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<span>The axioms</span> </div> </a> <button aria-controls="toc-The_axioms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle The axioms subsection</span> </button> <ul id="toc-The_axioms-sublist" class="vector-toc-list"> <li id="toc-Fundamental_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fundamental_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Fundamental relations</span> </div> </a> <ul id="toc-Fundamental_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruence_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Congruence axioms</span> </div> </a> <ul id="toc-Congruence_axioms-sublist" class="vector-toc-list"> <li id="toc-Commentary" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Commentary"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Commentary</span> </div> </a> <ul id="toc-Commentary-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Betweenness_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Betweenness_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Betweenness axioms</span> </div> </a> <ul id="toc-Betweenness_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Congruence_and_betweenness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Congruence_and_betweenness"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Congruence and betweenness</span> </div> </a> <ul id="toc-Congruence_and_betweenness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Discussion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Discussion"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Discussion</span> </div> </a> <ul id="toc-Discussion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Comparison_with_Hilbert's_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Comparison_with_Hilbert's_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Comparison with Hilbert's system</span> </div> </a> <ul id="toc-Comparison_with_Hilbert's_system-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Axiom set used in first-order logic</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about axioms for Euclidean geometry. For Tarski's axioms for the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a>, see <a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization of the reals</a>. For Tarski's axioms for <a href="/wiki/Set_theory" title="Set theory">set theory</a>, see <a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck set theory</a>.</div> <p><b>Tarski's axioms</b> are an <a href="/wiki/Axiom" title="Axiom">axiom</a> system for <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>, specifically for that portion of Euclidean geometry that is formulable in <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> with <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a> (i.e. is formulable as an <a href="/wiki/Elementary_theory" title="Elementary theory">elementary theory</a>). As such, it does not require an underlying <a href="/wiki/Set_theory" title="Set theory">set theory</a>. The only primitive objects of the system are "points" and the only primitive predicates are "betweenness" (expressing the fact that a point lies on a line segment between two other points) and "congruence" (expressing the fact that the distance between two points equals the distance between two other points). The system contains infinitely many axioms. </p><p>The axiom system is due to <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> who first presented it in 1926.<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> Other modern axiomizations of Euclidean geometry are <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a> (1899) and <a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's axioms</a> (1932). </p><p>Using his axiom system, Tarski was able to show that the first-order theory of Euclidean geometry is <a href="/wiki/Consistency" title="Consistency">consistent</a>, <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">complete</a> and <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>: every sentence in its language is either provable or disprovable from the axioms, and we have an algorithm which decides for any given sentence whether it is provable or not. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=1" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Early in his career Tarski taught geometry and researched set theory. His coworker Steven Givant (1999) explained Tarski's take-off point: </p> <dl><dd>From Enriques, Tarski learned of the work of <a href="/wiki/Mario_Pieri" title="Mario Pieri">Mario Pieri</a>, an Italian geometer who was strongly influenced by Peano. Tarski preferred Pieri's system [of his <i>Point and Sphere</i> memoir], where the logical structure and the complexity of the axioms were more transparent.</dd></dl> <p>Givant then says that "with typical thoroughness" Tarski devised his system: </p> <dl><dd>What was different about Tarski's approach to geometry? First of all, the axiom system was much simpler than any of the axiom systems that existed up to that time. In fact the length of all of Tarski's axioms together is not much more than just one of Pieri's 24 axioms. It was the first system of Euclidean geometry that was simple enough for all axioms to be expressed in terms of the <a href="/wiki/Primitive_notion" title="Primitive notion">primitive notions</a> only, without the help of defined notions. Of even greater importance, for the first time a clear distinction was made between full geometry and its elementary — that is, its first order — part.</dd></dl> <p>Like other modern axiomatizations of Euclidean geometry, Tarski's employs a <a href="/wiki/Formal_system" title="Formal system">formal system</a> consisting of symbol strings, called <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentences</a>, whose construction respects formal <a href="/wiki/Syntax_(logic)" title="Syntax (logic)">syntactical rules</a>, and rules of proof that determine the allowed manipulations of the sentences. Unlike some other modern axiomatizations, such as <a href="/wiki/Birkhoff%27s_axioms" title="Birkhoff's axioms">Birkhoff's</a> and <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a>, Tarski's axiomatization has no <a href="/wiki/Primitive_object" class="mw-redirect" title="Primitive object">primitive objects</a> other than <i>points</i>, so a variable or constant cannot refer to a line or an angle. Because points are the only primitive objects, and because Tarski's system is a <a href="/wiki/First-order_theory" class="mw-redirect" title="First-order theory">first-order theory</a>, it is not even possible to define lines as sets of points. The only primitive relations (<a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">predicates</a>) are "betweenness" and "congruence" among points. </p><p>Tarski's axiomatization is shorter than its rivals, in a sense Tarski and Givant (1999) make explicit. It is more concise than Pieri's because Pieri had only two primitive notions while Tarski introduced three: point, betweenness, and congruence. Such economy of primitive and defined notions means that Tarski's system is not very convenient for <i>doing</i> Euclidean geometry. Rather, Tarski designed his system to facilitate its analysis via the tools of <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, i.e., to facilitate deriving its metamathematical properties. Tarski's system has the unusual property that all sentences can be written in universal-existential form, a special case of the <a href="/wiki/Prenex_normal_form" title="Prenex normal form">prenex normal form</a>. This form has all <a href="/wiki/Universal_quantification" title="Universal quantification">universal quantifiers</a> preceding any <a href="/wiki/Existential_quantification" title="Existential quantification">existential quantifiers</a>, so that all sentences can be recast in the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall u\forall v\ldots \exists a\exists b\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>u</mi> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo>…</mo> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall u\forall v\ldots \exists a\exists b\dots .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a197b5eaf0af2289b3f37d75a9f37c9df78823d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.496ex; height:2.176ex;" alt="{\displaystyle \forall u\forall v\ldots \exists a\exists b\dots .}"></span> This fact allowed Tarski to prove that Euclidean geometry is <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>: there exists an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> which can determine the truth or falsity of any sentence. Tarski's axiomatization is also <a href="/wiki/Completeness_(logic)" title="Completeness (logic)">complete</a>. This does not contradict <a href="/wiki/G%C3%B6del%27s_first_incompleteness_theorem" class="mw-redirect" title="Gödel's first incompleteness theorem">Gödel's first incompleteness theorem</a>, because Tarski's theory lacks the expressive power needed to interpret <a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson arithmetic</a> (<a href="#CITEREFFranzén2005">Franzén 2005</a>, pp. 25–26). </p> <div class="mw-heading mw-heading2"><h2 id="The_axioms">The axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=2" title="Edit section: The axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> worked on the axiomatization and metamathematics of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> intermittently from 1926 until his death in 1983, with Tarski (1959) heralding his mature interest in the subject. The work of Tarski and his students on Euclidean geometry culminated in the monograph Schwabhäuser, Szmielew, and Tarski (1983), which set out the 10 <a href="/wiki/Axiom" title="Axiom">axioms</a> and one <a href="/wiki/Axiom_schema" title="Axiom schema">axiom schema</a> shown below, the associated <a href="/wiki/Metamathematics" title="Metamathematics">metamathematics</a>, and a fair bit of the subject. Gupta (1965) made important contributions, and Tarski and Givant (1999) discuss the history. </p> <div class="mw-heading mw-heading3"><h3 id="Fundamental_relations">Fundamental relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=3" title="Edit section: Fundamental relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These axioms are a more elegant version of a set Tarski devised in the 1920s as part of his investigation of the metamathematical properties of <a href="/wiki/Euclidean_plane_geometry" class="mw-redirect" title="Euclidean plane geometry">Euclidean plane geometry</a>. This objective required reformulating that geometry as a <a href="/wiki/First-order_logic" title="First-order logic">first-order theory</a>. Tarski did so by positing a <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universe</a> of <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>, with lower case letters denoting variables ranging over that universe. <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">Equality</a> is provided by the underlying logic (see <a href="/wiki/First-order_logic#Equality_and_its_axioms" title="First-order logic">First-order logic#Equality and its axioms</a>).