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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Mathematical structure</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">For the notion of "structure" in mathematical logic, see <a href="Structure_(mathematical_logic)" title="Structure (mathematical logic)">Structure (mathematical logic)</a>.</div>
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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a structure on a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an <a href="Operation_(mathematics)" title="Operation (mathematics)">operation</a>, <a href="Relation_(mathematics)" title="Relation (mathematics)">relation</a>, <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a>, or <a href="Topological_space" title="Topological space">topology</a>). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
</p><p>A partial list of possible structures are <a href="Measure_theory" class="mw-redirect" title="Measure theory">measures</a>, <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a> (<a href="Group_(mathematics)" title="Group (mathematics)">groups</a>, <a href="Field_(mathematics)" title="Field (mathematics)">fields</a>, etc.), <a href="Topology" title="Topology">topologies</a>, <a href="Metric_space" title="Metric space">metric structures</a> (<a href="Geometry" title="Geometry">geometries</a>), <a href="Order_theory" title="Order theory">orders</a>, <a href="Graph_theory" title="Graph theory">graphs</a>, <a href="Event_structure" title="Event structure">events</a>, <a href="Setoid" title="Setoid">Setoids</a>, <a href="Differential_structure" title="Differential structure">differential structures</a>, and <a href="Category_(mathematics)" title="Category (mathematics)">categories</a>.
</p><p>Sometimes, a set is endowed with more than one feature simultaneously, which allows mathematicians to study the interaction between the different structures more richly. For example, an ordering imposes a rigid form, shape, or topology on the set, and if a set has both a topology feature and a group feature, such that these two features are related in a certain way, then the structure becomes a <a href="Topological_group" title="Topological group">topological group</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p><a href="Map_(mathematics)" title="Map (mathematics)">Map</a> between two sets with the same type of structure, which preserve this structure [<a href="Morphism" title="Morphism">morphism</a>: structure in the <a href="Domain_of_a_function" title="Domain of a function">domain</a> is mapped properly to the (same type) structure in the <a href="Codomain" title="Codomain">codomain</a>] is of special interest in many fields of mathematics. Examples are <a href="Homomorphisms" class="mw-redirect" title="Homomorphisms">homomorphisms</a>, which preserve algebraic structures; <a href="Continuous_functions" class="mw-redirect" title="Continuous functions">continuous functions</a>, which preserve topological structures; and <a href="Differentiable_functions" class="mw-redirect" title="Differentiable functions">differentiable functions</a>, which preserve differential structures.
</p>
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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>
<p>In 1939, the French group with the pseudonym "<a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>" saw structures as the root of mathematics. They first mentioned them in their "Fascicule" of <i>Theory of Sets</i> and expanded it into Chapter IV of the 1957 edition.<sup id="cite_ref-Corry_2-0" class="reference"><a href="#cite_note-Corry-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> They identified <b>three <i>mother structures</i>: algebraic, topological, and order</b>.<sup id="cite_ref-Corry_2-1" class="reference"><a href="#cite_note-Corry-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wells_3-0" class="reference"><a href="#cite_note-Wells-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Example:_the_real_numbers">Example: the real numbers</h2></div>
<p>The set of <a href="Real_number" title="Real number">real numbers</a> has several standard structures:
</p>
<ul><li>An order: each number is either less than or greater than any other number.</li>
<li>Algebraic structure: there are operations of addition and multiplication, the first of which makes it into a <a href="Group_theory" title="Group theory">group</a> and the pair of which together make it into a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>.</li>
<li>A measure: <a href="Interval_(mathematics)" title="Interval (mathematics)">intervals</a> of the real line have a specific <a href="Length" title="Length">length</a>, which can be extended to the <a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> on many of its <a href="Subset" title="Subset">subsets</a>.</li>
