ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<title>Iterated forcing</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Iterated forcing</span></span>

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<p>In mathematics, <b>iterated forcing</b> is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (<a href="#CITEREFSolovayTennenbaum1971">1971</a>) in their construction of a model of set theory with no <a href="Suslin_tree" title="Suslin tree">Suslin tree</a>. They also showed that iterated forcing can construct models where <a href="Martin's_axiom" title="Martin's axiom">Martin's axiom</a> holds and the continuum is any given regular cardinal.

</p><p>In iterated forcing, one has a transfinite sequence <i>P</i><sub>α</sub> of forcing notions indexed by some ordinals α, which give a family of Boolean-valued models <i>V</i><sup><i>P</i><sub>α</sub></sup>. If α+1 is a successor ordinal then <i>P</i><sub>α+1</sub> is often constructed from <i>P</i><sub>α</sub> using a forcing notion in <i>V</i><sup><i>P</i><sub>α</sub></sup>, while if α is a limit ordinal then <i>P</i><sub>α</sub> is often constructed as some sort of limit (such as the direct limit) of the <i>P</i><sub>β</sub> for β<α.

</p><p>A key consideration is that, typically, it is necessary that <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 \omega _{1}}">

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</math></span><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"></span> is not collapsed. This is often accomplished by the use of a preservation theorem such as:

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<ul><li>Finite support iteration of c.c.c. forcings (see <a href="Countable_chain_condition" title="Countable chain condition">countable chain condition</a>) are c.c.c. and thus preserve <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 \omega _{1}}">

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

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</math></span><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"></span>.</li>

<li>Countable support iterations of proper forcings are proper (see <a href="Proper_forcing_axiom#The_Fundamental_Theorem_of_Proper_Forcing" title="Proper forcing axiom">Fundamental Theorem of Proper Forcing</a>) and thus preserve <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 \omega _{1}}">

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</math></span><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"></span>.</li>

<li>Revised countable support iterations of semi-proper forcings are semi-proper and thus preserve <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 \omega _{1}}">

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<p>Some non-semi-proper forcings, such as <a href="Namba_forcing" class="mw-redirect" title="Namba forcing">Namba forcing</a>, can be iterated with appropriate cardinal collapses while preserving <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 \omega _{1}}">

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</math></span><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"></span> using methods developed by <a href="Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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><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>

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

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Shelah, S., Proper and Improper Forcing, Springer 1992</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579–606</span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865–1883</span>

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

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</style><cite id="CITEREFJech2003" class="citation cs2"><a href="Thomas_Jech" title="Thomas Jech">Jech, Thomas</a> (2003), <i>Set Theory: Millennium Edition</i>, Springer Monographs in Mathematics, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-44085-7</bdi></cite></li>

<li><cite id="CITEREFKunen1980" class="citation cs2"><a href="Kenneth_Kunen" title="Kenneth Kunen">Kunen, Kenneth</a> (1980), <i><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></i>, Elsevier, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-444-86839-8</bdi></cite></li>

<li><cite id="CITEREFShelah1998" class="citation cs2">Shelah, Saharon (1998) [1982], <i>Proper and improper forcing</i>, Perspectives in Mathematical Logic (2 ed.), Berlin: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-51700-6</bdi>, <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=1623206">1623206</a></cite></li>

<li><cite id="CITEREFSolovayTennenbaum,_S.1971" class="citation journal cs1">Solovay, R. M.; Tennenbaum, S. (1971). "Iterated Cohen extensions and Souslin's problem". <i>Ann. of Math</i>. 2. <b>94</b> (2). Annals of Mathematics: <span class="nowrap">201–</span>245. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970860">10.2307/1970860</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970860">1970860</a>.</cite></li></ul>

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

<ul><li><cite id="CITEREFEisworthMoore2009" class="citation cs2">Eisworth, Todd; Moore, Justin Tatch (2009), Milovich, David (ed.), <a rel="nofollow" class="external text" href="http://www.math.cmu.edu/~eschimme/Appalachian/EisworthMooreNotes.pdf"><i>ITERATED FORCING AND THE CONTINUUM HYPOTHESIS</i></a> <span class="cs1-format">(PDF)</span>, Appalachian Set Theory Workshop lecture notes</cite></li></ul></div><div><div class="zim-footer">

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