right\}.}</annotation>

</semantics>

</math><img src="./c770319caee093e791ef7f2df8b245102f021695.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.517ex; height:2.843ex;" alt="{\displaystyle G=\left\{B~({\text{in }}M)\mid r\in B^{*}~({\text{in }}M[G])\right\}.}" loading="lazy"></dd></dl>

Because of this mutual inter-definability between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, one generally writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[r]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[r]}</annotation>

</semantics>

</math><img src="./d872771ce5ae4825013a9d3b133b04d24ed3b188.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.785ex; height:2.843ex;" alt="{\displaystyle M[r]}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy">.

A different interpretation of reals in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> was provided by <a href="Dana_Scott" title="Dana Scott">Dana Scott</a>. Rational numbers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> have names that correspond to countably-many distinct rational values assigned to a maximal antichain of Borel sets – in other words, a certain rational-valued function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=[0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=[0,1]}</annotation>

</semantics>

</math><img src="./87ec65159c44769434523e46928bc1b82681f842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.923ex; height:2.843ex;" alt="{\displaystyle I=[0,1]}" loading="lazy">. Real numbers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> then correspond to <a href="Dedekind_cut" title="Dedekind cut">Dedekind cuts</a> of such functions, that is, <a href="Measurable_function" title="Measurable function">measurable functions</a>.

<div class="mw-heading mw-heading2"><h2 id="Boolean-valued_models">Boolean-valued models</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Boolean-valued_model" title="Boolean-valued model">Boolean-valued model</a></div>

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In these, any statement is assigned a <a href="Truth_value" title="Truth value">truth value</a> from some complete atomless <a href="Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a>, rather than just a true/false value. Then an <a href="Ultrafilter" title="Ultrafilter">ultrafilter</a> is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model that contains this ultrafilter, which can be understood as a new model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in an appropriate way, we can get a model that has the desired property. In it, only statements that must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).

<div class="mw-heading mw-heading2"><h2 id="Meta-mathematical_explanation">Meta-mathematical explanation</h2></div>

In forcing, we usually seek to show that some <a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentence</a> is <a href="Consistency_proof" class="mw-redirect" title="Consistency proof">consistent</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> (or optionally some extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">). One way to interpret the argument is to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> is consistent and then prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> combined with the new <a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentence</a> is also consistent.

Each "condition" is a finite piece of information – the idea is that only finite pieces are relevant for consistency, since, by the <a href="Compactness_theorem" title="Compactness theorem">compactness theorem</a>, a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then we can pick an infinite set of consistent conditions to extend our model. Therefore, assuming the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">, we prove the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> extended by this infinite set.

<div class="mw-heading mw-heading2"><h2 id="Logical_explanation">Logical explanation</h2></div>

By <a href="G%C3%B6del's_second_incompleteness_theorem" class="mw-redirect" title="Gödel's second incompleteness theorem">Gödel's second incompleteness theorem</a>, one cannot prove the consistency of any sufficiently strong formal theory, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">, using only the axioms of the theory itself, unless the theory is inconsistent. Consequently, mathematicians do not attempt to prove the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> using only the axioms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">, or to prove that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}+H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+H}</annotation>

</semantics>

</math><img src="./552cc68597f5344a3fe175d92cafe90641a8cd6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.132ex; height:2.343ex;" alt="{\displaystyle {\mathsf {ZFC}}+H}" loading="lazy"> is consistent for any hypothesis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> using only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}+H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+H}</annotation>

</semantics>

</math><img src="./552cc68597f5344a3fe175d92cafe90641a8cd6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.132ex; height:2.343ex;" alt="{\displaystyle {\mathsf {ZFC}}+H}" loading="lazy">. For this reason, the aim of a consistency proof is to prove the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}+H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+H}</annotation>

</semantics>

</math><img src="./552cc68597f5344a3fe175d92cafe90641a8cd6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.132ex; height:2.343ex;" alt="{\displaystyle {\mathsf {ZFC}}+H}" loading="lazy"> relative to the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">. Such problems are known as problems of relative consistency, one of which proves

