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<title>Set theory</title>

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

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the branch of mathematics. For other uses, see <a href="Set_theory_(disambiguation)" class="mw-disambig" title="Set theory (disambiguation)">Set theory (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Set_theory_(music)" title="Set theory (music)">Set theory (music)</a>.</div>

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</style><table class="sidebar nomobile nowraplinks hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Mathematics" title="Mathematics">Mathematics</a></th></tr><tr><td class="sidebar-above" style="padding-bottom:0.35em;">

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Index</a></li></ul></td></tr><tr><td class="sidebar-content-with-subgroup">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)"><a href="Areas_of_mathematics" class="mw-redirect" title="Areas of mathematics">Areas</a></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Number_theory" title="Number theory">Number theory</a></li>

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

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

<li><a href="Calculus" title="Calculus">Calculus</a> and <a href="Mathematical_analysis" title="Mathematical analysis">Analysis</a></li>

<li><a href="Discrete_mathematics" title="Discrete mathematics">Discrete mathematics</a></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Logic</a></li>

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

<li><a href="Statistics" title="Statistics">Statistics</a> and <a href="Decision_theory" title="Decision theory">Decision theory</a></li></ul></div></div></td>

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<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="border-top:1px solid #aaa;background:#ddddff;text-align:center;;color: var(--color-base)">Relationship with sciences</div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Mathematical_physics" title="Mathematical physics">Physics</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Chemistry</a></li>

<li><a href="Geomathematics" title="Geomathematics">Geosciences</a></li>

<li><a href="Computational_mathematics" title="Computational mathematics">Computation</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Biology</a></li>

<li><a href="Mathematical_linguistics" title="Mathematical linguistics">Linguistics</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Economics</a></li>

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

<li><a href="Mathematics_education" title="Mathematics education">Education</a></li></ul></div></div></td>

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<p><b>Set theory</b> is the branch of <a href="Mathematical_logic" title="Mathematical logic">mathematical logic</a> that studies <a href="Set_(mathematics)" title="Set (mathematics)">sets</a>, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of <a href="Mathematics" title="Mathematics">mathematics</a> – is mostly concerned with those that are relevant to mathematics as a whole.

</p><p>The modern study of set theory was initiated by the German mathematicians <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> and <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a> in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of <i><a href="Naive_set_theory" title="Naive set theory">naive set theory</a></i>. After the discovery of <a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">paradoxes within naive set theory</a> (such as <a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a>, <a href="Cantor's_paradox" title="Cantor's paradox">Cantor's paradox</a> and the <a href="Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>), various <a href="Axiomatic_system" title="Axiomatic system">axiomatic systems</a> were proposed in the early twentieth century, of which <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (with or without the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>) is still the best-known and most studied.

</p><p>Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Besides its foundational role, set theory also provides the framework to develop a mathematical theory of <a href="Infinity" title="Infinity">infinity</a>, and has various applications in <a href="Computer_science" title="Computer science">computer science</a> (such as in the theory of <a href="Relational_algebra" title="Relational algebra">relational algebra</a>), <a href="Philosophy" title="Philosophy">philosophy</a>, <a href="Semantics_(computer_science)" title="Semantics (computer science)">formal semantics</a>, and <a href="Evolutionary_dynamics" title="Evolutionary dynamics">evolutionary dynamics</a>. Its foundational appeal, together with its <a href="Paradoxes" class="mw-redirect" title="Paradoxes">paradoxes</a>, and its implications for the concept of infinity and its multiple applications have made set theory an area of major interest for <a href="Logic" title="Logic">logicians</a> and <a href="Philosophy_of_mathematics" title="Philosophy of mathematics">philosophers of mathematics</a>. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the <a href="Real_number" title="Real number">real number</a> line to the study of the <a href="Consistency" title="Consistency">consistency</a> of <a href="Large_cardinal" title="Large cardinal">large cardinals</a>.

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

<div class="mw-heading mw-heading3"><h3 id="Early_history">Early history</h3></div>

<p>The basic notion of grouping objects has existed since at least the <a href="Natural_number#History" title="Natural number">emergence of numbers</a>, and the notion of treating sets as their own objects has existed since at least the <a href="Tree_of_Porphyry" class="mw-redirect" title="Tree of Porphyry">Tree of Porphyry</a> in 3rd-century AD. The simplicity and ubiquity of sets makes it hard to determine the origin of sets as now used in mathematics; however, <a href="Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a>'s <i><a href="Paradoxes_of_the_Infinite" title="Paradoxes of the Infinite">Paradoxes of the Infinite</a></i> (<i>Paradoxien des Unendlichen</i>, 1851) is generally considered the first rigorous introduction of sets to mathematics. In his work, he (among other things) expanded on <a href="Galileo's_paradox" title="Galileo's paradox">Galileo's paradox</a>, and introduced <a href="One-to-one_correspondence" class="mw-redirect" title="One-to-one correspondence">one-to-one correspondence</a> of infinite sets, for example between the <a href="Interval_(mathematics)" title="Interval (mathematics)">intervals</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,5]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,5]}</annotation>

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</math></span><img src="./466c74a63ec8b8422ccb96c6bc5140dca45b5593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,5]}" 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 [0,12]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,12]}</annotation>

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</math></span><img src="./89aabd85b3e389da4856956942ab5320c9bc586f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.815ex; height:2.843ex;" alt="{\displaystyle [0,12]}" loading="lazy"></span> by the relation <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 5y=12x}">

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<annotation encoding="application/x-tex">{\displaystyle 5y=12x}</annotation>

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</math></span><img src="./315b4b6038170411a2dcfabf59796ba0591d1533.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.071ex; height:2.509ex;" alt="{\displaystyle 5y=12x}" loading="lazy"></span>. However, he resisted saying these sets were <a href="Equinumerous" class="mw-redirect" title="Equinumerous">equinumerous</a>, and his work is generally considered to have been uninfluential in mathematics of his time.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Before mathematical set theory, basic concepts of <a href="Infinity" title="Infinity">infinity</a> were considered to be in the domain of philosophy (see: <i><a href="Infinity_(philosophy)" title="Infinity (philosophy)">Infinity (philosophy)</a></i> and <i><a href="Infinity#History" title="Infinity">Infinity § History</a></i>). Since the 5th century BC, beginning with Greek philosopher <a href="Zeno_of_Elea" title="Zeno of Elea">Zeno of Elea</a> in the West (and early <a href="Indian_mathematics" title="Indian mathematics">Indian mathematicians</a> in the East), mathematicians had struggled with the concept of infinity. With the <a href="History_of_calculus" title="History of calculus">development of calculus</a> in the late 17th century, philosophers began to generally distinguish between potential and <a href="Actual_infinity" class="mw-redirect" title="Actual infinity">actual infinity</a>, wherein mathematics was only considered in the latter.<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> <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> famously stated: "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics."<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>

</p><p>Development of mathematical set theory was motivated by several mathematicians. <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>'s lecture <i>On the Hypotheses which lie at the Foundations of Geometry</i> (1854) proposed new ideas about <a href="Topology" title="Topology">topology</a>. His lectures also introduced the concept of basing mathematics in terms of sets or <a href="Manifold" title="Manifold">manifolds</a> in the sense of a <a href="Class_(set_theory)" title="Class (set theory)">class</a> (which he called <i>Mannigfaltigkeit</i>) now called <a href="Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a>. The lecture was published by <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> in 1868, along with Riemann's paper on <a href="Trigonometric_series" title="Trigonometric series">trigonometric series</a> (which presented the <a href="Riemann_integral" title="Riemann integral">Riemann integral</a>), The latter was a starting point a movement in <a href="Real_analysis" title="Real analysis">real analysis</a> for the study of “seriously” <a href="Discontinuous_function" class="mw-redirect" title="Discontinuous function">discontinuous functions</a>. A young <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a> entered into this area, which led him to the study of <a href="Point_set" class="mw-redirect" title="Point set">point-sets</a>. Around 1871, influenced by Riemann, Dedekind began working with sets in his publications, which dealt very clearly and precisely with <a href="Equivalence_relations" class="mw-redirect" title="Equivalence relations">equivalence relations</a>, <a href="Partition_of_a_set" title="Partition of a set">partitions of sets</a>, and <a href="Homomorphisms" class="mw-redirect" title="Homomorphisms">homomorphisms</a>. Thus, many of the usual set-theoretic procedures of twentieth-century mathematics go back to his work. However, he did not publish a formal explanation of his set theory until 1888.

</p>

<div class="mw-heading mw-heading3"><h3 id="Naive_set_theory">Naive set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Naive_set_theory" title="Naive set theory">Naive set theory</a></div>

<p>Set theory, as understood by modern mathematicians, is generally considered to be founded by a single paper in 1874 by <a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a> titled <i><a href="On_a_Property_of_the_Collection_of_All_Real_Algebraic_Numbers" class="mw-redirect" title="On a Property of the Collection of All Real Algebraic Numbers">On a Property of the Collection of All Real Algebraic Numbers</a></i>.<sup id="cite_ref-cantor1874_5-0" class="reference"><a href="#cite_note-cantor1874-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In his paper, he developed the notion of <a href="Cardinality" title="Cardinality">cardinality</a>, comparing the sizes of two sets by setting them in one-to-one correspondence. His "revolutionary discovery" was that the set of all <a href="Real_number" title="Real number">real numbers</a> is <a href="Uncountable_set" title="Uncountable set">uncountable</a>, that is, one cannot put all real numbers in a list. This theorem is proved using <a href="Cantor's_first_set_theory_article#The_proofs" title="Cantor's first set theory article">Cantor's first uncountability proof</a>, which differs from the more familiar proof using his <a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a>.

