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<title>Empty set</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Empty set</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"∅" redirects here. For similar symbols, see <a href="%C3%98_(disambiguation)" class="mw-disambig" title="Ø (disambiguation)">Ø (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">For other uses of "Empty", see <a href="Empty_(disambiguation)" class="mw-redirect mw-disambig" title="Empty (disambiguation)">Empty (disambiguation)</a>.</div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>empty set</b> or <b>void set</b> is the unique <a href="Set_(mathematics)" title="Set (mathematics)">set</a> having no <a href="Element_(mathematics)" title="Element (mathematics)">elements</a>; its size or <a href="Cardinality" title="Cardinality">cardinality</a> (count of elements in a set) is <a href="0" title="0">zero</a>.<sup id="cite_ref-:1_1-0" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Some <a href="Axiomatic_set_theories" class="mw-redirect" title="Axiomatic set theories">axiomatic set theories</a> ensure that the empty set exists by including an <a href="Axiom_of_empty_set" title="Axiom of empty set">axiom of empty set</a>, while in other theories, its existence can be deduced. Many possible properties of sets are <a href="Vacuously_true" class="mw-redirect" title="Vacuously true">vacuously true</a> for the empty set.

</p><p>Any set other than the empty set is called <i>non-empty</i>.

</p><p>In some textbooks and popularizations, the empty set is referred to as the "null set".<sup id="cite_ref-:1_1-1" class="reference"><a href="#cite_note-:1-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, <a href="Null_set" title="Null set">null set</a> is a distinct notion within the context of <a href="Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a>, in which it describes a set of measure zero (which is not necessarily empty).

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Null_sign" title="Null sign">Null sign</a></div>

<p>Common notations for the empty set include "{ }", "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

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

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</math></span><img src="./6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy"></span>", and "∅". The latter two symbols were introduced by the <a href="Bourbaki_group" class="mw-redirect" title="Bourbaki group">Bourbaki group</a> (specifically <a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a>) in 1939, inspired by the letter <a href="%C3%98" title="Ø">Ø</a> (<span class="nowrap"><style data-mw-deduplicate="TemplateStyles:r886049734">

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</style><span class="monospaced">U+00D8</span> </span><span style="font-size:125%;line-height:1em">Ø</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">LATIN CAPITAL LETTER O WITH STROKE</span>) in the <a href="Danish_orthography" title="Danish orthography">Danish</a> and <a href="Norwegian_orthography" title="Norwegian orthography">Norwegian</a> alphabets.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the past, "0" (the numeral <a href="Zero" class="mw-redirect" title="Zero">zero</a>) was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>The symbol ∅ is available at <a href="Unicode" title="Unicode">Unicode</a> point <span class="nowrap"><span class="monospaced">U+2205</span> </span><span style="font-size:125%;line-height:1em">∅</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">EMPTY SET</span>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> It can be coded in <a href="HTML" title="HTML">HTML</a> as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">&empty;</code> and as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">&#8709;</code> or as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">&#x2205;</code>. It can be coded in <a href="LaTeX" title="LaTeX">LaTeX</a> as <code class="mw-highlight mw-highlight-lang-text mw-content-ltr" style="" dir="ltr">\varnothing</code>. The symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

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

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</p><p>When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character <span class="nowrap"><span class="monospaced">U+29B0</span> </span><span style="font-size:125%;line-height:1em">⦰</span> <span style="font-variant: small-caps; text-transform: lowercase; font-feature-settings: 'onum'">REVERSED EMPTY SET</span> may be used instead.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>In standard <a href="Axiomatic_set_theory" class="mw-redirect" title="Axiomatic set theory">axiomatic set theory</a>, by the <a href="Axiom_of_extensionality" title="Axiom of extensionality">principle of extensionality</a>, two sets are equal if they have the same elements (that is, neither of them has an element not in the other). As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".

</p><p>The only subset of the empty set is the empty set itself; equivalently, the <a href="Power_set" title="Power set">power set</a> of the empty set is the set containing only the empty set. The number of elements of the empty set (i.e., its <a href="Cardinality" title="Cardinality">cardinality</a>) is zero. The empty set is the only set with either of these properties.