<sup id="cite_ref-FOOTNOTETarskiGivant1999177_2-0" class="reference"><a href="#cite_note-FOOTNOTETarskiGivant1999177-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Tarski then posited two primitive relations: </p> <ul><li><i>Betweenness</i>, a <a href="/wiki/Triadic_relation" class="mw-redirect" title="Triadic relation">triadic relation</a>. The <a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic sentence</a> <i>Bxyz</i> denotes that the point <i>y</i> is "between" the points <i>x</i> and <i>z</i>, in other words, that <i>y</i> is a point on the <a href="/wiki/Line_segment" title="Line segment">line segment</a> <i>xz</i>. (This relation is interpreted inclusively, so that <i>Bxyz</i> is trivially true whenever <i>x=y</i> or <i>y=z</i>).</li> <li><i><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></i> (or "equidistance"), a <a href="/wiki/Polyadic_relation" class="mw-redirect" title="Polyadic relation">tetradic relation</a>. The <a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic sentence</a> <i>Cwxyz</i> or commonly <i>wx</i> ≡ <i>yz</i> can be interpreted as <i>wx</i> is <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> to <i>yz</i>, in other words, that the <a href="/wiki/Distance" title="Distance">length</a> of the line segment <i>wx</i> is equal to the length of the line segment <i>yz</i>.</li></ul> <p>Betweenness captures the <a href="/wiki/Affine_geometry" title="Affine geometry">affine</a> aspect (such as the parallelism of lines) of Euclidean geometry; congruence, its <a href="/wiki/Metric_space" title="Metric space">metric</a> aspect (such as angles and distances). The background logic includes <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identity</a>, a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> denoted by =. </p><p>The axioms below are grouped by the types of relation they invoke, then sorted, first by the number of existential quantifiers, then by the number of atomic sentences. The axioms should be read as <a href="/wiki/Universal_closure" class="mw-redirect" title="Universal closure">universal closures</a>; hence any <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free variables</a> should be taken as tacitly <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universally quantified</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_axioms">Congruence axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=4" title="Edit section: Congruence axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Reflexivity of Congruence</dt> <dd></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv yx\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>y</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv yx\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfaa4acc24869713ec4ed0bc067d77a2d8aab1de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.103ex; height:2.009ex;" alt="{\displaystyle xy\equiv yx\,.}"></span></dd></dl> <dl><dt>Identity of Congruence</dt> <dd></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zz\rightarrow x=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>z</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zz\rightarrow x=y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2c967b15930148595ab965597c60a118f71318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.604ex; height:2.176ex;" alt="{\displaystyle xy\equiv zz\rightarrow x=y.}"></span></dd></dl> <dl><dt><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a> of Congruence</dt> <dd></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xy\equiv zu\land xy\equiv vw)\rightarrow zu\equiv vw.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>u</mi> <mo>∧</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>v</mi> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>z</mi> <mi>u</mi> <mo>≡</mo> <mi>v</mi> <mi>w</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy\equiv zu\land xy\equiv vw)\rightarrow zu\equiv vw.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdd7f1dc82dc9db22756e59a1342e4dea997ad3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.338ex; height:2.843ex;" alt="{\displaystyle (xy\equiv zu\land xy\equiv vw)\rightarrow zu\equiv vw.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Commentary">Commentary</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=5" title="Edit section: Commentary"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While the congruence relation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zw}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zw}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87da06a4263885cd4fb58710fbd14c6c97a7085b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.336ex; height:2.009ex;" alt="{\displaystyle xy\equiv zw}"></span> is, formally, a 4-way relation among points, it may also be thought of, informally, as a binary relation between two line segments <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zw}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zw}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/330b5783edb6401740906e0ce2ae4bde03f4c50d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.752ex; height:1.676ex;" alt="{\displaystyle zw}"></span>. The "Reflexivity" and "Transitivity" axioms above, combined, prove both: </p> <ul><li>that this binary relation is in fact an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> <ul><li>it is reflexive: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac011cc750f96dc97359d8198f2ca806434b6829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.069ex; height:2.009ex;" alt="{\displaystyle xy\equiv xy}"></span>.</li> <li>it is symmetric <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zw\rightarrow zw\equiv xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> <mo stretchy="false">→</mo> <mi>z</mi> <mi>w</mi> <mo>≡</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zw\rightarrow zw\equiv xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4932c664e7e7269eee1123ffc8bb79258b9a8336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.286ex; height:2.176ex;" alt="{\displaystyle xy\equiv zw\rightarrow zw\equiv xy}"></span>.</li> <li>it is transitive <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xy\equiv zu\land zu\equiv vw)\rightarrow xy\equiv vw}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>u</mi> <mo>∧</mo> <mi>z</mi> <mi>u</mi> <mo>≡</mo> <mi>v</mi> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>v</mi> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy\equiv zu\land zu\equiv vw)\rightarrow xy\equiv vw}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7042bde3d85c14793766ac2169eddf86a3d43158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.691ex; height:2.843ex;" alt="{\displaystyle (xy\equiv zu\land zu\equiv vw)\rightarrow xy\equiv vw}"></span>.</li></ul></li> <li>and that the order in which the points of a line segment are specified is irrelevant. <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zw\rightarrow xy\equiv wz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>w</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zw\rightarrow xy\equiv wz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d05ff0526ce8688eac56401a717b0cc405dccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.286ex; height:2.176ex;" alt="{\displaystyle xy\equiv zw\rightarrow xy\equiv wz}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zw\rightarrow yx\equiv zw}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> <mo stretchy="false">→</mo> <mi>y</mi> <mi>x</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zw\rightarrow yx\equiv zw}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e0870ee7747c8372cb6a9be51585c6944263f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.286ex; height:2.176ex;" alt="{\displaystyle xy\equiv zw\rightarrow yx\equiv zw}"></span>.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\equiv zw\rightarrow yx\equiv wz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>z</mi> <mi>w</mi> <mo stretchy="false">→</mo> <mi>y</mi> <mi>x</mi> <mo>≡</mo> <mi>w</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\equiv zw\rightarrow yx\equiv wz}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0aef3f796ad79d70f1f1a9e36dc336c5b3ea8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.286ex; height:2.176ex;" alt="{\displaystyle xy\equiv zw\rightarrow yx\equiv wz}"></span>.</li></ul></li></ul> <p>The "transitivity" axiom asserts that congruence is <a href="/wiki/Euclidean_relation" title="Euclidean relation">Euclidean</a>, in that it respects the first of <a href="/wiki/Euclid%27s_elements" class="mw-redirect" title="Euclid's elements">Euclid's</a> "<a href="/wiki/Euclid%27s_axioms#Axiomatic_approach" class="mw-redirect" title="Euclid's axioms">common notions</a>". </p><p>The "Identity of Congruence" axiom states, intuitively, that if <i>xy</i> is congruent with a segment that begins and ends at the same point, <i>x</i> and <i>y</i> are the same point. This is closely related to the notion of <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexivity</a> for <a href="/wiki/Binary_relation" title="Binary relation">binary relations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Betweenness_axioms">Betweenness axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=6" title="Edit section: Betweenness axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tarski%27s_formulation_of_Pasch%27s_axiom.