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<title>Satisfiability</title>

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

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<p>In <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a>, a <a href="Well-formed_formula" title="Well-formed formula">formula</a> is <i>satisfiable</i> if it is true under some assignment of values to its <a href="Variable_(mathematics)" title="Variable (mathematics)">variables</a>. For example, the formula <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+3=y}">

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<annotation encoding="application/x-tex">{\displaystyle x+3=y}</annotation>

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</math></span><img src="./b73a8477c56cbf43698603746310b934fbc9d9a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x+3=y}" loading="lazy"></span> is satisfiable because it is true when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=3}">

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<annotation encoding="application/x-tex">{\displaystyle x=3}</annotation>

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</math></span><img src="./871a5063af170fa536b144fbcc5745146a42cc13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=3}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=6}">

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<annotation encoding="application/x-tex">{\displaystyle y=6}</annotation>

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</math></span><img src="./d1c6c2f9d0cc52e003c6be86fa8381c3a93ef363.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.416ex; height:2.509ex;" alt="{\displaystyle y=6}" loading="lazy"></span>, while the formula <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+1=x}">

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

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<annotation encoding="application/x-tex">{\displaystyle x+3=3+x}</annotation>

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</math></span><img src="./071b25e777e998026ea7c3b3c4ee01fe63bd88f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.764ex; height:2.343ex;" alt="{\displaystyle x+3=3+x}" loading="lazy"></span> is valid over the integers, but <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+3=y}">

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<annotation encoding="application/x-tex">{\displaystyle x+3=y}</annotation>

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</p><p>Formally, satisfiability is studied with respect to a fixed logic defining the <a href="Syntax_(logic)" title="Syntax (logic)">syntax</a> of allowed symbols, such as <a href="First-order_logic" title="First-order logic">first-order logic</a>, <a href="Second-order_logic" title="Second-order logic">second-order logic</a> or <a href="Propositional_calculus" class="mw-redirect" title="Propositional calculus">propositional logic</a>. Rather than being syntactic, however, satisfiability is a <a href="Semantics" title="Semantics">semantic</a> property because it relates to the <i>meaning</i> of the symbols, for example, the meaning 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 +}">

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

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</math></span><img src="./fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></span> in a formula such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+1=x}">

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

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</math></span><img src="./6f6a0298a9f29cdc29b664377cefcaef8eaae707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.761ex; height:2.343ex;" alt="{\displaystyle x+1=x}" loading="lazy"></span>. Formally, we define an <a href="Interpretation_(logic)" title="Interpretation (logic)">interpretation</a> (or <a href="Model_theory" title="Model theory">model</a>) to be an assignment of values to the variables and an assignment of meaning to all other non-logical symbols, and a formula is said to be satisfiable if there is some interpretation which makes it true.<sup id="cite_ref-FOOTNOTEBoolosBurgessJeffrey2007p._120:_"A_set_of_sentences_[...]_is_''satisfiable''_if_some_interpretation_[makes_it_true]."_1-0" class="reference"><a href="#cite_note-FOOTNOTEBoolosBurgessJeffrey2007p._120:_"A_set_of_sentences_[...]_is_''satisfiable''_if_some_interpretation_[makes_it_true]."-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> While this allows non-standard interpretations of symbols such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">

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

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</math></span><img src="./fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></span>, one can restrict their meaning by providing additional <a href="Axiom" title="Axiom">axioms</a>. The <a href="Satisfiability_modulo_theories" title="Satisfiability modulo theories">satisfiability modulo theories</a> problem considers satisfiability of a formula with respect to a <a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">formal theory</a>, which is a (finite or infinite) set of axioms.

</p><p>Satisfiability and validity are defined for a single formula, but can be generalized to an arbitrary theory or set of formulas: a theory is satisfiable if at least one interpretation makes every formula in the theory true, and valid if every formula is true in every interpretation. For example, theories of arithmetic such as <a href="Peano_axioms" title="Peano axioms">Peano arithmetic</a> are satisfiable because they are true in the natural numbers. This concept is closely related to the <a href="Consistency" title="Consistency">consistency</a> of a theory, and in fact is equivalent to consistency for first-order logic, a result known as <a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>. The negation of satisfiability is unsatisfiability, and the negation of validity is invalidity. These four concepts are related to each other in a manner exactly analogous to <a href="Aristotle" title="Aristotle">Aristotle</a>'s <a href="Square_of_opposition" title="Square of opposition">square of opposition</a>.