</p><p>Cantor introduced fundamental constructions in set theory, such as the <a href="Power_set" title="Power set">power set</a> of a set <i>A</i>, which is the set of all possible <a href="Subset" title="Subset">subsets</a> of <i>A</i>. He later proved that the size of the power set of <i>A</i> is strictly larger than the size of <i>A</i>, even when <i>A</i> is an infinite set; this result soon became known as <a href="Cantor's_theorem" title="Cantor's theorem">Cantor's theorem</a>. Cantor developed a theory of <a href="Transfinite_numbers" class="mw-redirect" title="Transfinite numbers">transfinite numbers</a>, called <a href="Cardinal_number" title="Cardinal number">cardinals</a> and <a href="Ordinal_number" title="Ordinal number">ordinals</a>, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter <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 \aleph }">

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

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</math></span><img src="./306c55e6bc96d94db729ff5821c8f45a34c72bce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }" loading="lazy"></span> (<a href="%E2%84%B5" class="mw-redirect" title="ℵ">ℵ</a>, <a href="Aleph" title="Aleph">aleph</a>) with a natural number subscript; for the ordinals he employed the Greek letter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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

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</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> (<style data-mw-deduplicate="TemplateStyles:r1301836668">

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

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</style><span class="polytonic">ω</span>, <a href="Omega" title="Omega">omega</a>).

</p><p>Set theory was beginning to become an essential ingredient of the new “modern” approach to mathematics. Originally, Cantor's theory of transfinite numbers was regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such as <a href="Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> and <a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> and later from <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> and <a href="L._E._J._Brouwer" title="L. E. J. Brouwer">L. E. J. Brouwer</a>, while <a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> raised <a href="Philosophical_objections_to_Cantor's_theory" class="mw-redirect" title="Philosophical objections to Cantor's theory">philosophical objections</a> (see: <i><a href="Controversy_over_Cantor's_theory" title="Controversy over Cantor's theory">Controversy over Cantor's theory</a></i>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> Dedekind's algebraic style only began to find followers in the 1890s

</p>

<p>Despite the controversy, Cantor's set theory gained remarkable ground around the turn of the 20th century with the work of several notable mathematicians and philosophers. Richard Dedekind, around the same time, began working with sets in his publications, and famously constructing the real numbers using <a href="Dedekind_cuts" class="mw-redirect" title="Dedekind cuts">Dedekind cuts</a>. He also worked with <a href="Giuseppe_Peano" title="Giuseppe Peano">Giuseppe Peano</a> in developing the <a href="Peano_axioms" title="Peano axioms">Peano axioms</a>, which formalized natural-number arithmetic, using set-theoretic ideas, which also introduced the <a href="Epsilon" title="Epsilon">epsilon</a> symbol for <a href="Element_(mathematics)" title="Element (mathematics)">set membership</a>. Possibly most prominently, <a href="Gottlob_Frege" title="Gottlob Frege">Gottlob Frege</a> began to develop his <i><a href="Foundations_of_Arithmetic" class="mw-redirect" title="Foundations of Arithmetic">Foundations of Arithmetic</a></i>.

</p><p>In his work, Frege tries to ground all mathematics in terms of logical axioms using Cantor's cardinality. For example, the sentence "the number of horses in the barn is four" means that four objects fall under the concept <i>horse in the barn</i>. Frege attempted to explain our grasp of numbers through cardinality ('the number of...', or <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 Nx:Fx}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Nx:Fx}</annotation>

</semantics>

</math></span><img src="./715cc0ad8bde7c0fd87bc54b8fbf4cf19ddb9ef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.401ex; height:2.176ex;" alt="{\displaystyle Nx:Fx}" loading="lazy"></span>), relying on <a href="Hume's_principle" title="Hume's principle">Hume's principle</a>.

</p>

<p>However, Frege's work was short-lived, as it was found by <a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a> that his axioms lead to a <a href="Contradiction" title="Contradiction">contradiction</a>. Specifically, Frege's <a href="Basic_Law_V" class="mw-redirect" title="Basic Law V">Basic Law V</a> (now known as the <a href="Axiom_schema_of_unrestricted_comprehension" class="mw-redirect" title="Axiom schema of unrestricted comprehension">axiom schema of unrestricted comprehension</a>). According to <a href="Basic_Law_V" class="mw-redirect" title="Basic Law V">Basic Law V</a>, for any sufficiently well-defined <a href="Property_(philosophy)" title="Property (philosophy)">property</a>, there is the set of all and only the objects that have that property. The contradiction, called <a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a>, is shown as follows:

</p><p>Let <i>R</i> be the set of all sets that are not members of themselves. (This set is sometimes called "the Russell set".) If <i>R</i> is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell's paradox. In symbols:

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}</annotation>

</semantics>

</math></span><img src="./1083e5691d2b959d103e2a6c3a9585a1b25b0438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.889ex; height:2.843ex;" alt="{\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}" loading="lazy"></span></dd></dl>

<p>This came around a time of several <a href="Paradox" title="Paradox">paradoxes</a> or counter-intuitive results. For example, that the <a href="Parallel_postulate" title="Parallel postulate">parallel postulate</a> cannot be proved, the existence of <a href="Mathematical_object" title="Mathematical object">mathematical objects</a> that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with <a href="Peano_arithmetic" class="mw-redirect" title="Peano arithmetic">Peano arithmetic</a>. The result was a <a href="Foundational_crisis_of_mathematics" class="mw-redirect" title="Foundational crisis of mathematics">foundational crisis of mathematics</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Basic_concepts_and_notation">Basic concepts and notation</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a> and <a href="Algebra_of_sets" title="Algebra of sets">Algebra of sets</a></div>

<p>Set theory begins with a fundamental <a href="Binary_relation" title="Binary relation">binary relation</a> between an object <span class="texhtml mvar" style="font-style:italic;">o</span> and a set <span class="texhtml mvar" style="font-style:italic;">A</span>. If <span class="texhtml mvar" style="font-style:italic;">o</span> is a <i><a href="Set_membership" class="mw-redirect" title="Set membership">member</a></i> (or <i>element</i>) of <span class="texhtml mvar" style="font-style:italic;">A</span>, the notation <span class="texhtml"><i>o</i> ∈ <i>A</i></span> is used. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces { }.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Since sets are objects, the membership relation can relate sets as well, i.e., sets themselves can be members of other sets.

</p><p>A derived binary relation between two sets is the subset relation, also called <i>set inclusion</i>. If all the members of set <span class="texhtml mvar" style="font-style:italic;">A</span> are also members of set <span class="texhtml mvar" style="font-style:italic;">B</span>, then <span class="texhtml mvar" style="font-style:italic;">A</span> is a <i><a href="Subset" title="Subset">subset</a></i> of <span class="texhtml mvar" style="font-style:italic;">B</span>, denoted <span class="texhtml"><i>A</i> ⊆ <i>B</i></span>. For example, <span class="texhtml">{1, 2}</span> is a subset of <span class="texhtml">{1, 2, 3}</span>, and so is <span class="texhtml">{2}</span> but <span class="texhtml">{1, 4}</span> is not. As implied by this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term <i><a href="Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a></i> is defined, variously denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subsetneq B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./7bf81e9a4a81df2d596b4db1cde6b9bdf82c73db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.606ex; height:2.676ex;" alt="{\displaystyle A\subsetneq B}" loading="lazy"></span>, or <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\subsetneqq B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./76b5046c0f318612eb9edecaa531d6360fea5c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.606ex; height:3.176ex;" alt="{\displaystyle A\subsetneqq B}" loading="lazy"></span> (note however that the notation <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\subset B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}" loading="lazy"></span> is sometimes used synonymously with <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\subseteq B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b09068bd2f7ba899aeb883ebe670b2ad07b0c851.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.606ex; height:2.343ex;" alt="{\displaystyle A\subseteq B}" loading="lazy"></span>; that is, allowing the possibility that <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> are equal). We call <span class="texhtml mvar" style="font-style:italic;">A</span> a <i>proper subset</i> of <span class="texhtml mvar" style="font-style:italic;">B</span> if and only if <span class="texhtml mvar" style="font-style:italic;">A</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">B</span>, but <span class="texhtml mvar" style="font-style:italic;">A</span> is not equal to <span class="texhtml mvar" style="font-style:italic;">B</span>. Also, 1, 2, and 3 are members (elements) of the set <span class="texhtml">{1, 2, 3}</span>, but are not subsets of it; and in turn, the subsets, such as <span class="texhtml">{1}</span>, are not members of the set <span class="texhtml">{1, 2, 3}</span>. More complicated relations can exist; for example, the set <span class="texhtml">{1}</span> is both a member and a proper subset of the set <span class="texhtml">{1, {1}}</span>.