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Tarski%27s_formulation_of_Pasch%27s_axiom.svg/220px-Tarski%27s_formulation_of_Pasch%27s_axiom.svg.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Tarski%27s_formulation_of_Pasch%27s_axiom.svg/330px-Tarski%27s_formulation_of_Pasch%27s_axiom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Tarski%27s_formulation_of_Pasch%27s_axiom.svg/440px-Tarski%27s_formulation_of_Pasch%27s_axiom.svg.png 2x" data-file-width="830" data-file-height="468" /></a><figcaption>Pasch's axiom</figcaption></figure> <dl><dt>Identity of Betweenness</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Bxyx\rightarrow x=y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mo stretchy="false">→</mo> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Bxyx\rightarrow x=y.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea24f598a560d12af73623ffd3d0e76d38df448e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.423ex; height:2.509ex;" alt="{\displaystyle Bxyx\rightarrow x=y.}"></span></dd></dl> <p>The only point on the line segment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01942dcb4131a025f2584b13809d039803492ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.659ex; height:1.676ex;" alt="{\displaystyle xx}"></span> is <span class="mwe-math-element"><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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> itself. </p> <dl><dt><a href="/wiki/Axiom_of_Pasch" class="mw-redirect" title="Axiom of Pasch">Axiom of Pasch</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Bxuz\land Byvz)\rightarrow \exists a\,(Buay\land Bvax).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>u</mi> <mi>z</mi> <mo>∧</mo> <mi>B</mi> <mi>y</mi> <mi>v</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>B</mi> <mi>u</mi> <mi>a</mi> <mi>y</mi> <mo>∧</mo> <mi>B</mi> <mi>v</mi> <mi>a</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Bxuz\land Byvz)\rightarrow \exists a\,(Buay\land Bvax).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44bd887cfe123f97a8a6a803fc368ae959252545" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.531ex; height:2.843ex;" alt="{\displaystyle (Bxuz\land Byvz)\rightarrow \exists a\,(Buay\land Bvax).}"></span></dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tarski%27s_continuity_axiom.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tarski%27s_continuity_axiom.svg/220px-Tarski%27s_continuity_axiom.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tarski%27s_continuity_axiom.svg/330px-Tarski%27s_continuity_axiom.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Tarski%27s_continuity_axiom.svg/440px-Tarski%27s_continuity_axiom.svg.png 2x" data-file-width="497" data-file-height="304" /></a><figcaption>Continuity: φ and ψ divide the ray into two halves and the axiom asserts the existence of a point b dividing those two halves</figcaption></figure> <dl><dt><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> of Continuity</dt></dl> <p>Let φ(<i>x</i>) and ψ(<i>y</i>) be <a href="/wiki/First_order_logic" class="mw-redirect" title="First order logic">first-order formulae</a> containing no <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free instances</a> of either <i>a</i> or <i>b</i>. Let there also be no free instances of <i>x</i> in ψ(<i>y</i>) or of <i>y</i> in φ(<i>x</i>). Then all instances of the following schema are axioms: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>B</mi> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>B</mi> <mi>x</mi> <mi>b</mi> <mi>y</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59ae3e756b25e36737ce1b1a821a1eccb189f93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.866ex; height:2.843ex;" alt="{\displaystyle \exists a\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Baxy]\rightarrow \exists b\,\forall x\,\forall y\,[(\phi (x)\land \psi (y))\rightarrow Bxby].}"></span></dd></dl> <p>Let <i>r</i> be a ray with endpoint <i>a</i>. Let the first order formulae φ and ψ define subsets <i>X</i> and <i>Y</i> of <i>r</i>, such that every point in <i>Y</i> is to the right of every point of <i>X</i> (with respect to <i>a</i>). Then there exists a point <i>b</i> in <i>r</i> lying between <i>X</i> and <i>Y</i>. This is essentially the <a href="/wiki/Dedekind_cut" title="Dedekind cut">Dedekind cut</a> construction, carried out in a way that avoids quantification over sets. </p><p>Note that the formulae φ(<i>x</i>) and ψ(<i>y</i>) may contain parameters, i.e. free variables different from <i>a</i>, <i>b</i>, <i>x,</i> <i>y</i>. And indeed, each instance of the axiom scheme that does not contain parameters can be proven from the other axioms.<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> </p> <dl><dt>Lower <a href="/wiki/Dimension" title="Dimension">Dimension</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists a\,\exists b\,\exists c\,[\neg Babc\land \neg Bbca\land \neg Bcab].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>c</mi> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mi>b</mi> <mi>c</mi> <mi>a</mi> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi>B</mi> <mi>c</mi> <mi>a</mi> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists a\,\exists b\,\exists c\,[\neg Babc\land \neg Bbca\land \neg Bcab].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd22fec38e8bcb55b3e60eb02646168494f74830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.024ex; height:2.843ex;" alt="{\displaystyle \exists a\,\exists b\,\exists c\,[\neg Babc\land \neg Bbca\land \neg Bcab].}"></span></dd></dl> <p>There exist three noncollinear points. Without this axiom, the theory could be <a href="/wiki/Model_theory" title="Model theory">modeled</a> by the one-dimensional <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a>, a single point, or even the empty set. </p> <div class="mw-heading mw-heading3"><h3 id="Congruence_and_betweenness">Congruence and betweenness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=7" title="Edit section: Congruence and betweenness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg/220px-Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg.png" decoding="async" width="220" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg/330px-Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg/440px-Points_in_a_plane_equidistant_to_two_given_points_lie_on_a_line.svg.png 2x" data-file-width="433" data-file-height="387" /></a><figcaption>Upper dimension axiom</figcaption></figure> <dl><dt>Upper <a href="/wiki/Dimension" title="Dimension">Dimension</a></dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (xu\equiv xv)\land (yu\equiv yv)\land (zu\equiv zv)\land (u\neq v)\rightarrow (Bxyz\lor Byzx\lor Bzxy).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>u</mi> <mo>≡</mo> <mi>x</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mi>u</mi> <mo>≡</mo> <mi>y</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mi>u</mi> <mo>≡</mo> <mi>z</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mo>≠</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>∨</mo> <mi>B</mi> <mi>y</mi> <mi>z</mi> <mi>x</mi> <mo>∨</mo> <mi>B</mi> <mi>z</mi> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xu\equiv xv)\land (yu\equiv yv)\land (zu\equiv zv)\land (u\neq v)\rightarrow (Bxyz\lor Byzx\lor Bzxy).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61bfd980fba80e98f76eb11e6cd9e3efde1ad9d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.602ex; height:2.843ex;" alt="{\displaystyle (xu\equiv xv)\land (yu\equiv yv)\land (zu\equiv zv)\land (u\neq v)\rightarrow (Bxyz\lor Byzx\lor Bzxy).}"></span></dd></dl> <p>Three points equidistant from two distinct points form a line. Without this axiom, the theory could be modeled by <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> or higher-dimensional space. </p> <dl><dt>Axiom of Euclid</dt></dl> <p>Three variants of this axiom can be given, labeled A, B and C below. They are equivalent to each other given the remaining Tarski's axioms, and indeed equivalent to Euclid's <a href="/wiki/Parallel_postulate" title="Parallel postulate">parallel postulate</a>. </p> <dl><dd><b>A</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((Bxyw\land xy\equiv yw)\land (Bxuv\land xu\equiv uv)\land (Byuz\land yu\equiv uz))\rightarrow yz\equiv vw.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>w</mi> <mo>∧</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <mi>y</mi> <mi>w</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>u</mi> <mi>v</mi> <mo>∧</mo> <mi>x</mi> <mi>u</mi> <mo>≡</mo> <mi>u</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>y</mi> <mi>u</mi> <mi>z</mi> <mo>∧</mo> <mi>y</mi> <mi>u</mi> <mo>≡</mo> <mi>u</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>y</mi> <mi>z</mi> <mo>≡</mo> <mi>v</mi> <mi>w</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((Bxyw\land xy\equiv yw)\land (Bxuv\land xu\equiv uv)\land (Byuz\land yu\equiv uz))\rightarrow yz\equiv vw.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/821186c08add6c296e147776931781b1296475b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.966ex; height:2.843ex;" alt="{\displaystyle ((Bxyw\land xy\equiv yw)\land (Bxuv\land xu\equiv uv)\land (Byuz\land yu\equiv uz))\rightarrow yz\equiv vw.}"></span></dd></dl> <p>Let a line segment join the midpoint of two sides of a given <a href="/wiki/Triangle" title="Triangle">triangle</a>. That line segment will be half as long as the third side. This is equivalent to the <a href="/wiki/Interior_angle" class="mw-redirect" title="Interior angle">interior angles</a> of any triangle summing to two <a href="/wiki/Right_angles" class="mw-redirect" title="Right angles">right angles</a>. </p> <dl><dd><b>B</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Bxyz\lor Byzx\lor Bzxy\lor \exists a\,(xa\equiv ya\land xa\equiv za).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>∨</mo> <mi>B</mi> <mi>y</mi> <mi>z</mi> <mi>x</mi> <mo>∨</mo> <mi>B</mi> <mi>z</mi> <mi>x</mi> <mi>y</mi> <mo>∨</mo> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>x</mi> <mi>a</mi> <mo>≡</mo> <mi>y</mi> <mi>a</mi> <mo>∧</mo> <mi>x</mi> <mi>a</mi> <mo>≡</mo> <mi>z</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Bxyz\lor Byzx\lor Bzxy\lor \exists a\,(xa\equiv ya\land xa\equiv za).