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<title>Undecidable problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Undecidable problem</span></span>

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<p>In <a href="Computability_theory" title="Computability theory">computability theory</a> and <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>, an <b>undecidable problem</b> is a <a href="Decision_problem" title="Decision problem">decision problem</a> for which it is proved to be impossible to construct an <a href="Algorithm" title="Algorithm">algorithm</a> that always leads to a correct yes-or-no answer. The <a href="Halting_problem" title="Halting problem">halting problem</a> is an example: it can be proven that there is no algorithm that correctly determines whether an arbitrary program eventually halts when run.<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>

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

<p>A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer.<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> Those inputs can be numbers (for example, the decision problem "is the input a <a href="Prime_number" title="Prime number">prime number</a>?") or values of some other kind, such as <a href="String_(computer_science)" title="String (computer science)">strings</a> of a <a href="Formal_language" title="Formal language">formal language</a>.

</p><p>The formal representation of a decision problem is a subset of the <a href="Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>. For decision problems on natural numbers, the set consists of those numbers that the decision problem answers "yes" to. For example, the decision problem "is the input even?" is formalized as the set of even numbers. A decision problem whose input consists of strings or more complex values is formalized as the set of numbers that, via a specific <a href="G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a>, correspond to inputs that satisfy the decision problem's criteria.

</p><p>A decision problem <i>A</i> is called decidable or effectively solvable if the formalized set of <i>A</i> is a <a href="Recursive_set" class="mw-redirect" title="Recursive set">recursive set</a>. Otherwise, <i>A</i> is called undecidable. A problem is called partially decidable, semi-decidable, solvable, or provable if <i>A</i> is a <a href="Recursively_enumerable_set" class="mw-redirect" title="Recursively enumerable set">recursively enumerable set</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Example:_the_halting_problem_in_computability_theory">Example: the halting problem in computability theory</h2></div>

<p>In <a href="Computability_theory" title="Computability theory">computability theory</a>, the <a href="Halting_problem" title="Halting problem">halting problem</a> is a <a href="Decision_problem" title="Decision problem">decision problem</a> which can be stated as follows:

</p>

<dl><dd>Given the description of an arbitrary <a href="Computer_program" title="Computer program">program</a> and a finite input, decide whether the program finishes running or will run forever.</dd></dl>

<p><a href="Alan_Turing" title="Alan Turing">Alan Turing</a> proved in 1936 that a general <a href="Algorithm" title="Algorithm">algorithm</a> running on a <a href="Turing_machine" title="Turing machine">Turing machine</a> that solves the halting problem for <i>all</i> possible program-input pairs necessarily cannot exist. Hence, the halting problem is <i>undecidable</i> for Turing machines.

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

<p>The concepts raised by <a href="G%C3%B6del's_incompleteness_theorem" class="mw-redirect" title="Gödel's incompleteness theorem">Gödel's incompleteness theorems</a> are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that an effective <a href="Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of the natural numbers that is both complete and <a href="Soundness" title="Soundness">sound</a> is impossible. The "sound" part is the weakening: it means that we require the axiomatic system in question to prove only <i>true</i> statements about natural numbers. Since soundness implies <a href="Consistency_proof" class="mw-redirect" title="Consistency proof">consistency</a>, this weaker form can be seen as a <a href="Corollary" title="Corollary">corollary</a> of the strong form. It is important to observe that the statement of the standard form of Gödel's First Incompleteness Theorem is completely unconcerned with the truth value of a statement, but only concerns the issue of whether it is possible to find it through a <a href="Mathematical_proof" title="Mathematical proof">mathematical proof</a>.