</p><p><a href="For_any" class="mw-redirect" title="For any">For any</a> set <i>A</i>:

</p>

<ul><li>The empty set is a <a href="Subset" title="Subset">subset</a> of <i>A</i></li>

<li>The <a href="Union_(set_theory)" title="Union (set theory)">union</a> of <i>A</i> with the empty set is <i>A</i></li>

<li>The <a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of <i>A</i> with the empty set is the empty set</li>

<li>The <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of <i>A</i> and the empty set is the empty set</li></ul>

<p>For any <a href="Property_(philosophy)" title="Property (philosophy)">property</a> <i>P</i>:

</p>

<ul><li>For every element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

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

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<li>There is no element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

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

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<p>Conversely, if for some property <i>P</i> and some set <i>V</i>, the following two statements hold:

</p>

<ul><li>For every element of <i>V</i> the property <i>P</i> holds</li>

<li>There is no element of <i>V</i> for which the property <i>P</i> holds</li></ul>

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

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</math></span><img src="./387b59575160880d2db83985b19c9a1a97e08b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.341ex; height:2.176ex;" alt="{\displaystyle V=\varnothing .}" loading="lazy"></span>

</p><p>By the definition of <a href="Subset" title="Subset">subset</a>, the empty set is a subset of any set <i>A</i>. That is, <em>every</em> element <i>x</i> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

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</math></span><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"></span> belongs to <i>A</i>. Indeed, if it were not true that every element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

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</math></span><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"></span> is in <i>A</i>, then there would be at least one element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

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</mrow>

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

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</math></span><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"></span> that is not present in <i>A</i>. Since there are <em>no</em> elements of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

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</mrow>

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

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</math></span><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"></span> at all, there is no element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

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</mrow>

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

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</math></span><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"></span> that is not in <i>A</i>. Any statement that begins "for every element of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

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</math></span><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"></span>" is not making any substantive claim; it is a <a href="Vacuous_truth" title="Vacuous truth">vacuous truth</a>. This is often paraphrased as "everything is true of the elements of the empty set."

</p><p>In the usual <a href="Set-theoretic_definition_of_natural_numbers" title="Set-theoretic definition of natural numbers">set-theoretic definition of natural numbers</a>, zero is modelled by the empty set.

</p>

<div class="mw-heading mw-heading3"><h3 id="Operations_on_the_empty_set">Operations on the empty set</h3></div>

<p>When speaking of the <a href="Summation" title="Summation">sum</a> of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set (the <a href="Empty_sum" title="Empty sum">empty sum</a>) is zero. The reason for this is that zero is the <a href="Identity_element" title="Identity element">identity element</a> for addition. Similarly, the <a href="Multiplication" title="Multiplication">product</a> of the elements of the empty set (the <a href="Empty_product" title="Empty product">empty product</a>) should be considered to be <a href="1_(number)" class="mw-redirect" title="1 (number)">one</a>, since one is the identity element for multiplication.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>A <a href="Derangement" title="Derangement">derangement</a> is a <a href="Permutation" title="Permutation">permutation</a> of a set without <a href="Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed points</a>. The empty set can be considered a derangement of itself, because it has only one permutation (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0!=1}">

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

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</math></span><img src="./22956a0fa255c6c9562eab440f8c23c2954a6cf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle 0!=1}" loading="lazy"></span>), and it is vacuously true that no element (of the empty set) can be found that retains its original position.

</p>

<div class="mw-heading mw-heading2"><h2 id="In_other_areas_of_mathematics">In other areas of mathematics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Extended_real_numbers">Extended real numbers</h3></div>

<p>Since the empty set has no member when it is considered as a subset of any <a href="Ordered_set" class="mw-redirect" title="Ordered set">ordered set</a>, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the <a href="Real_number_line" class="mw-redirect" title="Real number line">real number line</a>, every real number is both an upper and lower bound for the empty set.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> When considered as a subset of the <a href="Extended_reals" class="mw-redirect" title="Extended reals">extended reals</a> formed by adding two "numbers" or "points" to the real numbers (namely <a href="Negative_infinity" class="mw-redirect" title="Negative infinity">negative infinity</a>, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty \!\,,}">

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

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</math></span><img src="./7b438bbc5dfea93b5ef0d4f68bc2c14aa4dda81f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.779ex; height:2.343ex;" alt="{\displaystyle -\infty \!\,,}" loading="lazy"></span> which is defined to be less than every other extended real number, and <a href="Positive_infinity" class="mw-redirect" title="Positive infinity">positive infinity</a>, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty \!\,,}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle +\infty \!\,,}</annotation>