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}\vdash \operatorname {Con} ({\mathsf {ZFC}})\rightarrow \operatorname {Con} ({\mathsf {ZFC}}+H).}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}\vdash \operatorname {Con} ({\mathsf {ZFC}})\rightarrow \operatorname {Con} ({\mathsf {ZFC}}+H).}</annotation>

</semantics>

</math></td> <td></td> <td class="nowrap">⁎</td></tr></tbody></table>

The general schema of relative consistency proofs follows. As any proof is finite, it uses only a finite number of axioms:

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land (T\vdash \lnot H)).}</annotation>

</semantics>

</math><img src="./6b041fb4672833c3050fd78df5deba5df2da4ec0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.787ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land (T\vdash \lnot H)).}" loading="lazy"></dd></dl>

For any given proof, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> can verify the validity of this proof. This is provable by induction on the length of the proof.

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}\vdash (\forall T)((T\vdash \lnot H)\rightarrow ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))).}</annotation>

</semantics>

</math><img src="./98f0932ccf355540eae6f481335aef10d8b6afd3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.036ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}\vdash (\forall T)((T\vdash \lnot H)\rightarrow ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))).}" loading="lazy"></dd></dl>

Then resolve

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))).}</annotation>

</semantics>

</math><img src="./3a4d114595ed38e12f8b9c99ca987de37749d761.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.535ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))).}" loading="lazy"></dd></dl>

By proving the following

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}\vdash (\forall T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\rightarrow ({\mathsf {ZFC}}\vdash \operatorname {Con} (T+H))),}</annotation>

</semantics>

</math></td> <td></td> <td class="nowrap">⁎⁎</td></tr></tbody></table>

it can be concluded that

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))\land ({\mathsf {ZFC}}\vdash \operatorname {Con} (T+H))),}</annotation>

</semantics>

</math><img src="./d5909418980fa7b1ee6eeca7d764be60efe5ac54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:97.348ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash (\exists T)(\operatorname {Fin} (T)\land T\subseteq {\mathsf {ZFC}}\land ({\mathsf {ZFC}}\vdash (T\vdash \lnot H))\land ({\mathsf {ZFC}}\vdash \operatorname {Con} (T+H))),}" loading="lazy"></dd></dl>

which is equivalent to

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash \lnot \operatorname {Con} ({\mathsf {ZFC}}),}</annotation>

</semantics>

</math><img src="./5e45c551cc2ea134d670f187ef8e811edf35d83f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.546ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}+\lnot \operatorname {Con} ({\mathsf {ZFC}}+H)\vdash \lnot \operatorname {Con} ({\mathsf {ZFC}}),}" loading="lazy"></dd></dl>

which gives (*). The core of the relative consistency proof is proving (**). A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Con} (T+H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Con} (T+H)}</annotation>

</semantics>

</math><img src="./be2934d0e07082b5f6eb5f585602b17eab14497f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle \operatorname {Con} (T+H)}" loading="lazy"> can be constructed for any given finite subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms (by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> instruments of course). (No universal proof of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Con} (T+H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Con} (T+H)}</annotation>

</semantics>

</math><img src="./be2934d0e07082b5f6eb5f585602b17eab14497f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle \operatorname {Con} (T+H)}" loading="lazy"> of course.)

In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy">, it is provable that for any condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, the set of formulas (evaluated by names) forced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> is deductively closed. Furthermore, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axiom, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> proves that this axiom is forced by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} }</annotation>

</semantics>

</math><img src="./235ffc0f1788b720aef5caa7b97246a84421fd0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {1} }" loading="lazy">. Then it suffices to prove that there is at least one condition that forces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">.

In the case of Boolean-valued forcing, the procedure is similar: proving that the Boolean value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> is not <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation>

</semantics>

</math><img src="./62e8c650763635a93ddc69768c3c0c100afe985d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }" loading="lazy">.