</p><p>The weaker form of the theorem can be proved from the undecidability of the halting problem as follows.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Assume that we have a sound (and hence consistent) and complete effective <a href="Axiomatization" class="mw-redirect" title="Axiomatization">axiomatization</a> of all true <a href="First-order_logic" title="First-order logic">first-order logic</a> statements about <a href="Natural_number" title="Natural number">natural numbers</a>. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm <i>N</i>(<i>n</i>) that, given a natural number <i>n</i>, computes a true first-order logic statement about natural numbers, and that for all true statements, there is at least one <i>n</i> such that <i>N</i>(<i>n</i>) yields that statement. Now suppose we want to decide if the algorithm with representation <i>a</i> halts on input <i>i</i>. We know that this statement can be expressed with a first-order logic statement, say <i>H</i>(<i>a</i>, <i>i</i>). Since the axiomatization is complete it follows that either there is an <i>n</i> such that <i>N</i>(<i>n</i>) = <i>H</i>(<i>a</i>, <i>i</i>) or there is an <i>n<span class="nowrap" style="padding-left:0em;">′</span></i> such that <i>N</i>(<i>n<span class="nowrap" style="padding-left:0em;">′</span></i>) = ¬ <i>H</i>(<i>a</i>, <i>i</i>). So if we <a href="Iterate" class="mw-redirect" title="Iterate">iterate</a> over all <i>n</i> until we either find <i>H</i>(<i>a</i>, <i>i</i>) or its negation, we will always halt, and furthermore, the answer it gives us will be true (by soundness). This means that this gives us an algorithm to decide the halting problem. Since we know that there cannot be such an algorithm, it follows that the assumption that there is a sound and complete effective axiomatization of all true first-order logic statements about natural numbers must be false.

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_undecidable_problems">Examples of undecidable problems</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="List_of_undecidable_problems" title="List of undecidable problems">List of undecidable problems</a></div>

<p>Undecidable problems can be related to different topics, such as <a href="Logic" title="Logic">logic</a>, <a href="Abstract_machine" title="Abstract machine">abstract machines</a> or <a href="Topology" title="Topology">topology</a>. Since there are <a href="Uncountable_set" title="Uncountable set">uncountably</a> many undecidable problems,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> any list, even one of <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">infinite length</a>, is necessarily incomplete.

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<div class="mw-heading mw-heading2"><h2 id="Examples_of_undecidable_statements">Examples of undecidable statements</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">List of statements independent of ZFC</a> and <a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence (mathematical logic)</a></div>

<p>There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified <a href="Deductive_system" class="mw-redirect" title="Deductive system">deductive system</a>. The second sense is used in relation to <a href="Computability_theory" title="Computability theory">computability theory</a> and applies not to statements but to <a href="Decision_problem" title="Decision problem">decision problems</a>, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no <a href="Computable_function" title="Computable function">computable function</a> that correctly answers every question in the problem set. The connection between these two is that if a decision problem is undecidable (in the recursion theoretical sense) then there is no consistent, effective <a href="Formal_system" title="Formal system">formal system</a> which proves for every question <i>A</i> in the problem either "the answer to <i>A</i> is yes" or "the answer to <i>A</i> is no".

</p><p>Because of the two meanings of the word undecidable, the term <a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">independent</a> is sometimes used instead of undecidable for the "neither provable nor refutable" sense. The usage of "independent" is also ambiguous, however. It can mean just "not provable", leaving open whether an independent statement might be refuted.

</p><p>Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the <a href="Truth_value" title="Truth value">truth value</a> of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point among various <a href="Philosophy_of_mathematics" title="Philosophy of mathematics">philosophical schools</a>.

</p><p>One of the first problems suspected to be undecidable, in the second sense of the term, was the <a href="Word_problem_for_groups" title="Word problem for groups">word problem for groups</a>, first posed by <a href="Max_Dehn" title="Max Dehn">Max Dehn</a> in 1911, which asks if there is a finitely presented <a href="Group_(mathematics)" title="Group (mathematics)">group</a> for which no algorithm exists to determine whether two words are equivalent. This was shown to be the case in 1955.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>The combined work of Gödel and <a href="Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> has given two concrete examples of undecidable statements (in the first sense of the term): The <a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> can neither be proved nor refuted in <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> (the standard axiomatization of <a href="Set_theory" title="Set theory">set theory</a>), and the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> can neither be proved nor refuted in <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF</a> (which is all the ZFC axioms <i>except</i> the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proven from ZFC.