<li>A metric: there is a notion of <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">distance</a> between points.</li>
<li>A geometry: it is equipped with a <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> and is <a href="Flat_space" class="mw-redirect" title="Flat space">flat</a>.</li>
<li>A topology: there is a notion of <a href="Open_set" title="Open set">open sets</a>.</li></ul>
<p>There are interfaces among these:
</p>
<ul><li>Its order and, independently, its metric structure induce its topology.</li>
<li>Its order and algebraic structure make it into an <a href="Ordered_field" title="Ordered field">ordered field</a>.</li>
<li>Its algebraic structure and topology make it into a <a href="Lie_group" title="Lie group">Lie group</a>, a type of <a href="Topological_group" title="Topological group">topological group</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Abstract_structure" title="Abstract structure">Abstract structure</a></li>
<li><a href="Isomorphism" title="Isomorphism">Isomorphism</a></li>
<li><a href="Equivalent_definitions_of_mathematical_structures" title="Equivalent definitions of mathematical structures">Equivalent definitions of mathematical structures</a></li>
<li><a href="Forgetful_functor" title="Forgetful functor">Forgetful functor</a></li>
<li><a href="Intuitionistic_type_theory" title="Intuitionistic type theory">Intuitionistic type theory</a></li>
<li><a href="Mathematical_object" title="Mathematical object">Mathematical object</a></li>
<li><a href="Algebraic_structure" title="Algebraic structure">Algebraic structure</a></li>
<li><a href="Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a></li>
<li><a href="Category_(mathematics)" title="Category (mathematics)">Category (mathematics)</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFSaunders1996" class="citation journal cs1">Saunders, Mac Lane (1996). <a rel="nofollow" class="external text" href="http://www2.mat.ulaval.ca/fileadmin/Pages_personnelles_des_profs/hm/H14_Mac_Lane_Phil_Math_1996.pdf">"Structure in Mathematics"</a> <span class="cs1-format">(PDF)</span>. <i>Philosoph1A Mathemat1Ca</i>. <b>4</b> (3): 176.</cite></span>
</li>
<li id="cite_note-Corry-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Corry_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Corry_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFCorry1992" class="citation journal cs1">Corry, Leo (September 1992). "Nicolas Bourbaki and the concept of mathematical structure". <i>Synthese</i>. <b>92</b> (3): <span class="nowrap">315–</span>348. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00414286">10.1007/bf00414286</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/20117057">20117057</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16981077">16981077</a>.</cite></span>
</li>
<li id="cite_note-Wells-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wells_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWells2010" class="citation book cs1">Wells, Richard B. (2010). <a rel="nofollow" class="external text" href="http://www.mrc.uidaho.edu/~rwells/techdocs/Biological%20Signal%20Processing/Chapter%2010%20Mathematical%20Structures.pdf"><i>Biological signal processing and computational neuroscience</i></a> <span class="cs1-format">(PDF)</span>. pp. <span class="nowrap">296–</span>335<span class="reference-accessdate">. Retrieved <span class="nowrap">7 April</span> 2016</span>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li>Bourbaki, Nikolas (1968). "Elements of Mathematics: Theory of Sets". Hermann, Addison-Wesley. pp. 259–346, 383–385.</li>
<li><cite id="CITEREFFoldes1994" class="citation book cs1">Foldes, Stephan (1994). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalstruc0000fold"><i>Fundamental Structures of Algebra and Discrete Mathematics</i></a></span>. Hoboken: John Wiley & Sons. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781118031438</bdi>.</cite></li>
<li><cite id="CITEREFHegedusMoreno-Armella2011" class="citation journal cs1">Hegedus, Stephen John; Moreno-Armella, Luis (2011). "The emergence of mathematical structures". <i>Educational Studies in Mathematics</i>. <b>77</b> (2): <span class="nowrap">369–</span>388. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10649-010-9297-7">10.1007/s10649-010-9297-7</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119981368">119981368</a>.</cite></li>
<li><cite id="CITEREFKolmanBusbyRoss2000" class="citation book cs1">Kolman, Bernard; Busby, Robert C.; Ross, Sharon Cutler (2000). <i>Discrete mathematical structures</i> (4th ed.). Upper Saddle River, NJ: Prentice Hall. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-083143-9</bdi>.</cite></li>