Another approach uses the Reflection Theorem. For any given finite set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms, there is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> proof that this set of axioms has a countable transitive model. For any given finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms, there is a finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T'}</annotation>

</semantics>

</math><img src="./4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> proves that if a countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T'}</annotation>

</semantics>

</math><img src="./4b32d735f4fbb6eff3b35ed3dc1005a069d0b2e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.405ex; height:2.509ex;" alt="{\displaystyle T'}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">. By proving that there is finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T''}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T''}</annotation>

</semantics>

</math><img src="./b7376c7d3d25151dd6965ecc52ccf169873d0a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.857ex; height:2.509ex;" alt="{\displaystyle T''}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms such that if a countable transitive model <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T''}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T''}</annotation>

</semantics>

</math><img src="./b7376c7d3d25151dd6965ecc52ccf169873d0a43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.857ex; height:2.509ex;" alt="{\displaystyle T''}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M[G]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M[G]}</annotation>

</semantics>

</math><img src="./56895a360dcf11f18fc4d8ab115b6f9e608c847a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.563ex; height:2.843ex;" alt="{\displaystyle M[G]}" loading="lazy"> satisfies the hypothesis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">. Then, for any given finite set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> axioms, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> proves <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Con} (T+H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Con} (T+H)}</annotation>

</semantics>

</math><img src="./be2934d0e07082b5f6eb5f585602b17eab14497f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle \operatorname {Con} (T+H)}" loading="lazy">.

Sometimes in (**), a stronger theory <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> is used for proving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Con} (T+H)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Con} (T+H)}</annotation>

</semantics>

</math><img src="./be2934d0e07082b5f6eb5f585602b17eab14497f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.483ex; height:2.843ex;" alt="{\displaystyle \operatorname {Con} (T+H)}" loading="lazy">. Then we have proof of the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}+H}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}+H}</annotation>

</semantics>

</math><img src="./552cc68597f5344a3fe175d92cafe90641a8cd6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.132ex; height:2.343ex;" alt="{\displaystyle {\mathsf {ZFC}}+H}" loading="lazy"> relative to the consistency of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy">. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}\vdash \operatorname {Con} ({\mathsf {ZFC}})\leftrightarrow \operatorname {Con} ({\mathsf {ZFL}})}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}\vdash \operatorname {Con} ({\mathsf {ZFC}})\leftrightarrow \operatorname {Con} ({\mathsf {ZFL}})}</annotation>

</semantics>

</math><img src="./8b3520715be233a4a3580a50576ea128a05c00e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.669ex; height:2.843ex;" alt="{\displaystyle {\mathsf {ZFC}}\vdash \operatorname {Con} ({\mathsf {ZFC}})\leftrightarrow \operatorname {Con} ({\mathsf {ZFL}})}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFL}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFL}}}</annotation>

</semantics>

</math><img src="./a1118bbbc2dffb7d0357d46f7b2d2f9ce8f159c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.003ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFL}}}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZF}}+V=L}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZF}}+V=L}</annotation>

</semantics>

</math><img src="./e4884c39e1d8811d7c598dfe78cf8fef9ba41f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.052ex; height:2.343ex;" alt="{\displaystyle {\mathsf {ZF}}+V=L}" loading="lazy"> (the axiom of constructibility).

<ul><li><a href="List_of_forcing_notions" title="List of forcing notions">List of forcing notions</a></li>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-FOOTNOTECohen2008111-1">^ <a href="#cite_ref-FOOTNOTECohen2008111_1-0">a</a> <a href="#cite_ref-FOOTNOTECohen2008111_1-1">b</a> <a href="#cite_ref-FOOTNOTECohen2008111_1-2">c</a> <a href="#CITEREFCohen2008">Cohen 2008</a>, p. 111.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> As a concrete example, note that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{0}}</annotation>

</semantics>

</math><img src="./a214eff2fcc322f780dd8837e7472b0edb994a13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{0}}" loading="lazy">, the <a href="Order_type" title="Order type">order type</a> of all ordinals in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, is a countable ordinal (in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">) that is not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is taken to be a <a href="Well-ordering" class="mw-redirect" title="Well-ordering">well-ordering</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"> (as a <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation>