</p><p>Just as <a href="Arithmetic" title="Arithmetic">arithmetic</a> features <a href="Binary_operation" title="Binary operation">binary operations</a> on <a href="Number" title="Number">numbers</a>, set theory features binary operations on sets.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The following is a partial list of them:

</p>

<ul><li><i><a href="Union_(set_theory)" title="Union (set theory)">Union</a></i> of the sets <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, denoted <span class="texhtml"><i>A</i> ∪ <i>B</i></span>, is the set of all objects that are a member of <span class="texhtml mvar" style="font-style:italic;">A</span>, or <span class="texhtml mvar" style="font-style:italic;">B</span>, or both.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> For example, the union of <span class="texhtml">{1, 2, 3}</span> and <span class="texhtml">{2, 3, 4}</span> is the set <span class="texhtml">{1, 2, 3, 4}</span>.</li>

<li><i><a href="Intersection_(set_theory)" title="Intersection (set theory)">Intersection</a></i> of the sets <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, denoted <span class="texhtml"><i>A</i> ∩ <i>B</i></span>, is the set of all objects that are members of both <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> For example, the intersection of <span class="texhtml">{1, 2, 3}</span> and <span class="texhtml">{2, 3, 4}</span> is the set <span class="texhtml">{2, 3}</span>.</li>

<li><i><a href="Set_difference" class="mw-redirect" title="Set difference">Set difference</a></i> of <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">A</span>, denoted <span class="texhtml"><i>U</i> ∖ <i>A</i></span>, is the set of all members of <span class="texhtml mvar" style="font-style:italic;">U</span> that are not members of <span class="texhtml mvar" style="font-style:italic;">A</span>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The set difference <span class="texhtml">{1, 2, 3} ∖ {2, 3, 4} </span> is <span class="texhtml">{1}</span>, while conversely, the set difference <span class="texhtml">{2, 3, 4} ∖ {1, 2, 3}</span> is <span class="texhtml">{4}</span>. When <span class="texhtml mvar" style="font-style:italic;">A</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">U</span>, the set difference <span class="texhtml"><i>U</i> ∖ <i>A</i></span> is also called the <i><a href="Complement_(set_theory)" title="Complement (set theory)">complement</a></i> of <span class="texhtml mvar" style="font-style:italic;">A</span> in <span class="texhtml mvar" style="font-style:italic;">U</span>. In this case, if the choice of <span class="texhtml mvar" style="font-style:italic;">U</span> is clear from the context, the notation <span class="texhtml"><i>A</i><sup><i>c</i></sup></span> is sometimes used instead of <span class="texhtml"><i>U</i> ∖ <i>A</i></span>, particularly if <span class="texhtml mvar" style="font-style:italic;">U</span> is a <a href="Universal_set" title="Universal set">universal set</a> as in the study of <a href="Venn_diagram" title="Venn diagram">Venn diagrams</a>.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup></li>

<li><i><a href="Symmetric_difference" title="Symmetric difference">Symmetric difference</a></i> of sets <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, denoted <span class="texhtml"><i>A</i> △ <i>B</i></span> or <span class="texhtml"><i>A</i> ⊖ <i>B</i></span>, is the set of all objects that are a member of exactly one of <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> (elements which are in one of the sets, but not in both). For instance, for the sets <span class="texhtml">{1, 2, 3}</span> and <span class="texhtml">{2, 3, 4}</span>, the symmetric difference set is <span class="texhtml">{1, 4}</span>. It is the set difference of the union and the intersection, <span class="texhtml">(<i>A</i> ∪ <i>B</i>) ∖ (<i>A</i> ∩ <i>B</i>)</span> or <span class="texhtml">(<i>A</i> ∖ <i>B</i>) ∪ (<i>B</i> ∖ <i>A</i>)</span>.</li>

<li><i><a href="Cartesian_product" title="Cartesian product">Cartesian product</a></i> of <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span>, denoted <span class="texhtml"><i>A</i> × <i>B</i></span>, is the set whose members are all possible <a href="Ordered_pair" title="Ordered pair">ordered pairs</a> <span class="texhtml">(<i>a</i>, <i>b</i>)</span>, where <span class="texhtml mvar" style="font-style:italic;">a</span> is a member of <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> is a member of <span class="texhtml mvar" style="font-style:italic;">B</span>. For example, the Cartesian product of {1, 2} and {red, white} is <span class="nowrap">{(1, red), (1, white), (2, red), (2, white)}.</span><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Some basic sets of central importance are the set of <a href="Natural_number" title="Natural number">natural numbers</a>, the set of <a href="Real_number" title="Real number">real numbers</a> and the <a href="Empty_set" title="Empty set">empty set</a> – the unique set containing no elements. The empty set is also occasionally called the <i>null set</i>,<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> though this name is ambiguous and can lead to several interpretations. The empty set can be denoted with empty braces "<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 \{\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3e6f1caa524dfcc90158ad69a51b5f9577fe5f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.325ex; height:2.843ex;" alt="{\displaystyle \{\}}" loading="lazy"></span>" or the symbol "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varnothing }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./00595c5e33692e724937fdcc8870496acce1ac74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.009ex;" alt="{\displaystyle \varnothing }" loading="lazy"></span>" or "<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./6af50205f42bb2ec3c666b7b847d2c7f96e464c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.162ex; height:2.509ex;" alt="{\displaystyle \emptyset }" loading="lazy"></span>".

</p><p>The <a href="Power_set" title="Power set">power set</a> of a set <span class="texhtml mvar" style="font-style:italic;">A</span>, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}(A)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1757ec21abe0a22f8e91b51fe3e6ac4ea63a9122.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.256ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}(A)}" loading="lazy"></span>, is the set whose members are all of the possible subsets of <span class="texhtml mvar" style="font-style:italic;">A</span>. For example, the power set of <span class="texhtml">{1, 2}</span> is <span class="texhtml">{ {}, {1}, {2}, {1, 2} }</span>. Notably, <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 {\mathcal {P}}(A)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Von_Neumann_universe" title="Von Neumann universe">von Neumann universe</a></div>

<p>A set is <a href="Pure_set" class="mw-redirect" title="Pure set">pure</a> if all of its members are sets, all members of its members are sets, and so on. For example, the set containing only the empty set is a nonempty pure set. In modern set theory, it is common to restrict attention to the <i><a href="Von_Neumann_universe" title="Von Neumann universe">von Neumann universe</a></i> of pure sets, and many systems of axiomatic set theory are designed to axiomatize the pure sets only. There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a <a href="Cumulative_hierarchy" title="Cumulative hierarchy">cumulative hierarchy</a>, based on how deeply their members, members of members, etc. are nested. Each set in this hierarchy is assigned (by <a href="Transfinite_recursion" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a>) an <a href="Ordinal_number" title="Ordinal number">ordinal number</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, known as its <i>rank.</i> The rank of a pure set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"></span> is defined to be the least ordinal that is strictly greater than the rank of any of its elements. For example, the empty set is assigned rank 0, while the set <span class="texhtml"> </span> containing only the empty set is assigned rank 1. For each ordinal <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 \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, the set <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./582d40a9ff663187250948b07bb66456162c2042.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle V_{\alpha }}" loading="lazy"></span> is defined to consist of all pure sets with rank less than <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 \alpha }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>. The entire von Neumann universe is denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formalized_set_theory">Formalized set theory</h2></div>

<p>

Elementary set theory can be studied informally and intuitively, and so can be taught in primary schools using <a href="Venn_diagram" title="Venn diagram">Venn diagrams</a>. The intuitive approach tacitly assumes that a set may be formed from the class of all objects satisfying any particular defining condition. This assumption gives rise to paradoxes, the simplest and best known of which are <a href="Russell's_paradox" title="Russell's paradox">Russell's paradox</a> and the <a href="Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a>. <b>Axiomatic set theory</b> was originally devised to rid set theory of such paradoxes.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>

</p><p>The most widely studied systems of axiomatic set theory imply that all sets form a <a href="Cumulative_hierarchy" title="Cumulative hierarchy">cumulative hierarchy</a>.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup> Such systems come in two flavors, those whose <a href="Ontology" title="Ontology">ontology</a> consists of:

</p>

<ul><li><i>Sets alone</i>. This includes the most common axiomatic set theory, <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory"><b>Z</b>ermelo–<b>F</b>raenkel set theory</a> with the <a href="Axiom_of_choice" title="Axiom of choice">axiom of <b>c</b>hoice</a> (ZFC). Fragments of <b>ZFC</b> include:

<ul><li><a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo set theory</a>, which replaces the <a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">axiom schema of replacement</a> with that of <a href="Axiom_schema_of_separation" class="mw-redirect" title="Axiom schema of separation">separation</a>;</li>

<li><a href="General_set_theory" title="General set theory">General set theory</a>, a small fragment of Zermelo set theory sufficient for the <a href="Peano_axioms" title="Peano axioms">Peano axioms</a> and <a href="Finite_set" title="Finite set">finite sets</a>;</li>

<li><a href="Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek set theory</a>, which omits the axioms of infinity, <a href="Axiom_of_power_set" title="Axiom of power set">powerset</a>, and choice, and weakens the axiom schemata of <a href="Axiom_schema_of_separation" class="mw-redirect" title="Axiom schema of separation">separation</a> and <a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a>.</li></ul></li>

<li><i>Sets and <a href="Proper_class" class="mw-redirect" title="Proper class">proper classes</a></i>. These include <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 set theory</a>, which has the same <a href="Strength_(mathematical_logic)" title="Strength (mathematical logic)">strength</a> as <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> for theorems about sets alone, and <a href="Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley set theory</a> and <a href="Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck set theory</a>, both of which are stronger than ZFC.</li></ul>

<p>The above systems can be modified to allow <i><a href="Urelement" title="Urelement">urelements</a></i>, objects that can be members of sets but that are not themselves sets and do not have any members.