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a783d83e4f2b2992b5a9cb1ce4736a6cdc3d0a40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.727ex; height:2.843ex;" alt="{\displaystyle Bxyz\lor Byzx\lor Bzxy\lor \exists a\,(xa\equiv ya\land xa\equiv za).}"></span></dd></dl> <p>Given any <a href="/wiki/Triangle" title="Triangle">triangle</a>, there exists a <a href="/wiki/Circle" title="Circle">circle</a> that includes all of its vertices. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Tarski%27s_axiom_of_Euclid_C.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Tarski%27s_axiom_of_Euclid_C.svg/220px-Tarski%27s_axiom_of_Euclid_C.svg.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Tarski%27s_axiom_of_Euclid_C.svg/330px-Tarski%27s_axiom_of_Euclid_C.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Tarski%27s_axiom_of_Euclid_C.svg/440px-Tarski%27s_axiom_of_Euclid_C.svg.png 2x" data-file-width="514" data-file-height="509" /></a><figcaption>Axiom of Euclid: C</figcaption></figure> <dl><dd><b>C</b>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Bxuv\land Byuz\land x\neq u)\rightarrow \exists a\,\exists b\,(Bxya\land Bxzb\land Bavb).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>u</mi> <mi>v</mi> <mo>∧</mo> <mi>B</mi> <mi>y</mi> <mi>u</mi> <mi>z</mi> <mo>∧</mo> <mi>x</mi> <mo>≠</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∃</mi> <mi>b</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>a</mi> <mo>∧</mo> <mi>B</mi> <mi>x</mi> <mi>z</mi> <mi>b</mi> <mo>∧</mo> <mi>B</mi> <mi>a</mi> <mi>v</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Bxuv\land Byuz\land x\neq u)\rightarrow \exists a\,\exists b\,(Bxya\land Bxzb\land Bavb).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb661de7979f04e8ff227666a88a8ccec4f980d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.22ex; height:2.843ex;" alt="{\displaystyle (Bxuv\land Byuz\land x\neq u)\rightarrow \exists a\,\exists b\,(Bxya\land Bxzb\land Bavb).}"></span></dd></dl> <p>Given any <a href="/wiki/Angle" title="Angle">angle</a> and any point <i>v</i> in its interior, there exists a line segment including <i>v</i>, with an endpoint on each side of the angle. </p><p>Each variant has an advantage over the others: </p> <ul><li><b>A</b> dispenses with <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential quantifiers</a>;</li> <li><b>B</b> has the fewest variables and <a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic sentences</a>;</li> <li><b>C</b> requires but one primitive notion, betweenness. This variant is the usual one given in the literature.</li></ul> <dl><dt>Five Segment</dt></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Five_segment.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Five_segment.svg/220px-Five_segment.svg.png" decoding="async" width="220" height="289" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Five_segment.svg/330px-Five_segment.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Five_segment.svg/440px-Five_segment.svg.png 2x" data-file-width="578" data-file-height="760" /></a><figcaption>Five segment</figcaption></figure> <dl><dd><span class="mwe-math-element"><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\neq y\land Bxyz\land Bx'y'z'\land xy\equiv x'y'\land yz\equiv y'z'\land xu\equiv x'u'\land yu\equiv y'u')}\rightarrow zu\equiv z'u'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo>∧</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>∧</mo> <mi>B</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>∧</mo> <mi>x</mi> <mi>y</mi> <mo>≡</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>y</mi> <mo>′</mo> </msup> <mo>∧</mo> <mi>y</mi> <mi>z</mi> <mo>≡</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>z</mi> <mo>′</mo> </msup> <mo>∧</mo> <mi>x</mi> <mi>u</mi> <mo>≡</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>∧</mo> <mi>y</mi> <mi>u</mi> <mo>≡</mo> <msup> <mi>y</mi> <mo>′</mo> </msup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <mi>z</mi> <mi>u</mi> <mo>≡</mo> <msup> <mi>z</mi> <mo>′</mo> </msup> <msup> <mi>u</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {(x\neq y\land Bxyz\land Bx'y'z'\land xy\equiv x'y'\land yz\equiv y'z'\land xu\equiv x'u'\land yu\equiv y'u')}\rightarrow zu\equiv z'u'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23638f5c0f6281e6e94a5d3d3798a04f4a35bb7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:86.827ex; height:3.009ex;" alt="{\displaystyle {(x\neq y\land Bxyz\land Bx'y'z'\land xy\equiv x'y'\land yz\equiv y'z'\land xu\equiv x'u'\land yu\equiv y'u')}\rightarrow zu\equiv z'u'.}"></span></dd></dl> <p>Begin with two <a href="/wiki/Triangle" title="Triangle">triangles</a>, <i>xuz</i> and <i>x'u'z'.</i> Draw the line segments <i>yu</i> and <i>y'u',</i> connecting a vertex of each triangle to a point on the side opposite to the vertex. The result is two divided triangles, each made up of five segments. If four segments of one triangle are each <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> to a segment in the other triangle, then the fifth segments in both triangles must be congruent. </p><p>This is equivalent to the <a href="/wiki/Congruence_(geometry)#Determining_congruence" title="Congruence (geometry)">side-angle-side</a> rule for determining that two triangles are congruent; if the angles <i>uxz</i> and <i>u'x'z'</i> are congruent (there exist congruent triangles <i>xuz</i> and <i>x'u'z'</i>), and the two pairs of incident sides are congruent (<i>xu ≡ x'u'</i> and <i>xz ≡ x'z'</i>), then the remaining pair of sides is also congruent (<i>uz ≡ u'z'</i>). </p> <dl><dt>Segment Construction</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exists z\,[Bxyz\land yz\equiv ab].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>z</mi> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>∧</mo> <mi>y</mi> <mi>z</mi> <mo>≡</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists z\,[Bxyz\land yz\equiv ab].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af5dc21bc5573743744ec88249eb2043f46cc35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.198ex; height:2.843ex;" alt="{\displaystyle \exists z\,[Bxyz\land yz\equiv ab].}"></span></dd></dl> <p>For any point <i>y</i>, it is possible to draw in any direction (determined by <i>x</i>) a line congruent to any segment <i>ab</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Discussion">Discussion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=8" title="Edit section: Discussion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to Tarski and Givant (1999: 192-93), none of the above <a href="/wiki/Axiom" title="Axiom">axioms</a> are fundamentally new. The first four axioms establish some elementary properties of the two primitive relations. For instance, Reflexivity and Transitivity of Congruence establish that congruence is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> over line segments. The Identity of Congruence and of Betweenness govern the trivial case when those relations are applied to nondistinct points. The theorem <i>xy</i>≡<i>zz</i> ↔ <i>x</i>=<i>y</i> ↔ <i>Bxyx</i> extends these Identity axioms. </p><p>A number of other properties of Betweenness are derivable as theorems<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> including: </p> <ul><li><a href="/wiki/Reflexive_relation" title="Reflexive relation">Reflexivity</a>: <i>Bxxy</i> ;</li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a>: <i>Bxyz</i> → <i>Bzyx</i> ;</li> <li><a href="/wiki/Transitive_relation" title="Transitive relation">Transitivity</a>: (<i>Bxyw</i> ∧ <i>Byzw</i>) → <i>Bxyz</i> ;</li> <li><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connectivity</a>: (<i>Bxyw</i> ∧ <i>Bxzw</i>) → (<i>Bxyz</i> ∨ <i>Bxzy</i>).</li></ul> <p>The last two properties <a href="/wiki/Total_order" title="Total order">totally order</a> the points making up a line segment. </p><p>The Upper and Lower Dimension axioms together require that any model of these axioms have dimension 2, i.e. that we are axiomatizing the Euclidean plane. Suitable changes in these axioms yield axiom sets for <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> for <a href="/wiki/Dimension" title="Dimension">dimensions</a> 0, 1, and greater than 2 (Tarski and Givant 1999: Axioms 8<sup>(1)</sup>, 8<sup>(n)</sup>, 9<sup>(0)</sup>, 9<sup>(1)</sup>, 9<sup>(n)</sup> ). Note that <a href="/wiki/Solid_geometry" title="Solid geometry">solid geometry</a> requires no new axioms, unlike the case with <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a>. Moreover, Lower Dimension for <i>n</i> dimensions is simply the negation of Upper Dimension for <i>n</i> - 1 dimensions. </p><p>When the number of dimensions is greater than 1, Betweenness can be defined in terms of <a href="/wiki/Congruence_relation" title="Congruence relation">congruence</a> (Tarski and Givant, 1999). First define the relation "≤" (where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab\leq cd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>≤</mo> <mi>c</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab\leq cd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee4ecd431aa207cc329129c34c8c7dce41fc987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.343ex;" alt="{\displaystyle ab\leq cd}"></span> is interpreted "the length of line segment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"></span> is less than or equal to the length of line segment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e251933b871689a13ef7a1a14a0cead68454b8d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.223ex; height:2.176ex;" alt="{\displaystyle cd}"></span>"): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\leq zu\leftrightarrow \forall v(zv\equiv uv\rightarrow \exists w(xw\equiv yw\land yw\equiv uv)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>≤</mo> <mi>z</mi> <mi>u</mi> <mo stretchy="false">↔</mo> <mi mathvariant="normal">∀</mi> <mi>v</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mi>v</mi> <mo>≡</mo> <mi>u</mi> <mi>v</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∃</mi> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mi>w</mi> <mo>≡</mo> <mi>y</mi> <mi>w</mi> <mo>∧</mo> <mi>y</mi> <mi>w</mi> <mo>≡</mo> <mi>u</mi> <mi>v</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\leq zu\leftrightarrow \forall v(zv\equiv uv\rightarrow \exists w(xw\equiv yw\land yw\equiv uv)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd31360679e7c033d38626df330d8da91c76a45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.513ex; height:2.843ex;" alt="{\displaystyle xy\leq zu\leftrightarrow \forall v(zv\equiv uv\rightarrow \exists w(xw\equiv yw\land yw\equiv uv)).