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<title>Hilbert's second problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Hilbert's second problem</span></span>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>Hilbert's second problem</b> was posed by <a href="David_Hilbert" title="David Hilbert">David Hilbert</a> in 1900 as one of his <a href="Hilbert's_problems" title="Hilbert's problems">23 problems</a>. It asks for a proof that arithmetic is <a href="Consistency_proof" class="mw-redirect" title="Consistency proof">consistent</a> – free of any internal contradictions. Hilbert stated that the axioms he considered for arithmetic were the ones given in <a href="#CITEREFHilbert1900">Hilbert (1900)</a>, which include a second order completeness axiom.

</p><p>In the 1930s, <a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a> and <a href="Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a> proved results that cast new light on the problem. Some feel that Gödel's theorems give a negative solution to the problem, while others consider Gentzen's proof as a partial positive solution.

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<div class="mw-heading mw-heading2"><h2 id="Hilbert's_problem_and_its_interpretation">Hilbert's problem and its interpretation</h2></div>

<p>In one English translation, Hilbert asks:

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"When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. ... But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. ... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms."<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> </p></blockquote>

<p>Hilbert's statement is sometimes misunderstood, because by the "arithmetical axioms" he did not mean a system equivalent to Peano arithmetic, but a stronger system with a second-order completeness axiom. The system Hilbert asked for a completeness proof of is more like <a href="Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a> than first-order Peano arithmetic.

</p><p>As a nowadays common interpretation, a positive solution to Hilbert's second question would in particular provide a proof that <a href="Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> is consistent.

</p><p>There are many known proofs that Peano arithmetic is consistent that can be carried out in strong systems such as <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>. These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory (which are much stronger) to prove its consistency. Thus a satisfactory answer to Hilbert's problem must be carried out using principles that would be acceptable to someone who does not already believe PA is consistent. Such principles are often called <a href="Finitism" title="Finitism">finitistic</a> because they are completely constructive and do not presuppose a completed infinity of natural numbers. Gödel's second incompleteness theorem (see <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a>) places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.

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<div class="mw-heading mw-heading2"><h2 id="Gödel's_incompleteness_theorem">Gödel's incompleteness theorem</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a></div>

<p>Gödel's <a href="Second_incompleteness_theorem" class="mw-redirect" title="Second incompleteness theorem">second incompleteness theorem</a> shows that it is not possible for any proof that Peano Arithmetic is consistent to be carried out within Peano arithmetic itself. This theorem shows that if the only acceptable proof procedures are those that can be formalized within arithmetic then Hilbert's call for a consistency proof cannot be answered. However, as <a href="#CITEREFNagelNewman1958">Nagel & Newman (1958)</a> explain, there is still room for a proof that cannot be formalized in arithmetic:<sup id="cite_ref-FOOTNOTENagelNewman195896&ndash;99_2-0" class="reference"><a href="#cite_note-FOOTNOTENagelNewman195896&ndash;99-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<dl><dd>"This imposing result of Godel's analysis should not be misunderstood: it does not exclude a meta-mathematical proof of the consistency of arithmetic. What it excludes is a proof of consistency that can be mirrored by the formal deductions of arithmetic. Meta-mathematical proofs of the consistency of arithmetic have, in fact, been constructed, notably by <a href="Gerhard_Gentzen" title="Gerhard Gentzen">Gerhard Gentzen</a>, a member of the Hilbert school, in 1936, and by others since then. ... But these meta-mathematical proofs cannot be represented within the arithmetical calculus; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program. ... The possibility of constructing a finitistic absolute proof of consistency for arithmetic is not excluded by Gödel’s results. Gödel showed that no such proof is possible that can be represented within arithmetic. His argument does not eliminate the possibility of strictly finitistic proofs that cannot be represented within arithmetic. But no one today appears to have a clear idea of what a finitistic proof would be like that is not capable of formulation within arithmetic."<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></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Gentzen's_consistency_proof">Gentzen's consistency proof</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Gentzen's_consistency_proof" title="Gentzen's consistency proof">Gentzen's consistency proof</a></div>

<p>In 1936, Gentzen published a proof that Peano Arithmetic is consistent. Gentzen's result shows that a consistency proof can be obtained in a system that is much weaker than set theory.