</p><p>The <i><a href="New_Foundations" title="New Foundations">New Foundations</a></i> systems of <b>NFU</b> (allowing <a href="Urelement" title="Urelement">urelements</a>) and <b>NF</b> (lacking them), associate with <a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Van Orman Quine</a>, are not based on a cumulative hierarchy. NF and NFU include a "set of everything", relative to which every set has a complement. In these systems urelements matter, because NF, but not NFU, produces sets for which the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> does not hold. Despite NF's ontology not reflecting the traditional cumulative hierarchy and violating well-foundedness, <a href="Thomas_Forster_(mathematician)" title="Thomas Forster (mathematician)">Thomas Forster</a> has argued that it does reflect an iterative conception of set.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p><p>Systems of <a href="Constructive_set_theory" title="Constructive set theory">constructive set theory</a>, such as CST, CZF, and IZF, embed their set axioms in <a href="Intuitionistic_logic" title="Intuitionistic logic">intuitionistic</a> instead of <a href="Classical_logic" title="Classical logic">classical logic</a>. Yet other systems accept classical logic but feature a nonstandard membership relation. These include <a href="Rough_set" title="Rough set">rough set theory</a> and <a href="Fuzzy_set_theory" class="mw-redirect" title="Fuzzy set theory">fuzzy set theory</a>, in which the value of an <a href="Atomic_formula" title="Atomic formula">atomic formula</a> embodying the membership relation is not simply <b>True</b> or <b>False</b>. The <a href="Boolean-valued_model" title="Boolean-valued model">Boolean-valued models</a> of <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> are a related subject.

</p><p>An enrichment of ZFC called <a href="Internal_set_theory" title="Internal set theory">internal set theory</a> was proposed by <a href="Edward_Nelson" title="Edward Nelson">Edward Nelson</a> in 1977.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup>

</p>

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

<p>Many mathematical concepts can be defined precisely using only set theoretic concepts. For example, mathematical structures as diverse as <a href="Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, <a href="Manifolds" class="mw-redirect" title="Manifolds">manifolds</a>, <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="Vector_space" title="Vector space">vector spaces</a>, and <a href="Relational_algebra" title="Relational algebra">relational algebras</a> can all be defined as sets satisfying various (axiomatic) properties. <a href="Equivalence_relation" title="Equivalence relation">Equivalence</a> and <a href="Order_relation" class="mw-redirect" title="Order relation">order relations</a> are ubiquitous in mathematics, and the theory of mathematical <a href="Relation_(mathematics)" title="Relation (mathematics)">relations</a> can be described in set theory.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup>

</p><p>Set theory is also a promising foundational system for much of mathematics. Since the publication of the first volume of <i><a href="Principia_Mathematica" title="Principia Mathematica">Principia Mathematica</a></i>, it has been claimed that most (or even all) mathematical theorems can be derived using an aptly designed set of axioms for set theory, augmented with many definitions, using <a href="First-order_logic" title="First-order logic">first</a> or <a href="Second-order_logic" title="Second-order logic">second-order logic</a>. For example, properties of the <a href="Natural_number" title="Natural number">natural</a> and <a href="Real_number" title="Real number">real numbers</a> can be derived within set theory, as each of these number systems can be defined by representing their elements as sets of specific forms.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p><p>Set theory as a foundation for <a href="Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, <a href="Topology" title="Topology">topology</a>, <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>, and <a href="Discrete_mathematics" title="Discrete mathematics">discrete mathematics</a> is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present. One verification project, <a href="Metamath" title="Metamath">Metamath</a>, includes human-written, computer-verified derivations of more than 12,000 theorems starting from <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> set theory, <a href="First-order_logic" title="First-order logic">first-order logic</a> and <a href="Propositional_logic" title="Propositional logic">propositional logic</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Areas_of_study">Areas of study</h2></div>

<p>Set theory is a major area of research in mathematics with many interrelated subfields:

</p>

<div class="mw-heading mw-heading3"><h3 id="Combinatorial_set_theory">Combinatorial set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Infinitary_combinatorics" title="Infinitary combinatorics">Infinitary combinatorics</a></div>

<p><i>Combinatorial set theory</i> concerns extensions of finite <a href="Combinatorics" title="Combinatorics">combinatorics</a> to infinite sets. This includes the study of <a href="Cardinal_arithmetic" class="mw-redirect" title="Cardinal arithmetic">cardinal arithmetic</a> and the study of extensions of <a href="Ramsey's_theorem" title="Ramsey's theorem">Ramsey's theorem</a> such as the <a href="Erd%C5%91s%E2%80%93Rado_theorem" title="Erdős–Rado theorem">Erdős–Rado theorem</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Descriptive_set_theory">Descriptive set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Descriptive_set_theory" title="Descriptive set theory">Descriptive set theory</a></div>

<p><i>Descriptive set theory</i> is the study of subsets of the <a href="Real_line" class="mw-redirect" title="Real line">real line</a> and, more generally, subsets of <a href="Polish_space" title="Polish space">Polish spaces</a>. It begins with the study of <a href="Pointclass" title="Pointclass">pointclasses</a> in the <a href="Borel_hierarchy" title="Borel hierarchy">Borel hierarchy</a> and extends to the study of more complex hierarchies such as the <a href="Projective_hierarchy" title="Projective hierarchy">projective hierarchy</a> and the <a href="Wadge_hierarchy" title="Wadge hierarchy">Wadge hierarchy</a>. Many properties of <a href="Borel_set" title="Borel set">Borel sets</a> can be established in ZFC, but proving these properties hold for more complicated sets requires additional axioms related to determinacy and large cardinals.

</p><p>The field of <a href="Effective_descriptive_set_theory" title="Effective descriptive set theory">effective descriptive set theory</a> is between set theory and <a href="Recursion_theory" class="mw-redirect" title="Recursion theory">recursion theory</a>. It includes the study of <a href="Lightface_pointclass" class="mw-redirect" title="Lightface pointclass">lightface pointclasses</a>, and is closely related to <a href="Hyperarithmetical_theory" title="Hyperarithmetical theory">hyperarithmetical theory</a>. In many cases, results of classical descriptive set theory have effective versions; in some cases, new results are obtained by proving the effective version first and then extending ("relativizing") it to make it more broadly applicable.

</p><p>A recent area of research concerns <a href="Borel_equivalence_relation" title="Borel equivalence relation">Borel equivalence relations</a> and more complicated definable <a href="Equivalence_relation" title="Equivalence relation">equivalence relations</a>. This has important applications to the study of <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariants</a> in many fields of mathematics.

</p>

<div class="mw-heading mw-heading3"><h3 id="Fuzzy_set_theory">Fuzzy set theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Fuzzy_set_theory" class="mw-redirect" title="Fuzzy set theory">Fuzzy set theory</a></div>

<p>In set theory as Cantor defined and Zermelo and Fraenkel axiomatized, an object is either a member of a set or not. In <i><a href="Fuzzy_set_theory" class="mw-redirect" title="Fuzzy set theory">fuzzy set theory</a></i> this condition was relaxed by <a href="Lotfi_A._Zadeh" title="Lotfi A. Zadeh">Lotfi A. Zadeh</a> so an object has a <i>degree of membership</i> in a set, a number between 0 and 1. For example, the degree of membership of a person in the set of "tall people" is more flexible than a simple yes or no answer and can be a real number such as 0.75.

</p>

<div class="mw-heading mw-heading3"><h3 id="Inner_model_theory">Inner model theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Inner_model_theory" title="Inner model theory">Inner model theory</a></div>

<p>An <i>inner model</i> of Zermelo–Fraenkel set theory (ZF) is a transitive <a href="Proper_class" class="mw-redirect" title="Proper class">class</a> that includes all the ordinals and satisfies all the axioms of ZF. The canonical example is the <a href="Constructible_universe" title="Constructible universe">constructible universe</a> <i>L</i> developed by Gödel.

One reason that the study of inner models is of interest is that it can be used to prove consistency results. For example, it can be shown that regardless of whether a model <i>V</i> of ZF satisfies the <a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> or the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, the inner model <i>L</i> constructed inside the original model will satisfy both the generalized continuum hypothesis and the axiom of choice. Thus the assumption that ZF is consistent (has at least one model) implies that ZF together with these two principles is consistent.

</p><p>The study of inner models is common in the study of <a href="Axiom_of_determinacy" title="Axiom of determinacy">determinacy</a> and <a href="Large_cardinal" title="Large cardinal">large cardinals</a>, especially when considering axioms such as the axiom of determinacy that contradict the axiom of choice. Even if a fixed model of set theory satisfies the axiom of choice, it is possible for an inner model to fail to satisfy the axiom of choice. For example, the existence of sufficiently large cardinals implies that there is an inner model satisfying the axiom of determinacy (and thus not satisfying the axiom of choice).<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Large_cardinals">Large cardinals</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Large_cardinal_property" class="mw-redirect" title="Large cardinal property">Large cardinal property</a></div>

<p>A <i>large cardinal</i> is a cardinal number with an extra property. Many such properties are studied, including <a href="Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible cardinals</a>, <a href="Measurable_cardinal" title="Measurable cardinal">measurable cardinals</a>, and many more. These properties typically imply the cardinal number must be very large, with the existence of a cardinal with the specified property unprovable in <a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a>.