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</math></span><img src="./edd28ec65277221a7173e7db9049ced9dde8ef16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.779ex; height:2.343ex;" alt="{\displaystyle +\infty \!\,,}" loading="lazy"></span> which is defined to be greater than every other extended real number), we have that:

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sup \varnothing =\min(\{-\infty ,+\infty \}\cup \mathbb {R} )=-\infty ,}</annotation>

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</math></span></span>

and

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \inf \varnothing =\max(\{-\infty ,+\infty \}\cup \mathbb {R} )=+\infty .}</annotation>

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</math></span></span>

</p><p>That is, the least upper bound (sup or <a href="Supremum" class="mw-redirect" title="Supremum">supremum</a>) of the empty set is negative infinity, while the greatest lower bound (inf or <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a>) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.

</p>

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

<p>In any <a href="Topological_space" title="Topological space">topological space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>, the empty set is <a href="Open_set" title="Open set">open</a> by definition, as is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>. Since the <a href="Complement_(set_theory)" title="Complement (set theory)">complement</a> of an open set is <a href="Closed_set" title="Closed set">closed</a> and the empty set and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> are complements of each other, the empty set is also closed, making it a <a href="Clopen_set" title="Clopen set">clopen set</a>. Moreover, the empty set is <a href="Compact_set" class="mw-redirect" title="Compact set">compact</a> by the fact that every <a href="Finite_set" title="Finite set">finite set</a> is compact.

</p><p>A topological space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is said to have the <a href="Indiscrete_topology" class="mw-redirect" title="Indiscrete topology">indiscrete topology</a> if the only open sets are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</p><p>The <a href="Closure_(mathematics)" title="Closure (mathematics)">closure</a> of the empty set is empty. This is known as "preservation of <a href="Nullary" class="mw-redirect" title="Nullary">nullary</a> <a href="Union_(set_theory)" title="Union (set theory)">unions</a>".<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Category_theory">Category theory</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> is a set, then there exists precisely one <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> from <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./2746026864cc5896e3e52443a1c917be2df9d8ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}" loading="lazy"></span> the <a href="Empty_function" class="mw-redirect" title="Empty function">empty function</a>. As a result, the empty set is the unique <a href="Initial_object" class="mw-redirect" title="Initial object">initial object</a> of the <a href="Category_theory" title="Category theory">category</a> of sets and functions.

</p><p>The empty set can be turned into a <a href="Topological_space" title="Topological space">topological space</a>, called the empty space, in just one way: by defining the empty set to be <a href="Open_set" title="Open set">open</a>. This empty topological space is the unique initial object in the <a href="Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a> with <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous maps</a>. In fact, it is a <a href="Strict_initial_object" title="Strict initial object">strict initial object</a>: only the empty set has a function to the empty set.

</p>

<div class="mw-heading mw-heading3"><h3 id="Set_theory">Set theory</h3></div>

<p>In the <a href="Von_Neumann_ordinal" class="mw-redirect" title="Von Neumann ordinal">von Neumann construction of the ordinals</a>, 0 is defined as the empty set, and the successor of an ordinal is defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c549f8a2e06a23ca6805a284426976ac0a8e98a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.777ex; height:2.843ex;" alt="{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}}" loading="lazy"></span>. Thus, we have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f61ba2d4026babf89d0f6395b697adbbfece155c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.069ex; height:2.176ex;" alt="{\displaystyle 0=\varnothing }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=0\cup \{0\}=\{\varnothing \}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=0\cup \{0\}=\{\varnothing \}}</annotation>

</semantics>

</math></span><img src="./e0eff83f88cd02e2a00736d2c3cf1e7bed405e23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.725ex; height:2.843ex;" alt="{\displaystyle 1=0\cup \{0\}=\{\varnothing \}}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}</annotation>

</semantics>

</math></span><img src="./e0ec487b19c7dc1aa2905e258e9d89617dfa9891.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.892ex; height:2.843ex;" alt="{\displaystyle 2=1\cup \{1\}=\{\varnothing ,\{\varnothing \}\}}" loading="lazy"></span>, and so on. The von Neumann construction, along with the <a href="Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a>, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} _{0}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./77ab7e98123f0def29a1cd3df96a0b7a58f4202c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.732ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} _{0}}" loading="lazy"></span>, such that the <a href="Peano_axioms" title="Peano axioms">Peano axioms</a> of arithmetic are satisfied.