</p><p>Gentzen's proof proceeds by assigning to each proof in Peano arithmetic an <a href="Ordinal_number" title="Ordinal number">ordinal number</a>, based on the structure of the proof, with each of these ordinals less than <a href="Epsilon_numbers_(mathematics)" class="mw-redirect" title="Epsilon numbers (mathematics)">ε<sub>0</sub></a>.<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> He then proves by <a href="Transfinite_induction" title="Transfinite induction">transfinite induction</a> on these ordinals that no proof can conclude in a contradiction. The method used in this proof can also be used to prove a <a href="Cut_elimination" class="mw-redirect" title="Cut elimination">cut elimination</a> result for <a href="Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a> in a stronger logic than first-order logic, but the consistency proof itself can be carried out in ordinary first-order logic using the axioms of <a href="Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive arithmetic</a> and a transfinite induction principle. <a href="#CITEREFTait2005">Tait (2005)</a> gives a game-theoretic interpretation of Gentzen's method.

</p><p>Gentzen's consistency proof initiated the program of <a href="Ordinal_analysis" title="Ordinal analysis">ordinal analysis</a> in proof theory. In this program, formal theories of arithmetic or set theory are assigned <a href="Ordinal_numbers" class="mw-redirect" title="Ordinal numbers">ordinal numbers</a> that measure the <a href="Consistency_strength" class="mw-redirect" title="Consistency strength">consistency strength</a> of the theories. A theory will be unable to prove the consistency of another theory with a higher proof theoretic ordinal.

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<div class="mw-heading mw-heading2"><h2 id="Modern_viewpoints_on_the_status_of_the_problem">Modern viewpoints on the status of the problem</h2></div>

<p>While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether (or in what way) these theorems answer Hilbert's second problem. <a href="#CITEREFSimpson1988">Simpson (1988)</a> argues that Gödel's incompleteness theorem shows that it is not possible to produce finitistic consistency proofs of strong theories.<sup id="cite_ref-FOOTNOTESimpson1988sec._3_5-0" class="reference"><a href="#cite_note-FOOTNOTESimpson1988sec._3-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFKreisel1976">Kreisel (1976)</a> states that although Gödel's results imply that no finitistic syntactic consistency proof can be obtained, semantic (in particular, <a href="Second-order_logic" title="Second-order logic">second-order</a>) arguments can be used to give convincing consistency proofs. <a href="#CITEREFDetlefsen1990">Detlefsen (1990)</a> argues that Gödel's theorem does not prevent a consistency proof because its hypotheses might not apply to all the systems in which a consistency proof could be carried out.<sup id="cite_ref-FOOTNOTEDetlefsen199065_6-0" class="reference"><a href="#cite_note-FOOTNOTEDetlefsen199065-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <a href="#CITEREFDawson2006">Dawson (2006)</a> calls the belief that Gödel's theorem eliminates the possibility of a persuasive consistency proof "erroneous", citing the consistency proof given by Gentzen and <a href="Dialectica_interpretation" title="Dialectica interpretation">a later one given by Gödel in 1958</a>.<sup id="cite_ref-FOOTNOTEDawson2006sec._2_7-0" class="reference"><a href="#cite_note-FOOTNOTEDawson2006sec._2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Takeuti_conjecture" class="mw-redirect" title="Takeuti conjecture">Takeuti conjecture</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</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"><a href="#CITEREFAmerican_Mathematical_Society1902">American Mathematical Society (1902)</a>, translated by M. Newson. For the original version, see <a href="#CITEREFHilbert1901">Hilbert (1901)</a>.</span>