</p><p>The <a href="Decision_problem" title="Decision problem">problem</a> of determining whether a formula in <a href="Propositional_logic" title="Propositional logic">propositional logic</a> is satisfiable is <a href="Decidable_problem" class="mw-redirect" title="Decidable problem">decidable</a>, and is known as the <a href="Boolean_satisfiability_problem" title="Boolean satisfiability problem">Boolean satisfiability problem</a>, or SAT. In general, the problem of determining whether a sentence of <a href="First-order_logic" title="First-order logic">first-order logic</a> is satisfiable is not decidable. In <a href="Universal_algebra" title="Universal algebra">universal algebra</a>, <a href="Equational_theory" class="mw-redirect" title="Equational theory">equational theory</a>, and <a href="Automated_theorem_proving" title="Automated theorem proving">automated theorem proving</a>, the methods of <a href="Term_rewriting" class="mw-redirect" title="Term rewriting">term rewriting</a>, <a href="Congruence_closure" class="mw-redirect" title="Congruence closure">congruence closure</a> and <a href="Unification_(computer_science)" title="Unification (computer science)">unification</a> are used to attempt to decide satisfiability. Whether a particular <a href="Theory_(logic)" class="mw-redirect" title="Theory (logic)">theory</a> is decidable or not depends whether the theory is <a href="Variable-free" class="mw-redirect" title="Variable-free">variable-free</a> and on other conditions.<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>

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<div class="mw-heading mw-heading2"><h2 id="Reduction_of_validity_to_satisfiability">Reduction of validity to satisfiability</h2></div>

<p>For <a href="Classical_logic" title="Classical logic">classical logics</a> with negation, it is generally possible to re-express the question of the validity of a formula to one involving satisfiability, because of the relationships between the concepts expressed in the above square of opposition. In particular φ is valid if and only if ¬φ is unsatisfiable, which is to say it is false that ¬φ is satisfiable. Put another way, φ is satisfiable if and only if ¬φ is invalid.

</p><p>For logics without negation, such as the <a href="List_of_logic_systems" class="mw-redirect" title="List of logic systems">positive propositional calculus</a>, the questions of validity and satisfiability may be unrelated. In the case of the <a href="List_of_logic_systems" class="mw-redirect" title="List of logic systems">positive propositional calculus</a>, the satisfiability problem is trivial, as every formula is satisfiable, while the validity problem is <a href="Co-NP-complete" title="Co-NP-complete">co-NP complete</a>.

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<div class="mw-heading mw-heading2"><h2 id="Propositional_satisfiability_for_classical_logic">Propositional satisfiability for classical logic</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Propositional_satisfiability" class="mw-redirect" title="Propositional satisfiability">Propositional satisfiability</a></div>

<p>In the case of <a href="Classical_propositional_logic" class="mw-redirect" title="Classical propositional logic">classical propositional logic</a>, satisfiability is decidable for propositional formulae. In particular, satisfiability is an <a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> problem, and is one of the most intensively studied problems in <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a>.

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<div class="mw-heading mw-heading2"><h2 id="Satisfiability_in_first-order_logic">Satisfiability in first-order logic</h2></div>

<p>For <a href="First-order_logic" title="First-order logic">first-order logic</a> (FOL), satisfiability is <a href="Undecidable_problem" title="Undecidable problem">undecidable</a>. More specifically, it is a <a href="RE_(complexity)#co-RE-complete" title="RE (complexity)">co-RE-complete</a> problem and therefore not <a href="Semidecidable" class="mw-redirect" title="Semidecidable">semidecidable</a>.<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> This fact has to do with the undecidability of the validity problem for FOL. The question of the status of the validity problem was posed firstly by <a href="David_Hilbert" title="David Hilbert">David Hilbert</a>, as the so-called <a href="Entscheidungsproblem" title="Entscheidungsproblem">Entscheidungsproblem</a>. The universal validity of a formula is a semi-decidable problem by <a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a>. If satisfiability were also a semi-decidable problem, then the problem of the existence of counter-models would be too (a formula has counter-models iff its negation is satisfiable). So the problem of logical validity would be decidable, which contradicts the <a href="Entscheidungsproblem#Negative_answer" title="Entscheidungsproblem">Church–Turing theorem</a>, a result stating the negative answer for the Entscheidungsproblem.