</p>

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

<div class="mw-heading mw-heading3"><h3 id="Historical_issues">Historical issues</h3></div>

<p>In the context of sets of real numbers, Cantor used <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\equiv O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e62f6f69c2b408e8198ff17f990e20845d3d3df5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.617ex; height:2.176ex;" alt="{\displaystyle P\equiv O}" loading="lazy"></span> to denote "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> contains no single point". This <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./60d27f8374b57d760a9b18c0ddc1aeb9342ee14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.227ex; height:2.176ex;" alt="{\displaystyle \equiv O}" loading="lazy"></span> notation was utilized in definitions; for example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span> as an existent set on its own, or if Cantor merely used <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \equiv O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./60d27f8374b57d760a9b18c0ddc1aeb9342ee14b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.227ex; height:2.176ex;" alt="{\displaystyle \equiv O}" loading="lazy"></span> as an emptiness predicate. Zermelo accepted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span> itself as a set, but considered it an "improper set".<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Axiomatic_set_theory">Axiomatic set theory</h3></div>

<p>In <a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a>, the existence of the empty set is assured by the <a href="Axiom_of_empty_set" title="Axiom of empty set">axiom of empty set</a>, and its uniqueness follows from the <a href="Axiom_of_extensionality" title="Axiom of extensionality">axiom of extensionality</a>. However, the axiom of empty set can be shown redundant in at least two ways:

</p>

<ul><li>Standard <a href="First-order_logic" title="First-order logic">first-order logic</a> implies, merely from the <a href="Logical_axiom" class="mw-redirect" title="Logical axiom">logical axioms</a>, that <em>something</em> exists, and in the language of set theory, that thing must be a set. Now the existence of the empty set follows easily from the <a href="Axiom_of_separation" class="mw-redirect" title="Axiom of separation">axiom of separation</a>.</li>

<li>Even using <a href="Free_logic" title="Free logic">free logic</a> (which does not logically imply that something exists), there is already an axiom implying the existence of at least one set, namely the <a href="Axiom_of_infinity" title="Axiom of infinity">axiom of infinity</a>.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Philosophical_issues">Philosophical issues</h3></div>

<p>While the empty set is a standard and widely accepted mathematical concept, it remains an <a href="Ontological" class="mw-redirect" title="Ontological">ontological</a> curiosity, whose meaning and usefulness are debated by philosophers and logicians.

</p><p>The empty set is not the same thing as <em><a href="Nothing" title="Nothing">nothing</a></em>; rather, it is a set with nothing <em>inside</em> it and a set is always <em>something</em>. This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all <a href="Chess_opening" title="Chess opening">opening moves</a> in <a href="Chess" title="Chess">chess</a> that involve a <a href="King_(chess)" title="King (chess)">king</a>."<sup id="cite_ref-Darling_10-0" class="reference"><a href="#cite_note-Darling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p>The popular <a href="Syllogism" title="Syllogism">syllogism</a>

</p>

<dl><dd>Nothing is better than eternal happiness; a ham sandwich is better than nothing; therefore, a ham sandwich is better than eternal happiness</dd></dl>

<p>is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>" and the latter to "The set {ham sandwich} is better than the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span>". The first compares elements of sets, while the second compares the sets themselves.<sup id="cite_ref-Darling_10-1" class="reference"><a href="#cite_note-Darling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</p><p><a href="E._J._Lowe_(philosopher)" title="E. J. Lowe (philosopher)">Jonathan Lowe</a> argues that while the empty set

</p>

<dl><dd>was undoubtedly an important landmark in the history of mathematics, … we should not assume that its utility in calculation is dependent upon its actually denoting some object.</dd></dl>

<p>it is also the case that:

</p>

<dl><dd>"All that we are ever informed about the empty set is that it (1) is a set, (2) has no members, and (3) is unique amongst sets in having no members. However, there are very many things that 'have no members', in the set-theoretical sense—namely, all non-sets. It is perfectly clear why these things have no members, for they are not sets. What is unclear is how there can be, uniquely amongst sets, a <em>set</em> which has no members. We cannot conjure such an entity into existence by mere stipulation."<sup id="cite_ref-Lowe_11-0" class="reference"><a href="#cite_note-Lowe-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></dd></dl>

<p><a href="George_Boolos" title="George Boolos">George Boolos</a> argued that much of what has been heretofore obtained by set theory can just as easily be obtained by <a href="Plural_quantification" title="Plural quantification">plural quantification</a> over individuals, without <a href="https://en.wiktionary.org/wiki/reification" class="extiw external" title="wikt:reification">reifying</a> sets as singular entities having other entities as members.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="0" title="0">0</a> – Number<span style="display:none" class="category-spaceless-annotation">Pages displaying short descriptions with no spaces</span></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited set</a> – Property of sets used in constructive mathematics</li>