}"></span></dd></dl> <p>In the case of two dimensions, the intuition is as follows: For any line segment <i>xy</i>, consider the possible range of lengths of <i>xv</i>, where <i>v</i> is any point on the perpendicular bisector of <i>xy</i>. It is apparent that while there is no upper bound to the length of <i>xv</i>, there is a lower bound, which occurs when <i>v</i> is the midpoint of <i>xy</i>. So if <i>xy</i> is shorter than or equal to <i>zu</i>, then the range of possible lengths of <i>xv</i> will be a superset of the range of possible lengths of <i>zw</i>, where <i>w</i> is any point on the perpendicular bisector of <i>zu</i>. </p><p>Betweenness can then be defined by using the intuition that the shortest distance between any two points is a straight line: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Bxyz\leftrightarrow \forall u((ux\leq xy\land uz\leq zy)\rightarrow u=y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">↔</mo> <mi mathvariant="normal">∀</mi> <mi>u</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>≤</mo> <mi>x</mi> <mi>y</mi> <mo>∧</mo> <mi>u</mi> <mi>z</mi> <mo>≤</mo> <mi>z</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>u</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Bxyz\leftrightarrow \forall u((ux\leq xy\land uz\leq zy)\rightarrow u=y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97f699a77b32b57603628f760c19906dd7dc1b04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.622ex; height:2.843ex;" alt="{\displaystyle Bxyz\leftrightarrow \forall u((ux\leq xy\land uz\leq zy)\rightarrow u=y).}"></span></dd></dl> <p>The Axiom Schema of Continuity assures that the ordering of points on a line is <a href="/wiki/Dedekind_complete" class="mw-redirect" title="Dedekind complete">complete</a> (with respect to first-order definable properties). As was pointed out by Tarski, this first-order axiom schema may be replaced by a more powerful <a href="/wiki/Second-order_logic" title="Second-order logic">second-order</a> Axiom of Continuity if one allows for variables to refer to arbitrary sets of points. The resulting second-order system is equivalent to Hilbert's set of axioms. (Tarski and Givant 1999) </p><p>The Axioms of <a href="/wiki/Pasch%27s_axiom" title="Pasch's axiom">Pasch</a> and Euclid are well known. The <i>Segment Construction</i> axiom makes <a href="/wiki/Measurement" title="Measurement">measurement</a> and the <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">Cartesian coordinate system</a> possible—simply assign the length 1 to some arbitrary non-empty line segment. Indeed, it is shown in (Schwabhäuser 1983) that by specifying two distinguished points on a line, called 0 and 1, we can define an addition, multiplication and ordering, turning the set of points on that line into a <a href="/wiki/Real_closed_field" title="Real closed field">real-closed ordered field</a>. We can then introduce coordinates from this field, showing that every model of Tarski's axioms is isomorphic to the two-dimensional plane over some real-closed ordered field. </p><p>The standard geometric notions of parallelism and intersection of lines (where lines are represented by two distinct points on them), right angles, congruence of angles, similarity of triangles, tangency of lines and circles (represented by a center point and a radius) can all be defined in Tarski's system. </p><p>Let <i>wff</i> stand for a <a href="/wiki/Well-formed_formula" title="Well-formed formula">well-formed formula</a> (or syntactically correct first-order formula) in Tarski's system. Tarski and Givant (1999: 175) proved that Tarski's system is: </p> <ul><li><a href="/wiki/Consistency" title="Consistency">Consistent</a>: There is no wff such that it and its negation can both be proven from the axioms;</li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete</a>: Every wff or its negation is a theorem provable from the axioms;</li> <li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">Decidable</a>: There exists an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> that decides for every wff whether is it is provable or disprovable from the axioms. This follows from Tarski's: <ul><li><a href="/wiki/Decision_procedure" class="mw-redirect" title="Decision procedure">Decision procedure</a> for the <a href="/wiki/Real_closed_field" title="Real closed field">real closed field</a>, which he found by <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a> (the <a href="/wiki/Tarski%E2%80%93Seidenberg_theorem" title="Tarski–Seidenberg theorem">Tarski–Seidenberg theorem</a>);</li> <li>Axioms admitting the above-mentioned representation as a two-dimensional plane over a <a href="/wiki/Real_closed_field" title="Real closed field">real closed field</a>.</li></ul></li></ul> <p>This has the consequence that every statement of (second-order, general) Euclidean geometry which can be formulated as a first-order sentence in Tarski's system is true if and only if it is provable in Tarski's system, and this provability can be automatically checked with Tarski's algorithm. This, for instance, applies to all theorems in <a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Euclid's Elements</a>, Book I. An example of a theorem of Euclidean geometry which cannot be so formulated is the <a href="/wiki/Archimedean_property" title="Archimedean property">Archimedean property</a>: to any two positive-length line segments <i>S</i><sub>1</sub> and <i>S</i><sub>2</sub> there exists a natural number <i>n</i> such that <i>nS</i><sub>1</sub> is longer than <i>S</i><sub>2</sub>. (This is a consequence of the fact that there are real-closed fields that contain infinitesimals.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>) Other notions that cannot be expressed in Tarski's system are the <a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">constructability with straightedge and compass</a> and statements that talk about "all polygones" etc.<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> </p><p>Gupta (1965) proved the Tarski's axioms independent, excepting <i>Pasch</i> and <i>Reflexivity of Congruence</i>. </p><p>Negating the Axiom of Euclid yields <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, while eliminating it outright yields <a href="/wiki/Absolute_geometry" title="Absolute geometry">absolute geometry</a>. Full (as opposed to elementary) Euclidean geometry requires giving up a first order axiomatization: replace φ(<i>x</i>) and ψ(<i>y</i>) in the axiom schema of Continuity with <i>x</i> ∈ <i>A</i> and <i>y</i> ∈ <i>B</i>, where <i>A</i> and <i>B</i> are universally quantified variables ranging over sets of points. </p> <div class="mw-heading mw-heading2"><h2 id="Comparison_with_Hilbert's_system"><span id="Comparison_with_Hilbert.27s_system"></span>Comparison with Hilbert's system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=9" title="Edit section: Comparison with Hilbert's system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a> for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch. The only notion from intuitive geometry invoked in the remarks to Tarski's axioms is <a href="/wiki/Triangle" title="Triangle">triangle</a>. (Versions <b>B</b> and <b>C</b> of the Axiom of Euclid refer to "circle" and "angle," respectively.) Hilbert's axioms also require "ray," "angle," and the notion of a triangle "including" an angle. In addition to betweenness and congruence, Hilbert's axioms require a primitive <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> "on," linking a point and a line. </p><p>Hilbert uses two axioms of Continuity, and they require <a href="/wiki/Second-order_logic" title="Second-order logic">second-order logic</a>. By contrast, Tarski's <a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a> of Continuity consists of infinitely many first-order axioms. Such a schema is indispensable; Euclidean geometry in Tarski's (or equivalent) language cannot be finitely axiomatized as a <a href="/wiki/First-order_logic" title="First-order logic">first-order theory</a>. </p><p>Hilbert's system is therefore considerably stronger: every <a href="/wiki/Model_theory" title="Model theory">model</a> is isomorphic to the real plane <span class="mwe-math-element"><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} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> (using the standard notions of points and lines). By contrast, Tarski's system has many non-isomorphic models: for every real-closed field <i>F</i>, the plane <i>F<sup>2</sup></i> provides one such model (where betweenness and congruence are defined in the obvious way).<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> </p><p>The first four groups of axioms of <a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's axioms</a> for plane geometry are bi-interpretable with Tarski's axioms minus continuity. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li> <li><a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=11" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Tarski 1959, Tarski and Givant 1999</span> </li> <li id="cite_note-FOOTNOTETarskiGivant1999177-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETarskiGivant1999177_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTarskiGivant1999">Tarski & Givant 1999</a>, p. 177.</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">Schwabhäuser 1983, p. 287-288</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">Tarski and Givant 1999, p. 189</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Greenberg 2010</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMcNaughton,_Robert1953" class="citation journal cs1">McNaughton, Robert (1953). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1953-59-01/S0002-9904-1953-09664-1/S0002-9904-1953-09664-1.pdf">"Review: <i>A decision method for elementary algebra and geometry</i> by A. Tarski"</a> <span class="cs1-format">(PDF)</span>. <i>Bull. Amer. Math. Soc</i>. <b>59</b> (1): 91–93. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9904-1953-09664-1">10.1090/s0002-9904-1953-09664-1</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Review%3A+A+decision+method+for+elementary+algebra+and+geometry+by+A.+Tarski&rft.volume=59&rft.issue=1&rft.pages=91-93&rft.date=1953&rft_id=info%3Adoi%2F10.1090%2Fs0002-9904-1953-09664-1&rft.au=McNaughton%2C+Robert&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1953-59-01%2FS0002-9904-1953-09664-1%2FS0002-9904-1953-09664-1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Schwabhäuser 1983, section I.16</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Tarski%27s_axioms&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranzén2005" class="citation cs2"><a href="/wiki/Torkel_Franz%C3%A9n" title="Torkel Franzén">Franzén, Torkel</a> (2005), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/gdelstheoreminco0000fran"><i>Gödel's Theorem: An Incomplete Guide to Its Use and Abuse</i></a></span>, A K Peters, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56881-238-8" title="Special:BookSources/1-56881-238-8"><bdi>1-56881-238-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%B6del%27s+Theorem%3A+An+Incomplete+Guide+to+Its+Use+and+Abuse&rft.pub=A+K+Peters&rft.date=2005&rft.isbn=1-56881-238-8&rft.aulast=Franz%C3%A9n&rft.aufirst=Torkel&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgdelstheoreminco0000fran&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGivant1999" class="citation journal cs1">Givant, Steven (1 December 1999). <a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF03024832">"Unifying threads in Alfred Tarski's Work"</a>. <i><a href="/wiki/The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a></i>. <b>21</b> (1): 47–58. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03024832">10.1007/BF03024832</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1866-7414">1866-7414</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119716413">119716413</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Mathematical+Intelligencer&rft.atitle=Unifying+threads+in+Alfred+Tarski%27s+Work&rft.volume=21&rft.issue=1&rft.pages=47-58&rft.date=1999-12-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119716413%23id-name%3DS2CID&rft.issn=1866-7414&rft_id=info%3Adoi%2F10.1007%2FBF03024832&rft.aulast=Givant&rft.aufirst=Steven&rft_id=https%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2FBF03024832&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenberg2010" class="citation journal cs1">Greenberg, Marvin Jay (2010). <a rel="nofollow" class="external text" href="https://maa.org/sites/default/files/pdf/upload_library/22/Ford/Greenberg2011.pdf">"Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries"</a> <span class="cs1-format">(PDF)</span>. <i>The American Mathematical Monthly</i>. <b>117</b> (3): 198. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F000298910x480063">10.4169/000298910x480063</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Old+and+New+Results+in+the+Foundations+of+Elementary+Plane+Euclidean+and+Non-Euclidean+Geometries&rft.volume=117&rft.issue=3&rft.pages=198&rft.date=2010&rft_id=info%3Adoi%2F10.4169%2F000298910x480063&rft.aulast=Greenberg&rft.aufirst=Marvin+Jay&rft_id=https%3A%2F%2Fmaa.org%2Fsites%2Fdefault%2Ffiles%2Fpdf%2Fupload_library%2F22%2FFord%2FGreenberg2011.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGupta1965" class="citation thesis cs1">Gupta, H. N. (1965). <a rel="nofollow" class="external text" href="https://www.proquest.com/openview/afb5fc16333dd3ea8370405ab8cd5886/1?pq-origsite=gscholar&cbl=18750&diss=y"><i>Contributions to the Axiomatic Foundations of Geometry</i></a> (Ph.D. thesis). University of California-Berkeley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Contributions+to+the+Axiomatic+Foundations+of+Geometry&rft.inst=University+of+California-Berkeley&rft.date=1965&rft.aulast=Gupta&rft.aufirst=H.+N.&rft_id=https%3A%2F%2Fwww.proquest.com%2Fopenview%2Fafb5fc16333dd3ea8370405ab8cd5886%2F1%3Fpq-origsite%3Dgscholar%26cbl%3D18750%26diss%3Dy&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarski1959" class="citation cs2"><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1959), "What is elementary geometry?", in Leon Henkin, Patrick Suppes and Alfred Tarski (ed.), <i>The axiomatic method. With special reference to geometry and physics. Proceedings of an International Symposium held at the Univ. of Calif., Berkeley, Dec. 26, 1957-Jan. 4, 1958</i>, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, pp. 16–29, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0106185">0106185</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=What+is+elementary+geometry%3F&rft.btitle=The+axiomatic+method.+With+special+reference+to+geometry+and+physics.+Proceedings+of+an+International+Symposium+held+at+the+Univ.+of+Calif.%2C+Berkeley%2C+Dec.+26%2C+1957-Jan.+4%2C+1958&rft.place=Amsterdam&rft.series=Studies+in+Logic+and+the+Foundations+of+Mathematics&rft.pages=16-29&rft.pub=North-Holland&rft.date=1959&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0106185%23id-name%3DMR&rft.aulast=Tarski&rft.aufirst=Alfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span>. <ul><li>Available as a 2007 <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eVVKtnKzfnUC&pg=PA16">reprint</a>, Brouwer Press, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4437-2812-8" title="Special:BookSources/1-4437-2812-8">1-4437-2812-8</a></li></ul></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTarskiGivant1999" class="citation cs2"><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a>; Givant, Steven (1999), <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ae7daa12902dd37449fbfd21ff25e9a611e83212">"Tarski's system of geometry"</a>, <i>The Bulletin of Symbolic Logic</i>, <b>5</b> (2): 175–214, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012">10.1.1.27.9012</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F421089">10.2307/421089</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1079-8986">1079-8986</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/421089">421089</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1791303">1791303</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18551419">18551419</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Bulletin+of+Symbolic+Logic&rft.atitle=Tarski%27s+system+of+geometry&rft.volume=5&rft.issue=2&rft.pages=175-214&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18551419%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.2307%2F421089&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.27.9012%23id-name%3DCiteSeerX&rft.issn=1079-8986&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F421089%23id-name%3DJSTOR&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1791303%23id-name%3DMR&rft.aulast=Tarski&rft.aufirst=Alfred&rft.au=Givant%2C+Steven&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fdocument%3Frepid%3Drep1%26type%3Dpdf%26doi%3Dae7daa12902dd37449fbfd21ff25e9a611e83212&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwabhäuserSzmielewTarski1983" class="citation book cs1">Schwabhäuser, W.; <a href="/wiki/Wanda_Szmielew" title="Wanda Szmielew">Szmielew, W.</a>; <a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski, Alfred</a> (1983). <i>Metamathematische Methoden in der Geometrie</i>. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metamathematische+Methoden+in+der+Geometrie&rft.pub=Springer-Verlag&rft.date=1983&rft.aulast=Schwabh%C3%A4user&rft.aufirst=W.&rft.au=Szmielew%2C+W.&rft.au=Tarski%2C+Alfred&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATarski%27s+axioms" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzczerba1986" class="citation journal cs1">Szczerba, L. W. (1986). <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/pdf/2273904.pdf">"Tarski and Geometry"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Symbolic Logic</i>. <b>51</b> (4): 907–12. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2273904">10.2307/2273904</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2273904">2273904</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:35275962">35275962</a>.</cite><span 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href="/wiki/Template:Mathematical_logic" title="Template:Mathematical logic"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Mathematical_logic" title="Template talk:Mathematical logic"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Mathematical_logic" title="Special:EditPage/Template:Mathematical logic"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Mathematical_logic" style="font-size:114%;margin:0 4em"><a href="/wiki/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="/wiki/Axiom" title="Axiom">Axiom</a> <ul><li><a href="/wiki/List_of_axioms" title="List of axioms">list</a></li></ul></li> <li><a href="/wiki/Cardinality" title="Cardinality">Cardinality</a></li> <li><a href="/wiki/First-order_logic" title="First-order logic">First-order logic</a></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Formal_semantics_(logic)" class="mw-redirect" title="Formal semantics (logic)">Formal semantics</a></li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">Lemma</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems (<a href="/wiki/Category:Theorems_in_the_foundations_of_mathematics" title="Category:Theorems in the foundations of mathematics">list</a>)<br /> and <a href="/wiki/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="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel's completeness</a> and <a href="/wiki/G%C3%B6del%27s_incompleteness_theorems" title="Gödel's incompleteness theorems">incompleteness theorems</a></li> <li><a href="/wiki/Tarski%27s_undefinability_theorem" title="Tarski's undefinability theorem">Tarski's undefinability</a></li> <li><a href="/wiki/Banach%E2%80%93Tarski_paradox" title="Banach–Tarski paradox">Banach–Tarski paradox</a></li> <li>Cantor's <a href="/wiki/Cantor%27s_theorem" title="Cantor's theorem">theorem,</a> <a href="/wiki/Cantor%27s_paradox" title="Cantor's paradox">paradox</a> and <a href="/wiki/Cantor%27s_diagonal_argument" title="Cantor's diagonal argument">diagonal argument</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness</a></li> <li><a