</p>

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

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

<p><i>Determinacy</i> refers to the fact that, under appropriate assumptions, certain two-player games of perfect information are determined from the start in the sense that one player must have a winning strategy. The existence of these strategies has important consequences in descriptive set theory, as the assumption that a broader class of games is determined often implies that a broader class of sets will have a topological property. The <a href="Axiom_of_determinacy" title="Axiom of determinacy">axiom of determinacy</a> (AD) is an important object of study; although incompatible with the axiom of choice, AD implies that all subsets of the real line are well behaved (in particular, measurable and with the perfect set property). AD can be used to prove that the <a href="Wadge_degree" class="mw-redirect" title="Wadge degree">Wadge degrees</a> have an elegant structure.

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Forcing_(mathematics)" title="Forcing (mathematics)">Forcing (mathematics)</a></div>

<p><a href="Paul_Cohen_(mathematician)" class="mw-redirect" title="Paul Cohen (mathematician)">Paul Cohen</a> invented the method of <i><a href="Forcing_(mathematics)" title="Forcing (mathematics)">forcing</a></i> while searching for a <a href="Model_theory" title="Model theory">model</a> of <a href="ZFC" class="mw-redirect" title="ZFC">ZFC</a> in which the <a href="Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a> fails, or a model of ZF in which the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> fails. Forcing adjoins to some given model of set theory additional sets in order to create a larger model with properties determined (i.e. "forced") by the construction and the original model. For example, Cohen's construction adjoins additional subsets of the <a href="Natural_number" title="Natural number">natural numbers</a> without changing any of the <a href="Cardinal_number" title="Cardinal number">cardinal numbers</a> of the original model. Forcing is also one of two methods for proving <a href="Consistency_(mathematical_logic)" class="mw-redirect" title="Consistency (mathematical logic)">relative consistency</a> by finitistic methods, the other method being <a href="Boolean-valued_model" title="Boolean-valued model">Boolean-valued models</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Cardinal_invariants">Cardinal invariants</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cardinal_characteristics_of_the_continuum" class="mw-redirect" title="Cardinal characteristics of the continuum">Cardinal characteristics of the continuum</a></div>

<p>A <i>cardinal invariant</i> is a property of the real line measured by a cardinal number. For example, a well-studied invariant is the smallest cardinality of a collection of <a href="Meagre_set" title="Meagre set">meagre sets</a> of reals whose union is the entire real line. These are invariants in the sense that any two isomorphic models of set theory must give the same cardinal for each invariant. Many cardinal invariants have been studied, and the relationships between them are often complex and related to axioms of set theory.

</p>

<div class="mw-heading mw-heading3"><h3 id="Set-theoretic_topology">Set-theoretic topology</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic topology</a></div>

<p><i>Set-theoretic topology</i> studies questions of <a href="General_topology" title="General topology">general topology</a> that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof. A famous problem is the <a href="Moore_space_(topology)" title="Moore space (topology)">normal Moore space question</a>, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC.

</p>

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

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Controversy_over_Cantor's_theory" title="Controversy over Cantor's theory">Controversy over Cantor's theory</a></div>

<p>From set theory's inception, some mathematicians have objected to it as a <a href="Foundations_of_mathematics" title="Foundations of mathematics">foundation for mathematics</a>. The most common objection to set theory, one <a href="Leopold_Kronecker" title="Leopold Kronecker">Kronecker</a> voiced in set theory's earliest years, starts from the <a href="Mathematical_constructivism" class="mw-redirect" title="Mathematical constructivism">constructivist</a> view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in <a href="Naive_set_theory" title="Naive set theory">naive</a> and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. The feasibility of constructivism as a substitute foundation for mathematics was greatly increased by <a href="Errett_Bishop" title="Errett Bishop">Errett Bishop</a>'s influential book <i>Foundations of Constructive Analysis</i>.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup>

</p><p>A different objection put forth by <a href="Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> is that defining sets using the axiom schemas of <a href="Axiom_schema_of_specification" title="Axiom schema of specification">specification</a> and <a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a>, as well as the <a href="Axiom_of_power_set" title="Axiom of power set">axiom of power set</a>, introduces <a href="Impredicativity" title="Impredicativity">impredicativity</a>, a type of <a href="Circular_definition" title="Circular definition">circularity</a>, into the definitions of mathematical objects. The scope of predicatively founded mathematics, while less than that of the commonly accepted Zermelo–Fraenkel theory, is much greater than that of constructive mathematics, to the point that <a href="Solomon_Feferman" title="Solomon Feferman">Solomon Feferman</a> has said that "all of scientifically applicable analysis can be developed [using predicative methods]".<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Ludwig_Wittgenstein" title="Ludwig Wittgenstein">Ludwig Wittgenstein</a> condemned set theory philosophically for its connotations of <a href="Mathematical_platonism" class="mw-redirect" title="Mathematical platonism">mathematical platonism</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> He wrote that "set theory is wrong", since it builds on the "nonsense" of fictitious symbolism, has "pernicious idioms", and that it is nonsensical to talk about "all numbers".<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Wittgenstein identified mathematics with algorithmic human deduction;<sup id="cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteConsForm_§2.1]_30-0" class="reference"><a href="#cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteConsForm_§2.1]-30"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> the need for a secure foundation for mathematics seemed, to him, nonsensical.<sup id="cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittLateCritSetTheoNonEnumVsNonDenu_§3.4]_31-0" class="reference"><a href="#cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittLateCritSetTheoNonEnumVsNonDenu_§3.4]-31"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> Moreover, since human effort is necessarily finite, Wittgenstein's philosophy required an ontological commitment to radical <a href="Constructivism_(math)" class="mw-redirect" title="Constructivism (math)">constructivism</a> and <a href="Finitism" title="Finitism">finitism</a>. Meta-mathematical statements – which, for Wittgenstein, included any statement quantifying over infinite domains, and thus almost all modern set theory – are not mathematics.<sup id="cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteFini_§2.2]_32-0" class="reference"><a href="#cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteFini_§2.2]-32"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Few modern philosophers have adopted Wittgenstein's views after a spectacular blunder in <i><a href="Remarks_on_the_Foundations_of_Mathematics" title="Remarks on the Foundations of Mathematics">Remarks on the Foundations of Mathematics</a></i>: Wittgenstein attempted to refute <a href="G%C3%B6del's_incompleteness_theorems" title="Gödel's incompleteness theorems">Gödel's incompleteness theorems</a> after having only read the abstract. As reviewers <a href="Georg_Kreisel" title="Georg Kreisel">Kreisel</a>, <a href="Paul_Bernays" title="Paul Bernays">Bernays</a>, <a href="Michael_Dummett" title="Michael Dummett">Dummett</a>, and <a href="R._L._Goodstein" class="mw-redirect" title="R. L. Goodstein">Goodstein</a> all pointed out, many of his critiques did not apply to the paper in full. Only recently have philosophers such as <a href="Crispin_Wright" title="Crispin Wright">Crispin Wright</a> begun to rehabilitate Wittgenstein's arguments.<sup id="cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittGodeUndeMathProp_§3.6]_33-0" class="reference"><a href="#cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittGodeUndeMathProp_§3.6]-33"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup>

</p><p><a href="Category_theory" title="Category theory">Category theorists</a> have proposed <a href="Topos_theory" class="mw-redirect" title="Topos theory">topos theory</a> as an alternative to traditional axiomatic set theory. Topos theory can interpret various alternatives to that theory, such as <a href="Mathematical_constructivism" class="mw-redirect" title="Mathematical constructivism">constructivism</a>, finite set theory, and <a href="Turing_Machine" class="mw-redirect" title="Turing Machine">computable</a> set theory.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Topoi also give a natural setting for forcing and discussions of the independence of choice from ZF, as well as providing the framework for <a href="Pointless_topology" title="Pointless topology">pointless topology</a> and <a href="Stone_space" title="Stone space">Stone spaces</a>.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>

</p><p>An active area of research is the <a href="Univalent_foundations" title="Univalent foundations">univalent foundations</a> and related to it <a href="Homotopy_type_theory" title="Homotopy type theory">homotopy type theory</a>. Within homotopy type theory, a set may be regarded as a homotopy 0-type, with <a href="Universal_properties" class="mw-redirect" title="Universal properties">universal properties</a> of sets arising from the inductive and recursive properties of <a href="Higher_inductive_type" class="mw-redirect" title="Higher inductive type">higher inductive types</a>. Principles such as the <a href="Axiom_of_choice" title="Axiom of choice">axiom of choice</a> and the <a href="Law_of_the_excluded_middle" class="mw-redirect" title="Law of the excluded middle">law of the excluded middle</a> can be formulated in a manner corresponding to the classical formulation in set theory or perhaps in a spectrum of distinct ways unique to type theory. Some of these principles may be proven to be a consequence of other principles. The variety of formulations of these axiomatic principles allows for a detailed analysis of the formulations required in order to derive various mathematical results.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_education">Mathematical education</h2></div>

<p>As set theory gained popularity as a foundation for modern mathematics, there has been support for the idea of introducing the basics of <a href="Naive_set_theory" title="Naive set theory">naive set theory</a> early in <a href="Mathematics_education" title="Mathematics education">mathematics education</a>.