<li><a href="Nothing" title="Nothing">Nothing</a> – Complete absence of anything; the opposite of everything</li>

<li><a href="Power_set" title="Power set">Power set</a> – Mathematical set of all subsets of a set</li></ul>

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

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/EmptySet.html">"Empty Set"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-08-11</span></span>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/set.html">"Earliest Uses of Symbols of Set Theory and Logic"</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudin1976" class="citation book cs1">Rudin, Walter (1976). <a rel="nofollow" class="external text" href="https://archive.org/details/1979RudinW"><i>Principles of Mathematical Analysis</i></a> (3rd ed.). McGraw-Hill. p. 300. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>007054235X</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.unicode.org/charts/PDF/U2200.pdf">"Unicode Standard 5.2"</a> <span class="cs1-format">(PDF)</span>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">e.g. Nina Grønnum (2005, 2013) <i>Fonetik og Fonologi: Almen og dansk.</i> Akademisk forlag, Copenhagen.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFDavid_M._Bloom1979" class="citation book cs1">David M. Bloom (1979). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrageo0000bloo"><i>Linear Algebra and Geometry</i></a></span>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/linearalgebrageo0000bloo/page/45">45</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0521293243</bdi>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Bruckner, A.N., Bruckner, J.B., and Thomson, B.S. (2008). <i><a rel="nofollow" class="external text" href="http://classicalrealanalysis.info/com/documents/TBB-AllChapters-Portrait.pdf">Elementary Real Analysis</a></i>, 2nd edition, p. 9.</span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFMunkres2018" class="citation book cs1">Munkres, James Raymond (2018). <i>Topology</i> (Second, reissue ed.). New York, NY: Pearson. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0134689517</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">A. Kanamori, "<a rel="nofollow" class="external text" href="https://math.bu.edu/people/aki/8.pdf">The Empty Set, the Singleton, and the Ordered Pair</a>", p.275. Bulletin of Symbolic Logic vol. 9, no. 3, (2003). Accessed 21 August 2023.</span>

</li>

<li id="cite_note-Darling-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Darling_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Darling_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFD._J._Darling2004" class="citation book cs1">D. J. Darling (2004). <i>The Universal Book of Mathematics</i>. <a href="John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. p. 106. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-27047-4</bdi>.</cite></span>

</li>

<li id="cite_note-Lowe-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lowe_11-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFE._J._Lowe2005" class="citation book cs1">E. J. Lowe (2005). <i>Locke</i>. <a href="Routledge" title="Routledge">Routledge</a>. p. 87.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="George_Boolos" title="George Boolos">George Boolos</a> (1984), "To be is to be the value of a variable", <i><a href="The_Journal_of_Philosophy" title="The Journal of Philosophy">The Journal of Philosophy</a></i> 91: 430–49. Reprinted in 1998, <i>Logic, Logic and Logic</i> (<a href="Richard_Jeffrey" title="Richard Jeffrey">Richard Jeffrey</a>, and Burgess, J., eds.) <a href="Harvard_University_Press" title="Harvard University Press">Harvard University Press</a>, 54–72.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><a href="Paul_Halmos" title="Paul Halmos">Halmos, Paul</a>, <i><a href="Naive_Set_Theory_(book)" title="Naive Set Theory (book)">Naive Set Theory</a></i>. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90092-6</bdi> (Springer-Verlag edition). Reprinted by Martino Fine Books, 2011. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-61427-131-4</bdi> (paperback edition).</li>

<li><cite id="CITEREFJech2002" class="citation book cs1"><a href="Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2002). <i>Set Theory</i>. Springer Monographs in Mathematics (3rd millennium ed.). Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-44085-2</bdi>.</cite></li>

<li><cite id="CITEREFGraham1975" class="citation book cs1">Graham, Malcolm (1975). <i>Modern Elementary Mathematics</i> (2nd ed.). <a href="Harcourt_Brace_Jovanovich" class="mw-redirect" title="Harcourt Brace Jovanovich">Harcourt Brace Jovanovich</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0155610392</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="citation mathworld" id="Reference-Mathworld-Empty_Set"><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/EmptySet.html">"Empty Set"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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</style><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</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="Axiom" title="Axiom">Axiom</a>