href="/wiki/Halting_problem" title="Halting problem">Halting problem</a></li> <li><a href="/wiki/Lindstr%C3%B6m%27s_theorem" title="Lindström's theorem">Lindström's</a></li> <li><a href="/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem" title="Löwenheim–Skolem theorem">Löwenheim–Skolem</a></li> <li><a href="/wiki/Russell%27s_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="/wiki/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="Traditional" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Classical_logic" title="Classical logic">Classical logic</a></li> <li><a href="/wiki/Logical_truth" title="Logical truth">Logical truth</a></li> <li><a href="/wiki/Tautology_(logic)" title="Tautology (logic)">Tautology</a></li> <li><a href="/wiki/Proposition" title="Proposition">Proposition</a></li> <li><a href="/wiki/Inference" title="Inference">Inference</a></li> <li><a href="/wiki/Logical_equivalence" title="Logical equivalence">Logical equivalence</a></li> <li><a href="/wiki/Consistency" title="Consistency">Consistency</a> <ul><li><a href="/wiki/Equiconsistency" title="Equiconsistency">Equiconsistency</a></li></ul></li> <li><a href="/wiki/Argument" title="Argument">Argument</a></li> <li><a href="/wiki/Soundness" title="Soundness">Soundness</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li> <li><a href="/wiki/Syllogism" title="Syllogism">Syllogism</a></li> <li><a href="/wiki/Square_of_opposition" title="Square of opposition">Square of opposition</a></li> <li><a href="/wiki/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="/wiki/Propositional_calculus" 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="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a></li> <li><a href="/wiki/Boolean_function" title="Boolean function">Boolean functions</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connectives</a></li> <li><a href="/wiki/Propositional_calculus" title="Propositional calculus">Propositional calculus</a></li> <li><a href="/wiki/Propositional_formula" title="Propositional formula">Propositional formula</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth tables</a></li> <li><a href="/wiki/Many-valued_logic" title="Many-valued logic">Many-valued logic</a> <ul><li><a href="/wiki/Three-valued_logic" title="Three-valued logic">3</a></li> <li><a href="/wiki/Finite-valued_logic" title="Finite-valued logic">finite</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/First-order_logic" title="First-order logic">First-order</a> <ul><li><a href="/wiki/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="/wiki/Second-order_logic" title="Second-order logic">Second-order</a> <ul><li><a href="/wiki/Monadic_second-order_logic" title="Monadic second-order logic">Monadic</a></li></ul></li> <li><a href="/wiki/Higher-order_logic" title="Higher-order logic">Higher-order</a></li> <li><a href="/wiki/Fixed-point_logic" title="Fixed-point logic">Fixed-point</a></li> <li><a href="/wiki/Free_logic" title="Free logic">Free</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifiers</a></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Set</a> <ul><li><a href="/wiki/Hereditary_set" title="Hereditary set">hereditary</a></li></ul></li> <li><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Class</a></li> <li>(<a href="/wiki/Urelement" title="Urelement">Ur-</a>)<a href="/wiki/Element_(mathematics)" title="Element (mathematics)">Element</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal number</a></li> <li><a href="/wiki/Extensionality" title="Extensionality">Extensionality</a></li> <li><a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">Forcing</a></li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> <ul><li><a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a></li> <li><a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a></li></ul></li> <li>Set operations: <ul><li><a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a></li> <li><a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a></li> <li><a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a></li> <li><a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a></li> <li><a href="/wiki/Power_set" title="Power set">power set</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Countable_set" title="Countable set">Countable</a></li> <li><a href="/wiki/Uncountable_set" title="Uncountable set">Uncountable</a></li> <li><a href="/wiki/Empty_set" title="Empty set">Empty</a></li> <li><a href="/wiki/Inhabited_set" title="Inhabited set">Inhabited</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton</a></li> <li><a href="/wiki/Finite_set" title="Finite set">Finite</a></li> <li><a href="/wiki/Infinite_set" title="Infinite set">Infinite</a></li> <li><a href="/wiki/Transitive_set" title="Transitive set">Transitive</a></li> <li><a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">Ultrafilter</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive</a></li> <li><a href="/wiki/Fuzzy_set" title="Fuzzy set">Fuzzy</a></li> <li><a href="/wiki/Universal_set" title="Universal set">Universal</a></li> <li><a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">Universe</a> <ul><li><a href="/wiki/Constructible_universe" title="Constructible universe">constructible</a></li> <li><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Map_(mathematics)" title="Map (mathematics)">Maps</a> and <a href="/wiki/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="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a>/<a href="/wiki/Map_(mathematics)" title="Map (mathematics)">Map</a> <ul><li><a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a></li> <li><a href="/wiki/Codomain" title="Codomain">codomain</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></li> <li><a href="/wiki/Injective_function" title="Injective function">In</a>/<a href="/wiki/Surjective_function" title="Surjective function">Sur</a>/<a href="/wiki/Bijection" title="Bijection">Bi</a>-jection</li> <li><a href="/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem" title="Schröder–Bernstein theorem">Schröder–Bernstein theorem</a></li> <li><a href="/wiki/Isomorphism" title="Isomorphism">Isomorphism</a></li> <li><a href="/wiki/G%C3%B6del_numbering" title="Gödel numbering">Gödel numbering</a></li> <li><a href="/wiki/Enumeration" title="Enumeration">Enumeration</a></li> <li><a href="/wiki/Large_cardinal" title="Large cardinal">Large cardinal</a> <ul><li><a href="/wiki/Inaccessible_cardinal" title="Inaccessible cardinal">inaccessible</a></li></ul></li> <li><a href="/wiki/Aleph_number" title="Aleph number">Aleph number</a></li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> <ul><li><a href="/wiki/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="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel</a> <ul><li><a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a></li> <li><a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a></li></ul></li> <li><a href="/wiki/General_set_theory" title="General set theory">General</a></li> <li><a href="/wiki/Kripke%E2%80%93Platek_set_theory" title="Kripke–Platek set theory">Kripke–Platek</a></li> <li><a href="/wiki/Morse%E2%80%93Kelley_set_theory" title="Morse–Kelley set theory">Morse–Kelley</a></li> <li><a href="/wiki/Naive_set_theory" title="Naive set theory">Naive</a></li> <li><a href="/wiki/New_Foundations" title="New Foundations">New Foundations</a></li> <li><a href="/wiki/Tarski%E2%80%93Grothendieck_set_theory" title="Tarski–Grothendieck set theory">Tarski–Grothendieck</a></li> <li><a href="/wiki/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="/wiki/Ackermann_set_theory" title="Ackermann set theory">Ackermann</a></li> <li><a href="/wiki/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="/wiki/Formal_system" title="Formal system">Formal systems</a> (<a href="/wiki/List_of_formal_systems" title="List of formal systems"><span style="font-size:85%;">list</span></a>),<br /><a href="/wiki/Formal_language" title="Formal language">language</a> and <a href="/wiki/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="/wiki/Alphabet_(formal_languages)" title="Alphabet (formal languages)">Alphabet</a></li> <li><a href="/wiki/Arity" title="Arity">Arity</a></li> <li><a href="/wiki/Automata_theory" title="Automata theory">Automata</a></li> <li><a href="/wiki/Axiom_schema" title="Axiom schema">Axiom schema</a></li> <li><a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">Expression</a> <ul><li><a href="/wiki/Ground_expression" title="Ground expression">ground</a></li></ul></li> <li><a href="/wiki/Extension_by_new_constant_and_function_names" title="Extension by new constant and function names">Extension</a> <ul><li><a href="/wiki/Extension_by_definitions" title="Extension by definitions">by definition</a></li> <li><a href="/wiki/Conservative_extension" title="Conservative extension">conservative</a></li></ul></li> <li><a href="/wiki/Finitary_relation" title="Finitary relation">Relation</a></li> <li><a href="/wiki/Formation_rule" title="Formation rule">Formation rule</a></li> <li><a href="/wiki/Formal_grammar" title="Formal grammar">Grammar</a></li> <li><a href="/wiki/Well-formed_formula" title="Well-formed formula">Formula</a> <ul><li><a href="/wiki/Atomic_formula" title="Atomic formula">atomic</a></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">closed</a></li> <li><a href="/wiki/Ground_formula" class="mw-redirect" title="Ground formula">ground</a></li> <li><a href="/wiki/Open_formula" title="Open formula">open</a></li></ul></li> <li><a href="/wiki/Free_variables_and_bound_variables" title="Free variables and bound variables">Free/bound variable</a></li> <li><a href="/wiki/Formal_language" title="Formal language">Language</a></li> <li><a href="/wiki/Metalanguage" title="Metalanguage">Metalanguage</a></li> <li><a href="/wiki/Logical_connective" title="Logical connective">Logical connective</a> <ul><li><a href="/wiki/Negation" title="Negation">¬</a></li> <li><a href="/wiki/Logical_disjunction" title="Logical disjunction">∨</a></li> <li><a href="/wiki/Logical_conjunction" title="Logical conjunction">∧</a></li> <li><a href="/wiki/Material_conditional" title="Material conditional">→</a></li> <li><a href="/wiki/Logical_biconditional" title="Logical biconditional">↔</a></li> <li><a href="/wiki/Logical_equality" title="Logical equality">=</a></li></ul></li> <li><a href="/wiki/Predicate_(mathematical_logic)" title="Predicate (mathematical logic)">Predicate</a> <ul><li><a href="/wiki/Functional_predicate" title="Functional predicate">functional</a></li> <li><a href="/wiki/Predicate_variable" title="Predicate variable">variable</a></li> <li><a href="/wiki/Propositional_variable" title="Propositional variable">propositional variable</a></li></ul></li> <li><a href="/wiki/Formal_proof" title="Formal proof">Proof</a></li> <li><a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">Quantifier</a> <ul><li><a href="/wiki/Existential_quantification" title="Existential quantification">∃</a></li> <li><a href="/wiki/Uniqueness_quantification" title="Uniqueness quantification">!