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<li id="cite_note-FOOTNOTENagelNewman195896&ndash;99-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENagelNewman195896&ndash;99_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNagelNewman1958">Nagel & Newman (1958)</a>, p. 96–99.</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">A similar quotation with minor variations in wording appears in <a href="#CITEREFNagelNewman2001">Nagel & Newman (2001)</a>, p. 107–108, as revised by <a href="Douglas_R._Hofstadter" class="mw-redirect" title="Douglas R. Hofstadter">Douglas R. Hofstadter</a>.</span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Actually, the proof assigns a "notation" for an ordinal number to each proof. The notation is a finite string of symbols that intuitively stands for an ordinal number. By representing the ordinal in a finite way, Gentzen's proof does not presuppose strong axioms regarding ordinal numbers.</span>

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<li id="cite_note-FOOTNOTESimpson1988sec._3-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESimpson1988sec._3_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSimpson1988">Simpson (1988)</a>, sec. 3.</span>

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<li id="cite_note-FOOTNOTEDetlefsen199065-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDetlefsen199065_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDetlefsen1990">Detlefsen (1990)</a>, p. 65.</span>

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<li id="cite_note-FOOTNOTEDawson2006sec._2-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDawson2006sec._2_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDawson2006">Dawson (2006)</a>, sec. 2.</span>

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

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</style><cite id="CITEREFDawson2006" class="citation conference cs1">Dawson, John W. (2006). "Shaken foundations or groundbreaking realignment? A Centennial Assessment of Kurt Gödel's Impact on Logic, Mathematics, and Computer Science.". <i>2006 21st Annual IEEE Symposium on Logic in Computer Science</i>. IEEE. pp. <span class="nowrap">339–</span>341. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FLICS.2006.47">10.1109/LICS.2006.47</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7695-2631-4</bdi>.</cite></li>

<li><cite id="CITEREFDetlefsen1990" class="citation journal cs1"><a href="Michael_Detlefsen" title="Michael Detlefsen">Detlefsen, Michael</a> (1990). "On an alleged refutation of Hilbert's Program using Gödel's First Incompleteness Theorem". <i>Journal of Philosophical Logic</i>. <b>19</b> (4). Springer: <span class="nowrap">343–</span>377. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00263316">10.1007/BF00263316</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:44736805">44736805</a>.</cite></li>

<li><cite id="CITEREFFranzen2005" class="citation book cs1"><a href="Torkel_Franz%C3%A9n" title="Torkel Franzén">Franzen, Torkel</a> (2005). <i>Godel's theorem: An Incomplete Guide to its Use and Abuse</i>. <a href="Wellesley%2C_Massachusetts" title="Wellesley, Massachusetts">Wellesley MA</a>: <a href="A_K_Peters" title="A K Peters">A.K. Peters</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-56881-238-8</bdi>.</cite></li>

<li><cite id="CITEREFGentzen1936" class="citation journal cs1"><a href="Gerhard_Gentzen" title="Gerhard Gentzen">Gentzen, Gerhard</a> (1936). "Die Widerspruchsfreiheit der reinen Zahlentheorie". <i>Mathematische Annalen</i>. <b>112</b>. Springer: 493–565. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01565428">10.1007/BF01565428</a>.</cite></li>

<li><cite id="CITEREFGödel1931" class="citation journal cs1"><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Gödel, Kurt</a> (1931). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060705205103/http://home.ddc.net/ygg/etext/godel/">"Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I"</a>. <i>Monatshefte für Mathematik und Physik</i>. <b>38</b>: <span class="nowrap">173–</span>98. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01700692">10.1007/BF01700692</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:197663120">197663120</a>. Archived from <a rel="nofollow" class="external text" href="http://home.ddc.net/ygg/etext/godel/">the original</a> on 2006-07-05.</cite></li>

<li><cite id="CITEREFHilbert1900" class="citation journal cs1"><a href="David_Hilbert" title="David Hilbert">Hilbert, David</a> (1900). <a rel="nofollow" class="external text" href="http://resolver.sub.uni-goettingen.de/purl?PPN37721857X">"Über den Zahlbegriff"</a>. <i>Jahresbericht der Deutschen Mathematiker-Vereinigung</i>. <b>8</b>: <span class="nowrap">180–</span>184.</cite></li>