</p><p>In the US in the 1960s, the <a href="New_Math" title="New Math">New Math</a> experiment aimed to teach basic set theory, among other abstract concepts, to <a href="Primary_school" title="Primary school">primary school</a> students but was met with much criticism.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The math syllabus in European schools followed this trend and currently includes the subject at different levels in all grades. <a href="Venn_diagram" title="Venn diagram">Venn diagrams</a> are widely employed to explain basic set-theoretic relationships to primary school students (even though <a href="John_Venn" title="John Venn">John Venn</a> originally devised them as part of a procedure to assess the <a href="Validity_(logic)" title="Validity (logic)">validity</a> of <a href="Inference" title="Inference">inferences</a> in <a href="Term_logic" title="Term logic">term logic</a>).

</p><p>Set theory is used to introduce students to <a href="Logical_operators" class="mw-redirect" title="Logical operators">logical operators</a> (NOT, AND, OR), and semantic or rule description (technically <a href="Intensional_definition" class="mw-redirect" title="Intensional definition">intensional definition</a>)<sup id="cite_ref-Ruda2011_40-0" class="reference"><a href="#cite_note-Ruda2011-40"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> of sets (e.g. "months starting with the letter <i>A</i>"), which may be useful when learning <a href="Computer_programming" title="Computer programming">computer programming</a>, since <a href="Boolean_logic" class="mw-redirect" title="Boolean logic">Boolean logic</a> is used in various <a href="Programming_language" title="Programming language">programming languages</a>. Likewise, sets and other collection-like objects, such as <a href="Multiset" title="Multiset">multisets</a> and <a href="List_(abstract_data_type)" title="List (abstract data type)">lists</a>, are common <a href="Set_(abstract_data_type)" title="Set (abstract data type)">datatypes</a> in computer science and programming.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup>

</p><p>In addition to that, certain sets are commonly used in mathematical teaching, such as the sets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./fdf9a96b565ea202d0f4322e9195613fb26a9bed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" loading="lazy"></span> of <a href="Natural_numbers" class="mw-redirect" title="Natural numbers">natural numbers</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span> of <a href="Integer" title="Integer">integers</a>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> of <a href="Real_number" title="Real number">real numbers</a>, etc.). These are commonly used when defining a <a href="Mathematical_function" class="mw-redirect" title="Mathematical function">mathematical function</a> as a relation from one set (the <a href="Domain_of_a_function" title="Domain of a function">domain</a>) to another set (the <a href="Range_of_a_function" title="Range of a function">range</a>).<sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup>

</p>

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<ul><li><a href="Glossary_of_set_theory" title="Glossary of set theory">Glossary of set theory</a></li>

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

<li><a href="List_of_set_theory_topics" title="List of set theory topics">List of set theory topics</a></li>

<li><a href="Relational_model" title="Relational model">Relational model</a> – borrows from set theory</li>

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

<li><a href="Elementary_Theory_of_the_Category_of_Sets" title="Elementary Theory of the Category of Sets">Elementary Theory of the Category of Sets</a></li>

<li><a href="Structural_set_theory" title="Structural set theory">Structural set theory</a></li></ul>

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

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<li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">In his 1925 paper ""An Axiomatization of Set Theory", <a href="John_von_Neumann" title="John von Neumann">John von Neumann</a> observed that "set theory in its first, "naive" version, due to Cantor, led to contradictions. These are the well-known <a href="Antinomy" title="Antinomy">antinomies</a> of the set of all sets that do not contain themselves (Russell), of the set of all transfinite ordinal numbers (Burali-Forti), and the set of all finitely definable real numbers (Richard)." He goes on to observe that two "tendencies" were attempting to "rehabilitate" set theory. Of the first effort, exemplified by <a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a>, <a href="Julius_K%C3%B6nig" class="mw-redirect" title="Julius König">Julius König</a>, <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> and <a href="L._E._J._Brouwer" title="L. E. J. Brouwer">L. E. J. Brouwer</a>, von Neumann called the "overall effect of their activity . . . devastating". With regards to the axiomatic method employed by second group composed of Zermelo, Fraenkel and Schoenflies, von Neumann worried that "We see only that the known modes of inference leading to the antinomies fail, but who knows where there are not others?" and he set to the task, "in the spirit of the second group", to "produce, by means of a finite number of purely formal operations . . . all the sets that we want to see formed" but not allow for the antinomies. (All quotes from von Neumann 1925 reprinted in van Heijenoort, Jean (1967, third printing 1976), <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931</i>, Harvard University Press, Cambridge MA, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-674-32449-8</bdi> (pbk). A synopsis of the history, written by van Heijenoort, can be found in the comments that precede von Neumann's 1925 paper.</span>

</li>

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

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">The objections to Cantor's work were occasionally fierce: Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth". Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".</span>

</li>

<li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">This is the converse for ZFC; V is a model of ZFC.</span>

</li>

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

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFFerreirós2024" class="citation cs2">Ferreirós, José (2024), <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/settheory-early/">"The Early Development of Set Theory"</a>, in Zalta, Edward N.; Nodelman, Uri (eds.), <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2024 ed.), Metaphysics Research Lab, Stanford University, <a rel="nofollow" class="external text" href="https://archive.today/20230320205811/https://plato.stanford.edu/entries/settheory-early/">archived</a> from the original on 2023-03-20<span class="reference-accessdate">, retrieved <span class="nowrap">2025-01-04</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 id="CITEREFBolzano1975" class="citation cs2 cs1-prop-long-vol"><a href="Bernard_Bolzano" title="Bernard Bolzano">Bolzano, Bernard</a> (1975), Berg, Jan (ed.), <i>Einleitung zur Größenlehre und erste Begriffe der allgemeinen Größenlehre</i>, Bernard-Bolzano-Gesamtausgabe, edited by Eduard Winter et al., vol. II, A, 7, Stuttgart, Bad Cannstatt: Friedrich Frommann Verlag, p. 152, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-7728-0466-7</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="CITEREFZenkin2004" class="citation cs2">Zenkin, Alexander (2004), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.rml/1203431978">"Logic Of Actual Infinity And G. Cantor's Diagonal Proof Of The Uncountability Of The Continuum"</a>, <i>The Review of Modern Logic</i>, vol. 9, no. 30, pp. <span class="nowrap">27–</span>80, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200922022622/https://projecteuclid.org/euclid.rml/1203431978">archived</a> from the original on 2020-09-22<span class="reference-accessdate">, retrieved <span class="nowrap">2025-01-04</span></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="CITEREFDunham1991" class="citation book cs2">Dunham, William (1991), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/journeythroughge00dunh_359"><i>Journey through Genius: The Great Theorems of Mathematics</i></a></span>, Penguin, p. <a rel="nofollow" class="external text" href="https://archive.org/details/journeythroughge00dunh_359/page/n267">254</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780140147391</bdi></cite></span>

</li>

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<li id="cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteConsForm_§2.1]-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteConsForm_§2.1]_30-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRodych2018">Rodych 2018</a>, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteConsForm">§2.1</a>: "When we prove a theorem or decide a proposition, we operate in a purely formal, syntactical manner. In doing mathematics, we do not discover pre-existing truths that were 'already there without one knowing' (PG 481)—we invent mathematics, bit-by-little-bit." Note, however, that Wittgenstein does <i>not</i> identify such deduction with <a href="Philosophical_logic" title="Philosophical logic">philosophical logic</a>; cf. Rodych <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittMathTrac">§1</a>, paras. 7-12.</span>

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<li id="cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittLateCritSetTheoNonEnumVsNonDenu_§3.4]-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittLateCritSetTheoNonEnumVsNonDenu_§3.4]_31-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRodych2018">Rodych 2018</a>, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittLateCritSetTheoNonEnumVsNonDenu">§3.4</a>: "Given that mathematics is a '<style data-mw-deduplicate="TemplateStyles:r920966791">

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</style><span class="smallcaps">motley</span> of techniques of proof' (RFM III, §46), it does not require a foundation (RFM VII, §16) and it cannot be given a self-evident foundation (PR §160; WVC 34 & 62; RFM IV, §3). Since set theory was invented to provide mathematics with a foundation, it is, minimally, unnecessary."</span>

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<li id="cite_note-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteFini_§2.2]-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTERodych2018[httpsplatostanfordeduentrieswittgenstein-mathematicsWittInteFini_§2.2]_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFRodych2018">Rodych 2018</a>, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/wittgenstein-mathematics/#WittInteFini">§2.2</a>: "An expression quantifying over an infinite domain is never a meaningful proposition, not even when we have proved, for instance, that a particular number <span class="texhtml mvar" style="font-style:italic;">n</span> has a particular property."</span>