</semantics>

</math><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy">, i.e. a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \times \mathbb {N} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times \mathbb {N} }</annotation>

</semantics>

</math><img src="./5b25f8c4219e47350185be7ebdc5250c8d59ab35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \times \mathbb {N} }" loading="lazy">), then any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {ZFC}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathsf {ZFC}}}</annotation>

</semantics>

</math><img src="./3d397b9d5a1a8a9507a2ec5ab37c5f60da51f249.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.228ex; height:2.176ex;" alt="{\displaystyle {\mathsf {ZFC}}}" loading="lazy"> universe containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> must also contain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{0}}</annotation>

</semantics>

</math><img src="./a214eff2fcc322f780dd8837e7472b0edb994a13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{0}}" loading="lazy"> (thanks to the <a href="Axiom_of_replacement" class="mw-redirect" title="Axiom of replacement">axiom of replacement</a>).<a href="#cite_note-FOOTNOTECohen2008111-1">[1]</a> (Such a universe would also not resemble <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> in the sense that it would collapse all infinite cardinals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.)

</li>

<li id="cite_note-FOOTNOTEShoenfield1971-3">^ <a href="#cite_ref-FOOTNOTEShoenfield1971_3-0">a</a> <a href="#cite_ref-FOOTNOTEShoenfield1971_3-1">b</a> <a href="#cite_ref-FOOTNOTEShoenfield1971_3-2">c</a> <a href="#CITEREFShoenfield1971">Shoenfield 1971</a>.

</li>

<li id="cite_note-FOOTNOTEKunen1980-4"><a href="#cite_ref-FOOTNOTEKunen1980_4-0">^</a> <a href="#CITEREFKunen1980">Kunen 1980</a>.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Notably, if defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }}</annotation>

</semantics>

</math><img src="./02f0b28a063cf58cbc1e29b3b478814e74165de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.099ex; height:2.843ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }}" loading="lazy"> directly instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}</annotation>

</semantics>

</math><img src="./bbce74e05bba9146dc7d225134924ac1432941fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.099ex; height:3.176ex;" alt="{\displaystyle \Vdash _{M,\mathbb {P} }^{*}}" loading="lazy">, one would need to replace the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vee }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \vee }</annotation>

</semantics>

</math><img src="./7b76220c6805c9b465d6efbc7686c624f49f3023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \vee }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \wedge }</annotation>

</semantics>

</math><img src="./1caa4004cb216ef2930bb12fe805a76870caed94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.009ex;" alt="{\displaystyle \wedge }" loading="lazy"> in case 4 and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \exists }</annotation>

</semantics>

</math><img src="./77ed842b6b90b2fdd825320cf8e5265fa937b583.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \exists }" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \forall }</annotation>

</semantics>

</math><img src="./bfc1a1a9c4c0f8d5df989c98aa2773ed657c5937.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \forall }" loading="lazy"> in case 5 (in addition to making cases 1 and 2 more complicated) to make this internal definition agree with the external definition. However, then when trying to prove Truth inductively, case 4 will require the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">, as a <a href="Filter_(mathematics)" title="Filter (mathematics)">filter</a>, is downward <a href="Directed_set" title="Directed set">directed</a>, and case 5 will outright break down.

</li>

<li id="cite_note-FOOTNOTECohen2008Section_IV.8,_Lemma_2-6"><a href="#cite_ref-FOOTNOTECohen2008Section_IV.8,_Lemma_2_6-0">^</a> <a href="#CITEREFCohen2008">Cohen 2008</a>, Section IV.8, Lemma 2.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBell1985" class="citation book cs1">Bell, John Lane (1985). Boolean-Valued Models and Independence Proofs in Set Theory. Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780198532415</bdi>.</cite></li>

<li><cite id="CITEREFCohen2008" class="citation book cs1">Cohen, Paul Joseph (2008) [1966]. Set theory and the continuum hypothesis. Mineola, New York City: Dover Publications. p. 151. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-46921-8</bdi>.</cite></li>