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<div class="mw-heading mw-heading2"><h2 id="Satisfiability_in_model_theory">Satisfiability in model theory</h2></div>

<p>In <a href="Model_theory" title="Model theory">model theory</a>, an <a href="Atomic_formula" title="Atomic formula">atomic formula</a> is satisfiable if there is a collection of elements of a <a href="Structure_(logic)" class="mw-redirect" title="Structure (logic)">structure</a> that render the formula true.<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> If <i>A</i> is a structure, φ is a formula, and <i>a</i> is a collection of elements, taken from the structure, that satisfy φ, then it is commonly written that

</p>

<p>If φ has no free variables, that is, if φ is an <a href="Atomic_sentence" title="Atomic sentence">atomic sentence</a>, and it is satisfied by <i>A</i>, then one writes

</p>

<p>In this case, one may also say that <i>A</i> is a model for φ, or that φ is <i>true</i> in <i>A</i>. If <i>T</i> is a collection of atomic sentences (a theory) satisfied by <i>A</i>, one writes

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<div class="mw-heading mw-heading2"><h2 id="Finite_satisfiability">Finite satisfiability</h2></div>

<p>A problem related to satisfiability is that of <b>finite satisfiability</b>, which is the question of determining whether a formula admits a <i>finite</i> model that makes it true. For a logic that has the <a href="Finite_model_property" title="Finite model property">finite model property</a>, the problems of satisfiability and finite satisfiability coincide, as a formula of that logic has a model if and only if it has a finite model. This question is important in the mathematical field of <a href="Finite_model_theory" title="Finite model theory">finite model theory</a>.

</p><p>Finite satisfiability and satisfiability need not coincide in general. For instance, consider the <a href="First-order_logic" title="First-order logic">first-order logic</a> formula obtained as the <a href="Logical_conjunction" title="Logical conjunction">conjunction</a> of the following sentences, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}">

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

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</math></span><img src="./693ad9f934775838bd72406b41ada4a59785d7ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}">

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

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</math></span><img src="./bbf42ecda092975c9c69dae84e16182ba5fe2e07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}" loading="lazy"></span> are <a href="Logical_constant" title="Logical constant">constants</a>:

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<msub>

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<annotation encoding="application/x-tex">{\displaystyle R(a_{0},a_{1})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \forall xy(R(x,y)\rightarrow \exists zR(y,z))}</annotation>

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</math></span><img src="./39bf603b092dbee2430b59c2e491220f31fe743d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.525ex; height:2.843ex;" alt="{\displaystyle \forall xy(R(x,y)\rightarrow \exists zR(y,z))}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \forall xyz(R(y,x)\wedge R(z,x)\rightarrow y=z))}</annotation>

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</math></span><img src="./9d3a27605e12ae926e8d6926a922633d72ef005d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.236ex; height:2.843ex;" alt="{\displaystyle \forall xyz(R(y,x)\wedge R(z,x)\rightarrow y=z))}" loading="lazy"></span></li>

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<annotation encoding="application/x-tex">{\displaystyle \forall x\neg R(x,a_{0})}</annotation>

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</math></span><img src="./bf23e71d8376e87980d949857e54629f967f4e2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.394ex; height:2.843ex;" alt="{\displaystyle \forall x\neg R(x,a_{0})}" loading="lazy"></span></li></ul>

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The resulting formula has the infinite model <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 R(a_{0},a_{1}),R(a_{1},a_{2}),\ldots }">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle R(a_{0},a_{1}),R(a_{1},a_{2}),\ldots }</annotation>

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</math></span><img src="./9a25548672ac785b1d8ee9edc2dc5194f05806e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.142ex; height:2.843ex;" alt="{\displaystyle R(a_{0},a_{1}),R(a_{1},a_{2}),\ldots }" loading="lazy"></span>, but it can be shown that it has no finite model (starting at the fact <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 R(a_{0},a_{1})}">

<msub>

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</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(a_{0},a_{1})}</annotation>

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</math></span><img src="./65a106b22fe5092a653da7242da430b019300ea9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.175ex; height:2.843ex;" alt="{\displaystyle R(a_{0},a_{1})}" loading="lazy"></span> and following the chain 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 R}">

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

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> <a href="Atomic_formula" title="Atomic formula">atoms</a> that must exist by the second axiom, the finiteness of a model would require the existence of a loop, which would violate the third and fourth axioms, whether it loops back on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}">

<msub>

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

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</p><p>The <a href="Computational_complexity_theory" title="Computational complexity theory">computational complexity</a> of deciding satisfiability for an input formula in a given logic may differ from that of deciding finite satisfiability; in fact, for some logics, only one of them is <a href="Decidability_(logic)" title="Decidability (logic)">decidable</a>.