</p><p>In 1970, Russian mathematician <a href="Yuri_Matiyasevich" title="Yuri Matiyasevich">Yuri Matiyasevich</a> showed that <a href="Hilbert's_tenth_problem" title="Hilbert's tenth problem">Hilbert's Tenth Problem</a>, posed in 1900 as a challenge to the next century of mathematicians, cannot be solved. Hilbert's challenge sought an algorithm which finds all solutions of a <a href="Diophantine_equation" title="Diophantine equation">Diophantine equation</a>. A Diophantine equation is a more general case of <a href="Fermat's_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>; we seek the <a href="Integer_number" class="mw-redirect" title="Integer number">integer roots</a> of a <a href="Polynomial" title="Polynomial">polynomial</a> in any number of variables with integer coefficients. Since we have only one equation but <i>n</i> variables, infinitely many solutions exist (and are easy to find) in the <a href="Complex_plane" title="Complex plane">complex plane</a>; however, the problem becomes impossible if solutions are constrained to integer values only. Matiyasevich showed this problem to be unsolvable by mapping a Diophantine equation to a <a href="Recursively_enumerable_set" class="mw-redirect" title="Recursively enumerable set">recursively enumerable set</a> and invoking Gödel's Incompleteness Theorem.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>In 1936, <a href="Alan_Turing" title="Alan Turing">Alan Turing</a> proved that the <a href="Halting_problem" title="Halting problem">halting problem</a>—the question of whether or not a <a href="Turing_machine" title="Turing machine">Turing machine</a> halts on a given program—is undecidable, in the second sense of the term. This result was later generalized by <a href="Rice's_theorem" title="Rice's theorem">Rice's theorem</a>.

</p><p>In 1973, <a href="Saharon_Shelah" title="Saharon Shelah">Saharon Shelah</a> showed the <a href="Whitehead_problem" title="Whitehead problem">Whitehead problem</a> in <a href="Group_theory" title="Group theory">group theory</a> is undecidable, in the first sense of the term, in standard set theory.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>In 1977, Paris and Harrington proved that the <a href="Paris-Harrington_theorem" class="mw-redirect" title="Paris-Harrington theorem">Paris-Harrington principle</a>, a version of the <a href="Ramsey_theorem" class="mw-redirect" title="Ramsey theorem">Ramsey theorem</a>, is undecidable in the axiomatization of arithmetic given by the <a href="Peano_axioms" title="Peano axioms">Peano axioms</a> but can be proven to be true in the larger system of <a href="Second-order_arithmetic" title="Second-order arithmetic">second-order arithmetic</a>.

</p><p><a href="Kruskal's_tree_theorem" title="Kruskal's tree theorem">Kruskal's tree theorem</a>, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable on basis of a philosophy of mathematics called predicativism.

</p><p><a href="Goodstein's_theorem" title="Goodstein's theorem">Goodstein's theorem</a> is a statement about the <a href="Ramsey_theory" title="Ramsey theory">Ramsey theory</a> of the natural numbers that Kirby and Paris showed is undecidable in Peano arithmetic.

</p><p><a href="Gregory_Chaitin" title="Gregory Chaitin">Gregory Chaitin</a> produced undecidable statements in <a href="Algorithmic_information_theory" title="Algorithmic information theory">algorithmic information theory</a> and proved another incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough arithmetic, there is an upper bound <i>c</i> such that no specific number can be proven in that theory to have <a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a> greater than <i>c</i>. While Gödel's theorem is related to the <a href="Liar_paradox" title="Liar paradox">liar paradox</a>, Chaitin's result is related to <a href="Berry's_paradox" class="mw-redirect" title="Berry's paradox">Berry's paradox</a>.

</p><p>In 2007, researchers Kurtz and Simon, building on earlier work by <a href="John_Horton_Conway" title="John Horton Conway">J.H. Conway</a> in the 1970s, proved that a natural generalization of the <a href="Collatz_problem" class="mw-redirect" title="Collatz problem">Collatz problem</a> is undecidable.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>In 2019, Ben-David and colleagues constructed an example of a learning model (named EMX), and showed a family of functions whose learnability in EMX is undecidable in standard set theory.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

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

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

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

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

<li><a href="Wicked_problem" title="Wicked problem">Wicked problem</a></li></ul>

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

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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">This means that there exists an algorithm that halts eventually when the answer is <i>yes</i> but may run forever if the answer is <i>no</i>.</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">There are uncountably many subsets 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 \{0,1\}^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7c3eac9e230a7c68b89e70dd4ed545a3b2a80b1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.738ex; height:2.843ex;" alt="{\displaystyle \{0,1\}^{*}}" loading="lazy"></span>, only countably many of which can be decided by algorithms. However, also only countably many decision problems can be stated in any language.</span>