<li><cite id="CITEREFHilbert1901" class="citation journal cs1">——— (1901) [1900]. "Mathematische Probleme". <i>Archiv der Mathematik und Physik</i>. <b>3</b> (1): <span class="nowrap">44–</span>63, <span class="nowrap">213–</span>237.</cite></li>

<li><cite id="CITEREFKreisel1976" class="citation conference cs1"><a href="George_Kreisel" class="mw-redirect" title="George Kreisel">Kreisel, George</a> (1976). "What have we learnt from Hilbert's second problem?". <i>Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill.,)</i>. Providence, R. I.: Amer. Math. Soc. pp. <span class="nowrap">93–</span>130. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8218-1428-1</bdi>.</cite></li>

<li><cite id="CITEREFAmerican_Mathematical_Society1902" class="citation journal cs1"><a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1902-08-10/S0002-9904-1902-00923-3/S0002-9904-1902-00923-3.pdf">"Mathematical Problems"</a> <span class="cs1-format">(PDF)</span>. <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. <b>8</b>. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>: <span class="nowrap">437–</span>479. 1902.</cite></li>

<li><cite id="CITEREFNagelNewman1958" class="citation book cs1"><a href="Ernest_Nagel" title="Ernest Nagel">Nagel, Ernest</a>; <a href="James_R._Newman" title="James R. Newman">Newman, James R.</a> (1958). <i>Godel's Proof</i>. New York University Press.</cite></li>

<li><cite id="CITEREFNagelNewman2001" class="citation book cs1">———; ——— (2001). <a href="Douglas_R._Hofstadter" class="mw-redirect" title="Douglas R. Hofstadter">Hofstadter, Douglas R.</a> (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oQsUCgAAQBAJ&pg=PA107"><i>Godel's Proof</i></a>. New York University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780814758014</bdi>.</cite></li>

<li><cite id="CITEREFSimpson1988" class="citation journal cs1"><a href="Stephen_G._Simpson" class="mw-redirect" title="Stephen G. Simpson">Simpson, Stephen G.</a> (1988). "Partial realizations of Hilbert's Program". <i>Journal of Symbolic Logic</i>. <b>53</b> (2): <span class="nowrap">349–</span>363. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.5808">10.1.1.79.5808</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2274508">10.2307/2274508</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0022-4812">0022-4812</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/2274508">2274508</a>.</cite></li>

<li><cite id="CITEREFTait2005" class="citation journal cs1"><a href="William_W._Tait" title="William W. Tait">Tait, William W.</a> (2005). "Gödel's reformulation of Gentzen's first consistency proof of arithmetic: the no-counterexample interpretation". <i>Bulletin of Symbolic Logic</i>. <b>11</b> (2): <span class="nowrap">225–</span>238. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1556751">1556751</a>.</cite></li>

<li><cite id="CITEREFvan_Heijenoort1967" class="citation book cs1"><a href="Jean_van_Heijenoort" title="Jean van Heijenoort">van Heijenoort, Jean</a> (1967). <i>From Frege to Gödel: A Source Book on Mathematical Logic</i>. Harvard University Press. pp. <span class="nowrap">596–</span>616.</cite>.</li></ul>

</div>

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

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120205025851/http://www.mathematik.uni-bielefeld.de/~kersten/hilbert/rede.html">Original text of Hilbert's talk, in German</a></li>

<li><a rel="nofollow" class="external text" href="http://aleph0.clarku.edu/~djoyce/hilbert/toc.html">English translation of Hilbert's 1900 address</a></li></ul>

<p><br>

</p>

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</style><div id="Hilbert&#039;s_problems22" style="font-size:114%;margin:0 4em"><a href="Hilbert's_problems" title="Hilbert's problems">Hilbert's problems</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic344" 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="2"><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="Empty_set" title="Empty set">Empty</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>

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