</p><p>For classical <a href="First-order_logic" title="First-order logic">first-order logic</a>, finite satisfiability is <a href="Computably_enumerable" class="mw-redirect" title="Computably enumerable">recursively enumerable</a> (in class <a href="RE_(complexity)" title="RE (complexity)">RE</a>) and <a href="Undecidable_problem" title="Undecidable problem">undecidable</a> by <a href="Trakhtenbrot's_theorem" title="Trakhtenbrot's theorem">Trakhtenbrot's theorem</a> applied to the negation of the formula.

</p>

<div class="mw-heading mw-heading2"><h2 id="Numerical_constraints">Numerical constraints</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Satisfiability_modulo_theories" title="Satisfiability modulo theories">Satisfiability modulo theories</a> and <a href="Constraint_satisfaction_problem" title="Constraint satisfaction problem">Constraint satisfaction problem</a></div>

<p><span class="clarify-content" style="padding-left:0.1em; padding-right:0.1em; color:var(--color-subtle, #54595d); border:1px solid var(--border-color-subtle, #c8ccd1);">Numerical constraints</span> often appear in the field of <a href="Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a>, where one usually wants to maximize (or minimize) an objective function subject to some constraints. However, leaving aside the objective function, the basic issue of simply deciding whether the constraints are satisfiable can be challenging or undecidable in some settings. The following table summarizes the main cases.

</p>

<th>Constraints</th>

<th>over reals</th>

<th>over integers

</th></tr>

<tr>

<td>Linear</td>

<td><a href="PTIME" class="mw-redirect" title="PTIME">PTIME</a> (see <a href="Linear_programming" title="Linear programming">linear programming</a>)</td>

<td><a href="NP-complete" class="mw-redirect" title="NP-complete">NP-complete</a> (see <a href="Integer_programming" title="Integer programming">integer programming</a>)

</td></tr>

<tr>

<td>Polynomial</td>

<td><a href="Decision_problem" title="Decision problem">decidable</a> through e.g. <a href="Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">Cylindrical algebraic decomposition</a></td>

<td>undecidable (<a href="Hilbert's_tenth_problem" title="Hilbert's tenth problem">Hilbert's tenth problem</a>)

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

<p><i>Table source: Bockmayr and Weispfenning</i>.<sup id="cite_ref-BockmayrWeispfenning2001_5-0" class="reference"><a href="#cite_note-BockmayrWeispfenning2001-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 754">: 754 </span></sup>

</p><p>For linear constraints, a fuller picture is provided by the following table.

</p>

<th>Constraints over:</th>

<th>rationals</th>

<th>integers</th>

<th>natural numbers

</th></tr>

<tr>

<td><a href="System_of_linear_equations" title="System of linear equations">Linear equations</a></td>

<td>PTIME</td>

<td>NP-complete

</td></tr>

<tr>

<td><a href="Linear_inequality#Systems_of_linear_inequalities" title="Linear inequality">Linear inequalities</a></td>

<td>PTIME</td>

<td>NP-complete</td>

<td>NP-complete

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

<p><i>Table source: Bockmayr and Weispfenning</i>.<sup id="cite_ref-BockmayrWeispfenning2001_5-1" class="reference"><a href="#cite_note-BockmayrWeispfenning2001-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 755">: 755 </span></sup>

</p>

<ul><li><a href="2-satisfiability" title="2-satisfiability">2-satisfiability</a></li>

<li><a href="Boolean_satisfiability_problem" title="Boolean satisfiability problem">Boolean satisfiability problem</a></li>

<li><a href="Circuit_satisfiability" class="mw-redirect" title="Circuit satisfiability">Circuit satisfiability</a></li>

<li><a href="Karp's_21_NP-complete_problems" title="Karp's 21 NP-complete problems">Karp's 21 NP-complete problems</a></li>

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

<li><a href="Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a></li></ul>

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

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</style><div class="reflist">

<li id="cite_note-FOOTNOTEBoolosBurgessJeffrey2007p._120:_"A_set_of_sentences_[...]_is_''satisfiable''_if_some_interpretation_[makes_it_true]."-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEBoolosBurgessJeffrey2007p._120:_"A_set_of_sentences_[...]_is_''satisfiable''_if_some_interpretation_[makes_it_true]."_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBoolosBurgessJeffrey2007">Boolos, Burgess & Jeffrey 2007</a>, p. 120: "A set of sentences [...] is <i>satisfiable</i> if some interpretation [makes it true].".</span>