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

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cs.rochester.edu/u/nelson/courses/csc_173/computability/undecidable.html">"Formal Computational Models and Computability"</a>. <i>www.cs.rochester.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-06-12</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="https://www.oxfordreference.com/view/10.1093/oi/authority.20110803095705786">"decision problem"</a>. <i>Oxford Reference</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-06-12</span></span>.</cite></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"><cite id="CITEREFAaronson2011" class="citation web cs1">Aaronson, Scott (21 July 2011). <a rel="nofollow" class="external text" href="https://scottaaronson.blog/?p=710">"Rosser's Theorem via Turing machines"</a>. <i>Shtetl-Optimized</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2 November</span> 2022</span>.</cite></span>

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<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="CITEREFNovikov1955" class="citation cs2 cs1-prop-foreign-lang-source"><a href="Pyotr_Novikov" title="Pyotr Novikov">Novikov, Pyotr S.</a> (1955), "On the algorithmic unsolvability of the word problem in group theory", <i><a href="Proceedings_of_the_Steklov_Institute_of_Mathematics" class="mw-redirect" title="Proceedings of the Steklov Institute of Mathematics">Proceedings of the Steklov Institute of Mathematics</a></i> (in Russian), <b>44</b>: <span class="nowrap">1–</span>143, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0068.01301">0068.01301</a></cite></span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFMatiyasevich1970" class="citation journal cs1 cs1-prop-script cs1-prop-foreign-lang-source"><a href="Yuri_Matiyasevich" title="Yuri Matiyasevich">Matiyasevich, Yuri</a> (1970). <bdi lang="ru">Диофантовость перечислимых множеств</bdi> [Enumerable sets are Diophantine]. <i><a href="Doklady_Akademii_Nauk_SSSR" class="mw-redirect" title="Doklady Akademii Nauk SSSR">Doklady Akademii Nauk SSSR</a></i> (in Russian). <b>191</b>: <span class="nowrap">279–</span>282.</cite></span>

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<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="CITEREFShelah1974" class="citation journal cs1">Shelah, Saharon (1974). "Infinite Abelian groups, Whitehead problem and some constructions". <i><a href="Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a></i>. <b>18</b> (3): <span class="nowrap">243–</span>256. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02757281">10.1007/BF02757281</a>. <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=0357114">0357114</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:123351674">123351674</a>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Kurtz, Stuart A.; Simon, Janos, <a rel="nofollow" class="external text" href="http://people.cs.uchicago.edu/~simon/RES/collatz.pdf">"The Undecidability of the Generalized Collatz Problem"</a>, in Proceedings of the 4th International Conference on Theory and Applications of Models of Computation, TAMC 2007, held in Shanghai, China in May 2007. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-72503-2</bdi>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-72504-6_49">10.1007/978-3-540-72504-6_49</a></span>

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<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><cite id="CITEREFBen-DavidHrubešMoranShpilka2019" class="citation journal cs1">Ben-David, Shai; Hrubeš, Pavel; Moran, Shay; Shpilka, Amir; Yehudayoff, Amir (2019-01-07). <span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://www.nature.com/articles/s42256-018-0002-3">"Learnability can be undecidable"</a></span>. <i>Nature Machine Intelligence</i>. <b>1</b> (1): <span class="nowrap">44–</span>48. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fs42256-018-0002-3">10.1038/s42256-018-0002-3</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/2522-5839">2522-5839</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:257109887">257109887</a>.</cite></span>

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<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite id="CITEREFReyzin2019" class="citation journal cs1">Reyzin, Lev (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-019-00012-4">"Unprovability comes to machine learning"</a>. <i>Nature</i>. <b>565</b> (7738): <span class="nowrap">166–</span>167. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019Natur.565..166R">2019Natur.565..166R</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-019-00012-4">10.1038/d41586-019-00012-4</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0028-0836">0028-0836</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/30617250">30617250</a>.</cite></span>

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

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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="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="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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