</a></li> <li><a href="/wiki/Universal_quantification" title="Universal quantification">∀</a></li> <li><a href="/wiki/Quantifier_rank" title="Quantifier rank">rank</a></li></ul></li> <li><a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">Sentence</a> <ul><li><a href="/wiki/Atomic_sentence" title="Atomic sentence">atomic</a></li> <li><a href="/wiki/Spectrum_of_a_sentence" title="Spectrum of a sentence">spectrum</a></li></ul></li> <li><a href="/wiki/Signature_(logic)" title="Signature (logic)">Signature</a></li> <li><a href="/wiki/String_(formal_languages)" class="mw-redirect" title="String (formal languages)">String</a></li> <li><a href="/wiki/Substitution_(logic)" title="Substitution (logic)">Substitution</a></li> <li><a href="/wiki/Symbol_(formal)" title="Symbol (formal)">Symbol</a> <ul><li><a href="/wiki/Uninterpreted_function" title="Uninterpreted function">function</a></li> <li><a href="/wiki/Logical_constant" title="Logical constant">logical/constant</a></li> <li><a href="/wiki/Non-logical_symbol" title="Non-logical symbol">non-logical</a></li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">variable</a></li></ul></li> <li><a href="/wiki/Term_(logic)" title="Term (logic)">Term</a></li> <li><a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">Theory</a> <ul><li><a href="/wiki/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="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic<br />systems</a> <span style="font-size:85%;">(<a href="/wiki/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="/wiki/True_arithmetic" title="True arithmetic">arithmetic</a>: <ul><li><a href="/wiki/Peano_axioms" title="Peano axioms">Peano</a></li> <li><a href="/wiki/Second-order_arithmetic" title="Second-order arithmetic">second-order</a></li> <li><a href="/wiki/Elementary_function_arithmetic" title="Elementary function arithmetic">elementary function</a></li> <li><a href="/wiki/Primitive_recursive_arithmetic" title="Primitive recursive arithmetic">primitive recursive</a></li> <li><a href="/wiki/Robinson_arithmetic" title="Robinson arithmetic">Robinson</a></li> <li><a href="/wiki/Skolem_arithmetic" title="Skolem arithmetic">Skolem</a></li></ul></li> <li>of the <a href="/wiki/Construction_of_the_real_numbers" title="Construction of the real numbers">real numbers</a> <ul><li><a href="/wiki/Tarski%27s_axiomatization_of_the_reals" title="Tarski's axiomatization of the reals">Tarski's axiomatization</a></li></ul></li> <li>of <a href="/wiki/Axiomatization_of_Boolean_algebras" class="mw-redirect" title="Axiomatization of Boolean algebras">Boolean algebras</a> <ul><li><a href="/wiki/Boolean_algebras_canonically_defined" title="Boolean algebras canonically defined">canonical</a></li> <li><a href="/wiki/Minimal_axioms_for_Boolean_algebra" title="Minimal axioms for Boolean algebra">minimal axioms</a></li></ul></li> <li>of <a href="/wiki/Foundations_of_geometry" title="Foundations of geometry">geometry</a>: <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a>: <ul><li><a href="/wiki/Euclid%27s_Elements" title="Euclid's Elements"><i>Elements</i></a></li> <li><a href="/wiki/Hilbert%27s_axioms" title="Hilbert's axioms">Hilbert's</a></li> <li><a class="mw-selflink selflink">Tarski's</a></li></ul></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean</a></li></ul></li></ul> <ul><li><i><a href="/wiki/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="/wiki/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="/wiki/Formal_proof" title="Formal proof">Formal proof</a></li> <li><a href="/wiki/Natural_deduction" title="Natural deduction">Natural deduction</a></li> <li><a href="/wiki/Logical_consequence" title="Logical consequence">Logical consequence</a></li> <li><a href="/wiki/Rule_of_inference" title="Rule of inference">Rule of inference</a></li> <li><a href="/wiki/Sequent_calculus" title="Sequent calculus">Sequent calculus</a></li> <li><a href="/wiki/Theorem" title="Theorem">Theorem</a></li> <li><a href="/wiki/Formal_system" title="Formal system">Systems</a> <ul><li><a href="/wiki/Axiomatic_system" title="Axiomatic system">axiomatic</a></li> <li><a href="/wiki/Deductive_system" class="mw-redirect" title="Deductive system">deductive</a></li> <li><a href="/wiki/Hilbert_system" title="Hilbert system">Hilbert</a> <ul><li><a href="/wiki/List_of_Hilbert_systems" class="mw-redirect" title="List of Hilbert systems">list</a></li></ul></li></ul></li> <li><a href="/wiki/Complete_theory" title="Complete theory">Complete theory</a></li> <li><a href="/wiki/Independence_(mathematical_logic)" title="Independence (mathematical logic)">Independence</a> (<a href="/wiki/List_of_statements_independent_of_ZFC" title="List of statements independent of ZFC">from ZFC</a>)</li> <li><a href="/wiki/Proof_of_impossibility" title="Proof of impossibility">Proof of impossibility</a></li> <li><a href="/wiki/Ordinal_analysis" title="Ordinal analysis">Ordinal analysis</a></li> <li><a href="/wiki/Reverse_mathematics" title="Reverse mathematics">Reverse mathematics</a></li> <li><a href="/wiki/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="/wiki/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="/wiki/Interpretation_(logic)" title="Interpretation (logic)">Interpretation</a> <ul><li><a href="/wiki/Interpretation_function" class="mw-redirect" title="Interpretation function">function</a></li> <li><a href="/wiki/Interpretation_(model_theory)" title="Interpretation (model theory)">of models</a></li></ul></li> <li><a href="/wiki/Structure_(mathematical_logic)" title="Structure (mathematical logic)">Model</a> <ul><li><a href="/wiki/Elementary_equivalence" title="Elementary equivalence">equivalence</a></li> <li><a href="/wiki/Finite_model_theory" title="Finite model theory">finite</a></li> <li><a href="/wiki/Saturated_model" title="Saturated model">saturated</a></li> <li><a href="/wiki/Spectrum_of_a_theory" title="Spectrum of a theory">spectrum</a></li> <li><a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">submodel</a></li></ul></li> <li><a href="/wiki/Non-standard_model" title="Non-standard model">Non-standard model</a> <ul><li><a href="/wiki/Non-standard_model_of_arithmetic" title="Non-standard model of arithmetic">of arithmetic</a></li></ul></li> <li><a href="/wiki/Diagram_(mathematical_logic)" title="Diagram (mathematical logic)">Diagram</a> <ul><li><a href="/wiki/Elementary_diagram" title="Elementary diagram">elementary</a></li></ul></li> <li><a href="/wiki/Categorical_theory" title="Categorical theory">Categorical theory</a></li> <li><a href="/wiki/Model_complete_theory" title="Model complete theory">Model complete theory</a></li> <li><a href="/wiki/Satisfiability" title="Satisfiability">Satisfiability</a></li> <li><a href="/wiki/Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li> <li><a href="/wiki/Strength_(mathematical_logic)" title="Strength (mathematical logic)">Strength</a></li> <li><a href="/wiki/Theories_of_truth" class="mw-redirect" title="Theories of truth">Theories of truth</a> <ul><li><a href="/wiki/Semantic_theory_of_truth" title="Semantic theory of truth">semantic</a></li> <li><a href="/wiki/Tarski%27s_theory_of_truth" class="mw-redirect" title="Tarski's theory of truth">Tarski's</a></li> <li><a href="/wiki/Kripke%27s_theory_of_truth" class="mw-redirect" title="Kripke's theory of truth">Kripke's</a></li></ul></li> <li><a href="/wiki/T-schema" title="T-schema">T-schema</a></li> <li><a href="/wiki/Transfer_principle" title="Transfer principle">Transfer principle</a></li> <li><a href="/wiki/Truth_predicate" title="Truth predicate">Truth predicate</a></li> <li><a href="/wiki/Truth_value" title="Truth value">Truth value</a></li> <li><a href="/wiki/Type_(model_theory)" title="Type (model theory)">Type</a></li> <li><a href="/wiki/Ultraproduct" title="Ultraproduct">Ultraproduct</a></li> <li><a href="/wiki/Validity_(logic)" title="Validity (logic)">Validity</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/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="/wiki/Church_encoding" title="Church encoding">Church encoding</a></li> <li><a href="/wiki/Church%E2%80%93Turing_thesis" title="Church–Turing thesis">Church–Turing thesis</a></li> <li><a href="/wiki/Computably_enumerable_set" title="Computably enumerable set">Computably enumerable</a></li> <li><a href="/wiki/Computable_function" title="Computable function">Computable function</a></li> <li><a href="/wiki/Computable_set" title="Computable set">Computable set</a></li> <li><a href="/wiki/Decision_problem" title="Decision problem">Decision problem</a> <ul><li><a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a></li> <li><a href="/wiki/Undecidable_problem" title="Undecidable problem">undecidable</a></li> <li><a href="/wiki/P_(complexity)" title="P (complexity)">P</a></li> <li><a href="/wiki/NP_(complexity)" title="NP (complexity)">NP</a></li> <li><a href="/wiki/P_versus_NP_problem" title="P versus NP problem">P versus NP problem</a></li></ul></li> <li><a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li> <li><a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a></li> <li><a href="/wiki/Primitive_recursive_function" title="Primitive recursive function">Primitive recursive function</a></li> <li><a href="/wiki/Recursion" title="Recursion">Recursion</a></li> <li><a href="/wiki/Recursive_set" class="mw-redirect" title="Recursive set">Recursive set</a></li> <li><a href="/wiki/Turing_machine" title="Turing machine">Turing machine</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" 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