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<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite id="CITEREFTaylor1999" class="citation magazine cs2">Taylor, Melissa August, Harriet Barovick, Michelle Derrow, Tam Gray, Daniel S. Levy, Lina Lofaro, David Spitz, Joel Stein and Chris (14 June 1999), <a rel="nofollow" class="external text" href="https://time.com/archive/6735628/the-100-worst-ideas-of-the-century/">"The 100 Worst Ideas Of The Century"</a>, <i>TIME</i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20250412010256/https://time.com/archive/6735628/the-100-worst-ideas-of-the-century/">archived</a> from the original on 12 April 2025<span class="reference-accessdate">, retrieved <span class="nowrap">12 April</span> 2025</span></cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite magazine}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

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<li id="cite_note-Ruda2011-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ruda2011_40-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFFrank_Ruda2011" class="citation book cs2">Frank Ruda (6 October 2011), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VV0SBwAAQBAJ&pg=PA151"><i>Hegel's Rabble: An Investigation into Hegel's Philosophy of Right</i></a>, Bloomsbury Publishing, p. 151, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4411-7413-0</bdi></cite></span>

</li>

<li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><cite id="CITEREFAdams1993" class="citation journal cs2">Adams, Stephen (October 1993), <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/journals/journal-of-functional-programming/article/functional-pearls-efficient-setsa-balancing-act/0CAA1C189B4F7C15CE9B8C02D0D4B54E">"Functional Pearls Efficient sets—a balancing act"</a>, <i>Journal of Functional Programming</i>, <b>3</b> (4): <span class="nowrap">553–</span>561, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0956796800000885">10.1017/S0956796800000885</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/1469-7653">1469-7653</a><span class="reference-accessdate">, retrieved <span class="nowrap">12 April</span> 2025</span></cite></span>

</li>

<li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><cite id="CITEREFAbbott2015" class="citation book cs2">Abbott, Stephen (2015), <i>Understanding analysis</i> (Second ed.), New York: Springer, p. 3, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4939-2711-1</bdi></cite></span>

</li>

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

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

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<ul><li><cite id="CITEREFKunen1980" class="citation cs2"><a href="Kenneth_Kunen" title="Kenneth Kunen">Kunen, Kenneth</a> (1980), <i><a href="Set_Theory%3A_An_Introduction_to_Independence_Proofs" title="Set Theory: An Introduction to Independence Proofs">Set Theory: An Introduction to Independence Proofs</a></i>, North-Holland, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-444-85401-0</bdi></cite></li>

<li><cite id="CITEREFJohnson1972" class="citation cs2">Johnson, Philip (1972), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/historyofsettheo0000unse"><i>A History of Set Theory</i></a></span>, Prindle, Weber & Schmidt, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-87150-154-6</bdi></cite></li></ul>

</div>

<ul><li><cite id="CITEREFDevlin1993" class="citation cs2"><a href="Keith_Devlin" title="Keith Devlin">Devlin, Keith</a> (1993), <i>The Joy of Sets: Fundamentals of Contemporary Set Theory</i>, Undergraduate Texts in Mathematics (2nd ed.), Springer Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0903-4">10.1007/978-1-4612-0903-4</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94094-4</bdi></cite></li>

<li><cite id="CITEREFFerreirós2001" class="citation cs2">Ferreirós, Jose (2001), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DITy0nsYQQoC"><i>Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics</i></a>, Berlin: Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-5749-8</bdi></cite></li>

<li><cite id="CITEREFMonk1969" class="citation cs2">Monk, J. Donald (1969), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontose0000monk/page/n5/mode/2up"><i>Introduction to Set Theory</i></a></span>, McGraw-Hill Book Company, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-898-74006-6</bdi></cite></li>

<li><cite id="CITEREFPotter2004" class="citation cs2">Potter, Michael (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=FxRoPuPbGgUC"><i>Set Theory and Its Philosophy: A Critical Introduction</i></a>, <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-191-55643-2</bdi></cite></li>

<li><cite id="CITEREFSmullyanFitting2010" class="citation cs2"><a href="Raymond_Smullyan" title="Raymond Smullyan">Smullyan, Raymond M.</a>; Fitting, Melvin (2010), <i>Set Theory and the Continuum Problem</i>, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-47484-7</bdi></cite></li>

<li><cite id="CITEREFTiles2004" class="citation cs2"><a href="Mary_Tiles" title="Mary Tiles">Tiles, Mary</a> (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=02ASV8VB4gYC"><i>The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise</i></a>, <a href="Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-43520-6</bdi></cite></li>

<li><cite id="Dauben1977" class="citation journal cs2"><a href="Joseph_Dauben" title="Joseph Dauben">Dauben, Joseph W.</a> (1977), "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite", <i>Journal of the History of Ideas</i>, <b>38</b> (1): <span class="nowrap">85–</span>108, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2708842">10.2307/2708842</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2708842">2708842</a></cite></li>

<li><cite id="Dauben1979" class="citation book cs2">Dauben, Joseph W. (1979), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/georgcantorhisma0000daub"><i>[Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite</i></a></span>, Boston: Harvard University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-02447-9</bdi></cite></li></ul>

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

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<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <i><b><a href="https://en.wikibooks.org/wiki/Discrete_mathematics/Set_theory" class="extiw external" title="wikibooks:Discrete mathematics/Set theory">Discrete mathematics/Set theory</a></b></i></div></div>

</div>

<ul><li>Daniel Cunningham, <a rel="nofollow" class="external text" href="http://www.iep.utm.edu/set-theo/">Set Theory</a> article in the <i><a href="Internet_Encyclopedia_of_Philosophy" title="Internet Encyclopedia of Philosophy">Internet Encyclopedia of Philosophy</a></i>.</li>

<li>Jose Ferreiros, <a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/settheory-early/">"The Early Development of Set Theory"</a> article in the <i>[Stanford Encyclopedia of Philosophy]</i>.</li>

<li><a href="Matthew_Foreman" title="Matthew Foreman">Foreman, Matthew</a>, <a href="Akihiro_Kanamori" title="Akihiro Kanamori">Akihiro Kanamori</a>, eds. <i><a rel="nofollow" class="external text" href="http://handbook.assafrinot.com/">Handbook of Set Theory</a></i>. 3 vols., 2010. Each chapter surveys some aspect of contemporary research in set theory. Does not cover established elementary set theory, on which see Devlin (1993).</li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Axiomatic_set_theory">"Axiomatic set theory"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Set_theory">"Set theory"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li>

<li><a href="Arthur_Schoenflies" class="mw-redirect" title="Arthur Schoenflies">Schoenflies, Arthur</a> (1898). <a rel="nofollow" class="external text" href="https://archive.org/stream/encyklomath101encyrich#page/n229">Mengenlehre</a> in <a href="Klein's_encyclopedia" class="mw-redirect" title="Klein's encyclopedia">Klein's encyclopedia</a>.</li>

<li><a class="external text external" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Set+theory&library=OLBP">Online books</a>, and library resources <a class="external text external" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Set+theory">in your library</a> and <a class="external text external" href="https://ftl.toolforge.org/cgi-bin/ftl?st=&su=Set+theory&library=0CHOOSE0">in other libraries</a> about set theory</li>

<li><cite id="CITEREFRudin1990" class="citation web cs2"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter B.</a> (April 6, 1990), <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=hBcWRZMP6xs&list=PLvAAmIFroksMKHv5O4lwpJJzfmUL0cQ7A&index=3">"Set Theory: An Offspring of Analysis"</a>, <i>Marden Lecture in Mathematics</i>, <a href="University_of_Wisconsin-Milwaukee" class="mw-redirect" title="University of Wisconsin-Milwaukee">University of Wisconsin-Milwaukee</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211031/hBcWRZMP6xs">archived</a> from the original on 2021-10-31 – via <a href="YouTube" title="YouTube">YouTube</a></cite></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Set_theory409" style="padding:3px"><table class="nowraplinks mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Set_theory409" style="font-size:114%;margin:0 4em"></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Set_(mathematics)" title="Set (mathematics)">Set (mathematics)</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="Venn_diagram" title="Venn diagram"></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Axiom" title="Axiom">Axioms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Axiom_of_adjunction" title="Axiom of adjunction">Adjunction</a></li>

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

<ul><li><a href="Axiom_of_countable_choice" title="Axiom of countable choice">countable</a></li>

<li><a href="Axiom_of_dependent_choice" title="Axiom of dependent choice">dependent</a></li>

<li><a href="Axiom_of_global_choice" title="Axiom of global choice">global</a></li></ul></li>

<li><a href="Axiom_of_constructibility" title="Axiom of constructibility">Constructibility (V=L)</a></li>

<li><a href="Axiom_of_determinacy" title="Axiom of determinacy">Determinacy</a>

<ul><li><a href="Axiom_of_projective_determinacy" class="mw-redirect" title="Axiom of projective determinacy">projective</a></li></ul></li>

<li><a href="Axiom_of_extensionality" title="Axiom of extensionality">Extensionality</a></li>

<li><a href="Axiom_of_infinity" title="Axiom of infinity">Infinity</a></li>

<li><a href="Axiom_of_limitation_of_size" title="Axiom of limitation of size">Limitation of size</a></li>

<li><a href="Axiom_of_pairing" title="Axiom of pairing">Pairing</a></li>

<li><a href="Axiom_of_power_set" title="Axiom of power set">Power set</a></li>

<li><a href="Axiom_of_regularity" title="Axiom of regularity">Regularity</a></li>