<li><cite id="CITEREFGrishin2001" class="citation cs2">Grishin, V. N. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Forcing_Method&oldid=51543">"Forcing Method"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite></li>

<li><cite id="CITEREFJech2013" class="citation book cs1">Jech, Thomas J. (2013) [1978]. Set Theory: The Third Millennium Edition. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer Verlag</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783642078996</bdi>.</cite></li>

<li><cite id="CITEREFKunen1980" class="citation book cs1">Kunen, Kenneth (1980). <a href="Set_Theory%3A_An_Introduction_to_Independence_Proofs" title="Set Theory: An Introduction to Independence Proofs">Set Theory: An Introduction to Independence Proofs</a>. <a href="Elsevier" title="Elsevier">North-Holland Publishing Company</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-85401-8</bdi>.</cite></li>

<li><cite id="CITEREFShoenfield1971" class="citation book cs1 cs1-prop-long-vol">Shoenfield, J. R. (1971). "Unramified forcing". Axiomatic Set Theory. Proc. Sympos. Pure Math. Vol. XIII, Part I. Providence, R.I.: Amer. Math. Soc. pp. 357–381. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0280359">0280359</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFChow2008" class="citation arxiv cs1">Chow, Timothy (2008). "A Beginner's Guide to Forcing". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0712.1320v2">0712.1320v2</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>]. <q>A good introduction to the concepts of forcing that avoids a lot of technical detail. It includes a section on Boolean-valued models.</q></cite></li>

<li><cite id="CITEREFCohen1963" class="citation journal cs1">Cohen, Paul Joseph (December 1963). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287">"The independence of the continuum hypothesis"</a>. <a href="Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">Proceedings of the National Academy of Sciences of the United States of America</a>. 50 (6): 1143–1148. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PNAS...50.1143C">1963PNAS...50.1143C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.50.6.1143">10.1073/pnas.50.6.1143</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC221287">221287</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16578557">16578557</a>.</cite></li>

<li><cite id="CITEREFCohen1964" class="citation journal cs1">Cohen, Paul Joseph (January 1964). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611">"The independence of the continuum hypothesis, II"</a>. <a href="Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America" title="Proceedings of the National Academy of Sciences of the United States of America">Proceedings of the National Academy of Sciences of the United States of America</a>. 51 (1): 105–110. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1964PNAS...51..105C">1964PNAS...51..105C</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.51.1.105">10.1073/pnas.51.1.105</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300611">300611</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16591132">16591132</a>.</cite></li>

<li><cite id="CITEREFCohen2002" class="citation journal cs1">Cohen, Paul Joseph (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1216%2Frmjm%2F1181070010">"The Discovery of Forcing"</a>. Rocky Mountain J. Math. 32 (4): 1071–1100. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1216%2Frmjm%2F1181070010">10.1216/rmjm/1181070010</a>. <q>A historical lecture about how he developed his independence proof.</q></cite></li>

<li><cite id="CITEREFEaswaran2007" class="citation arxiv cs1"><a href="Kenny_Easwaran" title="Kenny Easwaran">Easwaran, Kenny</a> (2007). "A Cheerful Introduction to Forcing and the Continuum Hypothesis". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0712.2279">0712.2279</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.LO">math.LO</a>]. <q>The article is also aimed at the beginner but includes more technical details than <a href="#CITEREFChow2008">Chow (2008)</a></q></cite></li>

<li><cite id="CITEREFGuntherPaganoSánchez_TerrafSteinberg2020" class="citation journal cs1">Gunther, Emmanuel; Pagano, Miguel; Sánchez Terraf, Pedro; Steinberg, Matías (May 2020). <a rel="nofollow" class="external text" href="https://www.isa-afp.org/entries/Forcing.html">"Formalization of Forcing in Isabelle/ZF"</a>. Archive of Formal Proofs. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2001.09715">2001.09715</a>. Retrieved 20 August 2023.</cite></li>