<li><a href="Cardinality" title="Cardinality">Cardinality</a></li>

<li><a href="First-order_logic" title="First-order logic">First-order logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li>

<li><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a></li>

<li><a href="Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</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="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

<li><a href="Tarski's_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li>

<li><a href="Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li>

<li>Cantor's <a href="Cantor's_theorem" title="Cantor's theorem">theorem,</a> <a href="Cantor's_paradox" title="Cantor's paradox">paradox</a> and <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li>

<li><a href="Compactness_theorem" title="Compactness theorem">Compactness</a></li>

<li><a href="Halting_problem" title="Halting problem">Halting problem</a></li>

<li><a href="Lindstr%C3%B6m's_theorem" title="Lindström's theorem">Lindström's</a></li>

<li><a href="L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li>

<li><a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Logic" title="Logic">Logics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Traditional95" scope="row" class="navbox-group" style="width:1%"><a href="Term_logic" title="Term logic">Traditional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical logic</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Tautology_(logic)" title="Tautology (logic)">Tautology</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li>

<li><a href="Consistency" title="Consistency">Consistency</a>

<ul><li><a href="Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li>

<li><a href="Argument" title="Argument">Argument</a></li>

<li><a href="Soundness" title="Soundness">Soundness</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li>

<li><a href="Syllogism" title="Syllogism">Syllogism</a></li>

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</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="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li>

<li><a href="Boolean_function" title="Boolean function">Boolean functions</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connectives</a></li>

<li><a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">Propositional calculus</a></li>

<li><a href="Propositional_formula" title="Propositional formula">Propositional formula</a></li>

<li><a href="Truth_table" title="Truth table">Truth tables</a></li>

<li><a href="Many-valued_logic" title="Many-valued logic">Many-valued logic</a>

<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Predicate_logic" class="mw-redirect" title="Predicate logic">Predicate</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="First-order_logic" title="First-order logic">First-order</a>

<li><a href="Second-order_logic" title="Second-order logic">Second-order</a>

<ul><li><a href="Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li>

<li><a href="Higher-order_logic" title="Higher-order logic">Higher-order</a></li>

<li><a href="Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>

<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li>

<li><a href="Class_(set_theory)" title="Class (set theory)">Class</a></li>

<li>(<a href="Urelement" title="Urelement">Ur-</a>)<a href="Element_(mathematics)" title="Element (mathematics)">Element</a></li>

<li><a href="Ordinal_number" title="Ordinal number">Ordinal number</a></li>

<li><a href="Extensionality" title="Extensionality">Extensionality</a></li>

<li><a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li>

<li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a>

<ul><li><a href="Equivalence_relation" title="Equivalence relation">equivalence</a></li>

<li><a href="Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li>

<li>Set operations:

<ul><li><a href="Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li>

<li><a href="Union_(set_theory)" title="Union (set theory)">union</a></li>

<li><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></li>

<li><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></li>

<li><a href="Power_set" title="Power set">power set</a></li>

<li><a href="List_of_set_identities_and_relations" title="List of set identities and relations">identities</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of <a href="Set_(mathematics)" title="Set (mathematics)">sets</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Countable_set" title="Countable set">Countable</a></li>

<li><a href="Uncountable_set" title="Uncountable set">Uncountable</a></li>

<li><a href="Inhabited_set" title="Inhabited set">Inhabited</a></li>

<li><a href="Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a></li>

<li><a href="Infinite_set" title="Infinite set">Infinite</a></li>

<li><a href="Transitive_set" title="Transitive set">Transitive</a></li>

<li><a href="Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li>

<li><a href="Fuzzy_set" title="Fuzzy set">Fuzzy</a></li>

<li><a href="Universal_set" title="Universal set">Universal</a></li>

<li><a href="Universe_(mathematics)" title="Universe (mathematics)">Universe</a>

<ul><li><a href="Constructible_universe" title="Constructible universe">constructible</a></li>

<li><a href="Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li>

<li><a href="Von_Neumann_universe" title="Von Neumann universe">Von Neumann</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="Cardinality" title="Cardinality">cardinality</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="Map_(mathematics)" title="Map (mathematics)">Map</a>

<ul><li><a href="Domain_of_a_function" title="Domain of a function">domain</a></li>