<li><a href="Axiom_of_union" title="Axiom of union">Union</a></li>

<li><a href="Martin's_axiom" title="Martin's axiom">Martin's axiom</a></li></ul>

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

<ul><li><a href="Axiom_schema_of_replacement" title="Axiom schema of replacement">replacement</a></li>

<li><a href="Axiom_schema_of_specification" title="Axiom schema of specification">specification</a></li></ul></li></ul>

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

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

<li><a href="Complement_(set_theory)" title="Complement (set theory)">Complement</a> (i.e. set difference)</li>

<li><a href="De_Morgan's_laws" title="De Morgan's laws">De Morgan's laws</a></li>

<li><a href="Disjoint_union" title="Disjoint union">Disjoint union</a></li>

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

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

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

<li><a href="Symmetric_difference" title="Symmetric difference">Symmetric difference</a></li>

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

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

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

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

<li><a href="Cardinal_number" title="Cardinal number">Cardinal number</a> (<a href="Large_cardinal" title="Large cardinal">large</a>)</li>

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

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

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

<li><a href="Cantor's_diagonal_argument" title="Cantor's diagonal argument">Diagonal argument</a></li>

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

<ul><li><a href="Ordered_pair" title="Ordered pair">ordered pair</a></li>

<li><a href="Tuple" title="Tuple">tuple</a></li></ul></li>

<li><a href="Family_of_sets" title="Family of sets">Family</a></li>

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

<li><a href="Bijection" title="Bijection">One-to-one correspondence</a></li>

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

<li><a href="Set-builder_notation" title="Set-builder notation">Set-builder notation</a></li>

<li><a href="Transfinite_induction" title="Transfinite induction">Transfinite induction</a></li>

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

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

<ul><li><a href="Amorphous_set" title="Amorphous set">Amorphous</a></li>

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

<li><a href="Empty_set" title="Empty set">Empty</a></li>

<li><a href="Finite_set" title="Finite set">Finite</a> (<a href="Hereditarily_finite_set" title="Hereditarily finite set">hereditarily</a>)</li>

<li><a href="Filter_(set_theory)" title="Filter (set theory)">Filter</a>

<li><a href="Filter_(set_theory)#Filters_and_prefilters" title="Filter (set theory)">subbase</a></li>

<li><a href="Ultrafilter_on_a_set" title="Ultrafilter on a set">Ultrafilter</a></li></ul></li>

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

<li><a href="Infinite_set" title="Infinite set">Infinite</a> (<a href="Dedekind-infinite_set" title="Dedekind-infinite set">Dedekind-infinite</a>)</li>

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

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

<li><a href="Subset" title="Subset">Subset <b>·</b> Superset</a></li>

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

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

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

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

<ul><li><a href="Alternative_set_theory" class="mw-redirect" title="Alternative set theory">Alternative</a></li>

<li><a class="mw-selflink-fragment" href="#Formalized_set_theory">Axiomatic</a></li>

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

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

<ul><li><a href="Zermelo_set_theory" title="Zermelo set theory">Zermelo</a>

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

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

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

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

<ul><li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">von Neumann–Bernays–Gödel </a>

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div class="hlist"><ul><li><a href="Paradoxes_of_set_theory" title="Paradoxes of set theory">Paradoxes</a></li><li>Problems</li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

<li><a href="Suslin's_problem" title="Suslin's problem">Suslin's problem</a></li>

<li><a href="Burali-Forti_paradox" title="Burali-Forti paradox">Burali-Forti paradox</a></li></ul>

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

<ul><li><a href="Paul_Bernays" title="Paul Bernays">Paul Bernays</a></li>

<li><a href="Georg_Cantor" title="Georg Cantor">Georg Cantor</a></li>

<li><a href="Paul_Cohen" title="Paul Cohen">Paul Cohen</a></li>

<li><a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a></li>

<li><a href="Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></li>

<li><a href="Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a></li>

<li><a href="Thomas_Jech" title="Thomas Jech">Thomas Jech</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li>

<li><a href="Willard_Van_Orman_Quine" title="Willard Van Orman Quine">Willard Quine</a></li>

<li><a href="Bertrand_Russell" title="Bertrand Russell">Bertrand Russell</a></li>

<li><a href="Thoralf_Skolem" title="Thoralf Skolem">Thoralf Skolem</a></li>

<li><a href="Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a></li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Major_mathematics_areas1069" style="padding:3px"><table class="nowraplinks hlist 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="Major_mathematics_areas1069" style="font-size:114%;margin:0 4em">Major <a href="Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_mathematics" title="History of mathematics">History</a>

<ul><li><a href="Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li>

<li><a href="Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li>

<li><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li>

<li><a href="Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Foundations_of_mathematics" title="Foundations of mathematics">Foundations</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="Category_theory" title="Category theory">Category theory</a></li>

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

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li>

<li><a href="Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</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%"><a href="Algebra" title="Algebra">Algebra</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="Abstract_algebra" title="Abstract algebra">Abstract</a></li>

<li><a href="Commutative_algebra" title="Commutative algebra">Commutative</a></li>

<li><a href="Elementary_algebra" title="Elementary algebra">Elementary</a></li>

<li><a href="Group_theory" title="Group theory">Group theory</a></li>

<li><a href="Linear_algebra" title="Linear algebra">Linear</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li>

<li><a href="Universal_algebra" title="Universal algebra">Universal</a></li>

<li><a href="Homological_algebra" title="Homological algebra">Homological</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematical_analysis" title="Mathematical analysis">Analysis</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="Calculus" title="Calculus">Calculus</a></li>

<li><a href="Real_analysis" title="Real analysis">Real analysis</a></li>

<li><a href="Complex_analysis" title="Complex analysis">Complex analysis</a></li>

<li><a href="Hypercomplex_analysis" title="Hypercomplex analysis">Hypercomplex analysis</a></li>

<li><a href="Differential_equation" title="Differential equation">Differential equations</a></li>

<li><a href="Functional_analysis" title="Functional analysis">Functional analysis</a></li>

<li><a href="Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></li>

<li><a href="Measure_(mathematics)" title="Measure (mathematics)">Measure theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Discrete_mathematics" title="Discrete mathematics">Discrete</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="Combinatorics" title="Combinatorics">Combinatorics</a></li>

<li><a href="Graph_theory" title="Graph theory">Graph theory</a></li>

<li><a href="Order_theory" title="Order theory">Order theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Geometry" title="Geometry">Geometry</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="Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li>

<li><a href="Analytic_geometry" title="Analytic geometry">Analytic</a></li>

<li><a href="Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential</a></li>

<li><a href="Discrete_geometry" title="Discrete geometry">Discrete</a></li>

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

<li><a href="Finite_geometry" title="Finite geometry">Finite</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Number_theory" title="Number theory">Number theory</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="Arithmetic" title="Arithmetic">Arithmetic</a></li>

<li><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li>

<li><a href="Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li>

<li><a href="Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Topology" title="Topology">Topology</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="General_topology" title="General topology">General</a></li>

<li><a href="Algebraic_topology" title="Algebraic topology">Algebraic</a></li>

<li><a href="Differential_topology" title="Differential topology">Differential</a></li>

<li><a href="Geometric_topology" title="Geometric topology">Geometric</a></li>

<li><a href="Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Applied_mathematics" title="Applied mathematics">Applied</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="Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li>

<li><a href="Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li>

<li><a href="Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li>

<li><a href="Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li>

<li><a href="Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li>

<li><a href="Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li>

<li><a href="Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li>

<li><a href="Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li>

<li><a href="Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li>

<li><a href="Probability_theory" title="Probability theory">Probability</a></li>

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

<li><a href="Systems_science" title="Systems science">Systems science</a>

<ul><li><a href="Control_theory" title="Control theory">Control theory</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computational_mathematics" title="Computational mathematics">Computational</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="Computer_science" title="Computer science">Computer science</a></li>

<li><a href="Theory_of_computation" title="Theory of computation">Theory of computation</a></li>

<li><a href="Computational_complexity_theory" title="Computational complexity theory">Computational complexity theory</a></li>

<li><a href="Numerical_analysis" title="Numerical analysis">Numerical analysis</a></li>

<li><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a></li>

<li><a href="Computer_algebra" title="Computer algebra">Computer algebra</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lists_of_mathematics_topics" title="Lists of mathematics topics">Related topics</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="Mathematicians" class="mw-redirect" title="Mathematicians">Mathematicians</a>

<ul><li><a href="List_of_mathematicians" class="mw-redirect" title="List of mathematicians">lists</a></li></ul></li>

<li><a href="Informal_mathematics" title="Informal mathematics">Informal mathematics</a></li>

<li><a href="List_of_films_about_mathematicians" title="List of films about mathematicians">Films about mathematicians</a></li>

<li><a href="Recreational_mathematics" title="Recreational mathematics">Recreational mathematics</a></li>

<li><a href="Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li>

<li><a href="Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><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></li>

<li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> <b>Category</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span> <b><a href="https://commons.wikimedia.org/wiki/Category:Mathematics" class="extiw external" title="commons:Category:Mathematics">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> <b>WikiProject</b></li></ul>

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

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Mathematical_logic344" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li>Set operations:

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

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

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

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

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

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

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

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

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

<li><a href="Empty_set" title="Empty set">Empty</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<li><a href="Von_Neumann%E2%80%93Bernays%E2%80%93G%C3%B6del_set_theory" title="Von Neumann–Bernays–Gödel set theory">Von Neumann–Bernays–Gödel</a></li>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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