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFFranz_BaaderTobias_Nipkow1998" class="citation book cs1"><a href="Franz_Baader" title="Franz Baader">Franz Baader</a>; <a href="Tobias_Nipkow" title="Tobias Nipkow">Tobias Nipkow</a> (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=N7BvXVUCQk8C&q=satisfiability+OR+satisfiable"><i>Term Rewriting and All That</i></a>. Cambridge University Press. pp. <span class="nowrap">58–</span>92. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-77920-0</bdi>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFBaier,_Christel2012" class="citation web cs1"><a href="Christel_Baier" title="Christel Baier">Baier, Christel</a> (2012). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201014044350/http://www.inf.tu-dresden.de/index.php?node_id=404">"Chapter 1.3 Undecidability of FOL"</a>. <i>Lecture Notes — Advanced Logics</i>. Technische Universität Dresden — Institute for Technical Computer Science. pp. <span class="nowrap">28–</span>32. Archived from <a rel="nofollow" class="external text" href="https://www.inf.tu-dresden.de/content/institutes/thi/algi/lehre/SS12/AL12/skript/script120413.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 14 October 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">21 July</span> 2012</span>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilifrid_Hodges1997" class="citation book cs1">Wilifrid Hodges (1997). <i>A Shorter Model Theory</i>. Cambridge University Press. p. 12. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-58713-1</bdi>.</cite></span>

</li>

<li id="cite_note-BockmayrWeispfenning2001-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-BockmayrWeispfenning2001_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BockmayrWeispfenning2001_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFAlexander_BockmayrVolker_Weispfenning2001" class="citation book cs1">Alexander Bockmayr; Volker Weispfenning (2001). "Solving Numerical Constraints". In John Alan Robinson; Andrei Voronkov (eds.). <i>Handbook of Automated Reasoning Volume I</i>. Elsevier and MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-82949-0</bdi>. (Elsevier) (MIT Press).</cite></span>

</li>

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

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

<ul><li><cite id="CITEREFBoolosBurgessJeffrey2007" class="citation book cs1">Boolos, George; Burgess, John; Jeffrey, Richard (2007). <i>Computability and Logic</i> (5th ed.). Cambridge University Press.</cite></li></ul>

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

<ul><li><cite id="CITEREFDaniel_KroeningOfer_Strichman2008" class="citation book cs1"><a href="Daniel_Kroening" title="Daniel Kroening">Daniel Kroening</a>; <a href="Ofer_Strichman" title="Ofer Strichman">Ofer Strichman</a> (2008). <i>Decision Procedures: An Algorithmic Point of View</i>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-74104-6</bdi>.</cite></li>

<li><cite id="CITEREFA._BiereM._HeuleH._van_MaarenT._Walsh2009" class="citation book cs1">A. Biere; M. Heule; H. van Maaren; T. Walsh, eds. (2009). <i>Handbook of Satisfiability</i>. IOS Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-60750-376-7</bdi>.</cite></li></ul>

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</style><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (list)<br> and <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li><a href="Square_of_opposition" title="Square of opposition">Square of opposition</a></li>

<li><a href="Venn_diagram" title="Venn diagram">Venn diagram</a></li></ul>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li>Set operations:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computability_theory" title="Computability theory">Computability theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Church_encoding" title="Church encoding">Church encoding</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<ul><li><a href="Abstract_logic" title="Abstract logic">Abstract logic</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Metalogic_and_metamathematics37" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Metalogic_and_metamathematics37" style="font-size:114%;margin:0 4em"><a href="Metalogic" title="Metalogic">Metalogic</a> and <a href="Metamathematics" title="Metamathematics">metamathematics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0;padding-left:2.0em;padding-right:2.0em;"><div style="padding:0 0.25em">

<ul><li><a href="Cantor's_theorem" title="Cantor's theorem">Cantor's theorem</a></li>

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

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

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

<li><a href="Effective_method" title="Effective method">Effective method</a></li>

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

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

<li><a href="G%C3%B6del's_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness theorem</a></li>

<li><a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a></li>

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

<li><a href="Completeness_(logic)" title="Completeness (logic)">Completeness</a></li>

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

<li><a href="Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a></li>

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

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

<li><a href="Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a></li>

<li><a href="Type%E2%80%93token_distinction" title="Type–token distinction">Type–token distinction</a></li>

<li><a href="Use%E2%80%93mention_distinction" title="Use–mention distinction">Use–mention distinction</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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