<li><a href="Codomain" title="Codomain">codomain</a></li>

<li><a href="Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li>

<li><a href="Injective_function" title="Injective function">In</a>/<a href="Surjective_function" title="Surjective function">Sur</a>/<a href="Bijection" title="Bijection">Bi</a>-jection</li>

<li><a href="Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li>

<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>

<li><a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li>

<li><a href="Enumeration" title="Enumeration">Enumeration</a></li>

<li><a href="Large_cardinal" title="Large cardinal">Large cardinal</a>

<ul><li><a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li>

<li><a href="Aleph_number" title="Aleph number">Aleph number</a></li>

<li><a href="Operation_(mathematics)" title="Operation (mathematics)">Operation</a>

<ul><li><a href="Binary_operation" title="Binary operation">binary</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Set theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a>

<ul><li><a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li>

<li><a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li>

<li><a href="General_set_theory" title="General set theory">General</a></li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li>

<li><a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li>

<li><a href="Naive_set_theory" title="Naive set theory">Naive</a></li>

<li><a href="New_Foundations" title="New Foundations">New Foundations</a></li>

<li><a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li>

<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></li>

<li><a href="Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li>

<li><a href="Constructive_set_theory" title="Constructive set theory">Constructive</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Formal_system" title="Formal system">Formal systems</a> (<a href="List_of_formal_systems" title="List of formal systems"><span style="font-size: 85%;">list</span></a>),<br><a href="Formal_language" title="Formal language">language</a> and <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li>

<li><a href="Arity" title="Arity">Arity</a></li>

<li><a href="Automata_theory" title="Automata theory">Automata</a></li>

<li><a href="Axiom_schema" title="Axiom schema">Axiom schema</a></li>

<li><a href="Expression_(mathematics)" title="Expression (mathematics)">Expression</a>

<ul><li><a href="Ground_expression" title="Ground expression">ground</a></li></ul></li>

<li><a href="Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a>

<ul><li><a href="Extension_by_definitions" class="mw-redirect" title="Extension by definitions">by definition</a></li>

<li><a href="Conservative_extension" title="Conservative extension">conservative</a></li></ul></li>

<li><a href="Finitary_relation" title="Finitary relation">Relation</a></li>

<li><a href="Formation_rule" title="Formation rule">Formation rule</a></li>

<li><a href="Formal_grammar" title="Formal grammar">Grammar</a></li>

<li><a href="Well-formed_formula" title="Well-formed formula">Formula</a>

<ul><li><a href="Atomic_formula" title="Atomic formula">atomic</a></li>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li>

<li><a href="Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li>

<li><a href="Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li>

<li><a href="Formal_language" title="Formal language">Language</a></li>

<li><a href="Metalanguage" title="Metalanguage">Metalanguage</a></li>

<li><a href="Logical_connective" title="Logical connective">Logical connective</a>

<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a>

<ul><li><a href="Functional_predicate" title="Functional predicate">functional</a></li>

<li><a href="Predicate_variable" title="Predicate variable">variable</a></li>

<li><a href="Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a>

<li><a href="Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a>

<ul><li><a href="Atomic_sentence" title="Atomic sentence">atomic</a></li>

<li><a href="Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li>

<li><a href="Signature_(logic)" title="Signature (logic)">Signature</a></li>

<li><a href="String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Symbol_(formal)" title="Symbol (formal)">Symbol</a>

<ul><li><a href="Uninterpreted_function" title="Uninterpreted function">function</a></li>

<li><a href="Logical_constant" title="Logical constant">logical/constant</a></li>

<li><a href="Non-logical_symbol" title="Non-logical symbol">non-logical</a></li>

<li><a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li>

<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><span class="nowrap">Example <a href="Axiomatic_system" title="Axiomatic system">axiomatic<br>systems</a> <span style="font-size: 85%;">(<a href="List_of_first-order_theories" title="List of first-order theories">list</a>)</span></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>of <a href="True_arithmetic" title="True arithmetic">arithmetic</a>:

<ul><li><a href="Peano_axioms" title="Peano axioms">Peano</a></li>

<li><a href="Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li>

<li><a href="Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li>

<li><a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li>

<li><a href="Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li>

<li><a href="Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li>

<li>of the <a href="Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a>

<ul><li><a href="Tarski's_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li>

<li>of <a href="Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a>

<ul><li><a href="Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li>

<li><a href="Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li>

<li>of <a href="Foundations_of_geometry" title="Foundations of geometry">geometry</a>:

<ul><li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean</a>:

<ul><li><a href="Euclid's_Elements" title="Euclid's Elements"><i>Elements</i></a></li>