<li><cite id="CITEREFKanamori2007" class="citation web cs1"><a href="Akihiro_Kanamori" title="Akihiro Kanamori">Kanamori, Akihiro</a> (2007). <a rel="nofollow" class="external text" href="http://math.bu.edu/people/aki/16.pdf">"Set theory from Cantor to Cohen"</a> (PDF).</cite></li>

<li><cite id="CITEREFWeaver2014" class="citation book cs1">Weaver, Nik (2014). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/8962">Forcing for Mathematicians</a>. World Scientific Publishing Co. p. 153. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F8962">10.1142/8962</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-9814566001</bdi>. <q>Written for mathematicians who want to learn the basic machinery of forcing. No background in logic is assumed, beyond the facility with formal syntax which should be second nature to any well-trained mathematician.</q></cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Forcing.html">"Forcing"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li></ul>

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</style><div id="Set_theory409" style="font-size:114%;margin:0 4em"><a href="Set_theory" title="Set theory">Set theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><a href="Venn_diagram" title="Venn diagram"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li>

<li><a href="Axiom_of_choice" title="Axiom of choice">Choice</a>

<ul><li><a href="Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li>

<li><a href="Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li>

<li><a href="Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li>

<li><a href="Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li>

<li><a href="Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a>

<ul><li><a href="Axiom_of_projective_determinacy" class="mw-redirect" title="Axiom of projective determinacy">projective</a></li></ul></li>

<li><a href="Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li>

<li><a href="Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li>

<li><a href="Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li>

<li><a href="Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li>

<li><a href="Axiom_of_power_set" title="Axiom of power set">Power set</a></li>

<li><a href="Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li>

<li><a href="Axiom_of_union" title="Axiom of union">Union</a></li>

<li><a href="Martin's_axiom" title="Martin's axiom">Martin's axiom</a></li></ul>

<ul><li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a>

<ul><li><a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li>

<li><a href="Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_(mathematics)#Basic_operations" title="Set (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li>

<li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

<li><a href="Disjoint_union" title="Disjoint union">Disjoint union</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">Identities</a></li>

<li><a href="Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></li>

<li><a href="Power_set" title="Power set">Power set</a></li>

<li><a href="Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">Union</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li>Concepts</li><li>Methods</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost" title="Almost">Almost</a></li>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="Large_cardinal" title="Large cardinal">large</a>)</li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li><a href="Constructible_universe" title="Constructible universe">Constructible universe</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">Continuum hypothesis</a></li>

<li><a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li>

<li><a href="Element_(mathematics)" title="Element (mathematics)">Element</a>

<ul><li><a href="Ordered_pair" title="Ordered pair">ordered pair</a></li>

<li><a href="Tuple" title="Tuple">tuple</a></li></ul></li>

<li><a href="Family_of_sets" title="Family of sets">Family</a></li>

<li><a href="Bijection" title="Bijection">One-to-one correspondence</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li>

<li><a href="Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_(mathematics)" title="Set (mathematics)">Set</a> types</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amorphous_set" title="Amorphous set">Amorphous</a></li>

<li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a> (<a href="Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li>

<li><a href="Filter_(set_theory)" title="Filter (set theory)">Filter</a>

<li><a href="Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li>

<li><a href="Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a> (<a href="Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li>

<li><a href="Computable_set" title="Computable set">Recursive</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Subset" title="Subset">Subset · Superset</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li>

<li><a href="Set_theory#Formalized_set_theory" title="Set theory">Axiomatic</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="Cantor's_theorem" title="Cantor's theorem">Cantor's theorem</a></li></ul>

<ul><li><a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo</a>

<ul><li><a href="General_set_theory" title="General set theory">General</a></li></ul></li>

<li><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a>

<ul><li><a href="New_Foundations" title="New Foundations">New Foundations</a></li></ul></li>

<li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel </a>

<ul><li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a>

<ul><li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li></ul></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li>

<li><a href="Suslin's_problem" title="Suslin's problem">Suslin's problem</a></li>

<li><a href="Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li>

<li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li>

<li><a href="Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li>

<li><a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li>

<li><a href="Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li>

<li><a href="Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li>

<li><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li>

<li><a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li>

<li><a href="Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li>

<li><a href="Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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