<li><cite id="CITEREFMalikSen2004" class="citation book cs1">Malik, D.S.; Sen, M.K. (2004). <i>Discrete mathematical structures : theory and applications</i>. Australia: Thomson/Course Technology. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-619-21558-3</bdi>.</cite></li>
<li><cite id="CITEREFPudlák2013" class="citation book cs1">Pudlák, Pavel (2013). "Mathematical structures". <i>Logical foundations of mathematics and computational complexity a gentle introduction</i>. Cham: Springer. pp. <span class="nowrap">2–</span>24. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783319001197</bdi>.</cite></li>
<li><cite id="CITEREFSenechal1993" class="citation journal cs1"><a href="Marjorie_Senechal" title="Marjorie Senechal">Senechal, M.</a> (21 May 1993). "Mathematical Structures". <i>Science</i>. <b>260</b> (5111): <span class="nowrap">1170–</span>1173. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.260.5111.1170">10.1126/science.260.5111.1170</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17806355">17806355</a>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/Structure">"Structure"</a>. <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a></i>.</cite> <i>(provides a model theoretic definition.)</i></li>
<li><a rel="nofollow" class="external text" href="http://journals.cambridge.org/action/displayJournal?jid=MSC">Mathematical structures in computer science</a> (journal)</li></ul>
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</style><div id="Mathematical_logic344" style="font-size:114%;margin:0 4em"><a href="Mathematical_logic" title="Mathematical logic">Mathematical logic</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Axiom" title="Axiom">Axiom</a>
<ul><li><a href="List_of_axioms" title="List of axioms">list</a></li></ul></li>
<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>
<ul><li><a href="Three-valued_logic" title="Three-valued logic">3</a></li>
<li><a href="Finite-valued_logic" title="Finite-valued logic">finite</a></li>
<li><a href="Infinite-valued_logic" title="Infinite-valued logic">∞</a></li></ul></li></ul>
</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>
<ul><li><a href="List_of_first-order_theories" title="List of first-order theories"><span style="font-size: 85%;">list</span></a></li></ul></li>
<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="Free_logic" title="Free logic">Free</a></li>
<li><a href="Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li>
<li><a href="Predicate_(mathematical_logic)" class="mw-redirect" title="Predicate (mathematical logic)">Predicate</a></li>
<li><a href="Monadic_predicate_calculus" title="Monadic predicate calculus">Monadic predicate calculus</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Set_theory" title="Set theory">Set theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><td colspan="2" class="navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a>
<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="Open_formula" title="Open formula">open</a></li></ul></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>
<ul><li><a href="Negation" title="Negation">¬</a></li>
<li><a href="Logical_disjunction" title="Logical disjunction">∨</a></li>
<li><a href="Logical_conjunction" title="Logical conjunction">∧</a></li>
<li><a href="Material_conditional" title="Material conditional">→</a></li>
<li><a href="Logical_biconditional" title="Logical biconditional">↔</a></li>
<li><a href="Logical_equality" title="Logical equality">=</a></li></ul></li>
<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>
<ul><li><a href="Existential_quantification" title="Existential quantification">∃</a></li>
<li><a href="Uniqueness_quantification" title="Uniqueness quantification">!</a></li>
<li><a href="Universal_quantification" title="Universal quantification">∀</a></li>
<li><a href="Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li>
<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="Term_(logic)" title="Term (logic)">Term</a></li>
<li><a href="Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a>
<ul><li><a href="List_of_mathematical_theories" title="List of mathematical theories"><span style="font-size: 85%;">list</span></a></li></ul></li></ul>
</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>
<ul><li><a href="List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li>
<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="Type_(model_theory)" title="Type (model theory)">Type</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_(complexity)" title="P (complexity)">P</a></li>
<li><a href="NP_(complexity)" title="NP (complexity)">NP</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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