<li><a href="Hilbert's_axioms" title="Hilbert's axioms">Hilbert's</a></li>

<li><a href="Tarski's_axioms" title="Tarski's axioms">Tarski's</a></li></ul></li>

<li><a href="Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul>

<ul><li><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Proof_theory" title="Proof theory">Proof theory</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="Formal_proof" title="Formal proof">Formal proof</a></li>

<li><a href="Natural_deduction" title="Natural deduction">Natural deduction</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Logical consequence</a></li>

<li><a href="Rule_of_inference" title="Rule of inference">Rule of inference</a></li>

<li><a href="Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li>

<li><a href="Theorem" title="Theorem">Theorem</a></li>

<li><a href="Formal_system" title="Formal system">Systems</a>

<ul><li><a href="Axiomatic_system" title="Axiomatic system">axiomatic</a></li>

<li><a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li>

<li><a href="Hilbert_system" title="Hilbert system">Hilbert</a>

<li><a href="Complete_theory" title="Complete theory">Complete theory</a></li>

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li>

<li><a href="Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li>

<li><a href="Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li>

<li><a href="Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li>

<li><a href="Self-verifying_theories" title="Self-verifying theories">Self-verifying theories</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Model_theory" title="Model theory">Model theory</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="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a>

<ul><li><a href="Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li>

<li><a href="Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li>

<li><a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a>

<ul><li><a href="Elementary_equivalence" title="Elementary equivalence">equivalence</a></li>

<li><a href="Finite_model_theory" title="Finite model theory">finite</a></li>

<li><a href="Saturated_model" title="Saturated model">saturated</a></li>

<li><a href="Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li>

<li><a href="Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li>

<li><a href="Non-standard_model" title="Non-standard model">Non-standard model</a>

<ul><li><a href="Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li>

<li><a href="Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a>

<ul><li><a href="Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li>

<li><a href="Categorical_theory" title="Categorical theory">Categorical theory</a></li>

<li><a href="Model_complete_theory" title="Model complete theory">Model complete theory</a></li>

<li><a href="Satisfiability" title="Satisfiability">Satisfiability</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li>

<li><a href="Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a>

<ul><li><a href="Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li>

<li><a href="Tarski's_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li>

<li><a href="Kripke's_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li>

<li><a href="T-schema" title="T-schema">T-schema</a></li>

<li><a href="Transfer_principle" title="Transfer principle">Transfer principle</a></li>

<li><a href="Truth_predicate" title="Truth predicate">Truth predicate</a></li>

<li><a href="Truth_value" title="Truth value">Truth value</a></li>

<li><a href="Ultraproduct" title="Ultraproduct">Ultraproduct</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</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="Church_encoding" title="Church encoding">Church encoding</a></li>

<li><a href="Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li>

<li><a href="Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li>

<li><a href="Computable_function" title="Computable function">Computable function</a></li>

<li><a href="Computable_set" title="Computable set">Computable set</a></li>

<li><a href="Decision_problem" title="Decision problem">Decision problem</a>

<ul><li><a href="Decidability_(logic)" title="Decidability (logic)">decidable</a></li>

<li><a href="Undecidable_problem" title="Undecidable problem">undecidable</a></li>

<li><a href="P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li>

<li><a href="Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li>

<li><a href="Recursion" title="Recursion">Recursion</a></li>

<li><a href="Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li>

<li><a href="Turing_machine" title="Turing machine">Turing machine</a></li>

<li><a href="Type_theory" title="Type theory">Type theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

<li><a href="Algebraic_logic" title="Algebraic logic">Algebraic logic</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Category_theory" title="Category theory">Category theory</a></li>

<li><a href="Concrete_category" title="Concrete category">Concrete</a>/<a href="Category_(mathematics)" title="Category (mathematics)">Abstract category</a></li>

<li><a href="Category_of_sets" title="Category of sets">Category of sets</a></li>

<li><a href="History_of_logic" title="History of logic">History of logic</a></li>

<li><a href="History_of_mathematical_logic" class="mw-redirect" title="History of mathematical logic">History of mathematical logic</a>

<ul><li><a href="Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li>

<li><a href="Logicism" title="Logicism">Logicism</a></li>

<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li>

<li><a href="Supertask" title="Supertask">Supertask</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Set_theory409" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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><span typeof="mw:File"><a href="Venn_diagram" title="Venn diagram"></a></span></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="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</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="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 <b>·</b> 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><i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>

<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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