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class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eindeutigkeit" title="Eindeutigkeit – German" lang="de" hreflang="de" data-title="Eindeutigkeit" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AB%D8%A8%D8%A7%D8%AA_%DB%8C%DA%A9%D8%AA%D8%A7%DB%8C%DB%8C" title="اثبات یکتایی – Persian" lang="fa" hreflang="fa" data-title="اثبات یکتایی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Unicit%C3%A9_(math%C3%A9matiques)" title="Unicité (mathématiques) – French" lang="fr" hreflang="fr" data-title="Unicité (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ketunggalan_(matematika)" title="Ketunggalan (matematika) – Indonesian" lang="id" hreflang="id" data-title="Ketunggalan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Unicit%C3%A0" title="Unicità – Italian" lang="it" hreflang="it" data-title="Unicità" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%99%D7%95%D7%9D_%D7%95%D7%99%D7%97%D7%99%D7%93%D7%95%D7%AA" title="קיום ויחידות – Hebrew" lang="he" hreflang="he" data-title="קיום ויחידות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Uniciteit" title="Uniciteit – Dutch" lang="nl" hreflang="nl" data-title="Uniciteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%B8%80%E6%84%8F%E6%80%A7_(%E6%95%B0%E5%AD%A6)" title="一意性 (数学) – Japanese" lang="ja" hreflang="ja" data-title="一意性 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eintydig" title="Eintydig – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Eintydig" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Quantifica%C3%A7%C3%A3o_de_singularidade" title="Quantificação de singularidade – Portuguese" lang="pt" hreflang="pt" data-title="Quantificação de singularidade" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%95%D0%B4%D0%B8%D0%BD%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Единственность – Russian" lang="ru" hreflang="ru" data-title="Единственность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Entydighet" title="Entydighet – Swedish" lang="sv" hreflang="sv" data-title="Entydighet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9A%E0%B9%88%E0%B8%87%E0%B8%9B%E0%B8%A3%E0%B8%B4%E0%B8%A1%E0%B8%B2%E0%B8%93_(%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87%E0%B8%95%E0%B8%B1%E0%B8%A7)" title="ตัวบ่งปริมาณ (หนึ่งตัว) – Thai" lang="th" hreflang="th" data-title="ตัวบ่งปริมาณ (หนึ่งตัว)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Biricik" title="Biricik – Turkish" lang="tr" hreflang="tr" data-title="Biricik" data-language-autonym="Türkçe" 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<div class="vector-body-before-content"> <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">Logical property of being the one and only object satisfying a condition</div><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html 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.ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Uniqueness+quantification%22">"Uniqueness quantification"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Uniqueness+quantification%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Uniqueness+quantification%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Uniqueness+quantification%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Uniqueness+quantification%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Uniqueness+quantification%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">November 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table><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">"Unique (mathematics)" redirects here. For other uses, see <a href="/wiki/Unique_(disambiguation)" class="mw-redirect mw-disambig" title="Unique (disambiguation)">Unique (disambiguation)</a>.</div><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Logic" title="Logic">logic</a>, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition.<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> This sort of <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantification</a> is known as <b>uniqueness quantification</b> or <b>unique existential quantification</b>, and is often denoted with the symbols "<a href="/wiki/Existential_quantification" title="Existential quantification">∃</a>!"<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or "∃<sub>=1</sub>". </p><p>For example, the formal statement </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 !n\in \mathbb {N} \,(n-2=4)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !n\in \mathbb {N} \,(n-2=4)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf60f9cafc6075f5a9356ce7519c16de28bcdbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.708ex; height:2.843ex;" alt="{\displaystyle \exists !n\in \mathbb {N} \,(n-2=4)}"></span></dd></dl> <p>may be read as "there is exactly one natural number <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> such that <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 n-2=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-2=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176ff5a878d2f873302f2ef6c407bfad6c158884" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.658ex; height:2.343ex;" alt="{\displaystyle n-2=4}"></span>". </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proving_uniqueness">Proving uniqueness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniqueness_quantification&action=edit&section=1" title="Edit section: Proving uniqueness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, <i><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></i> and <i><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></i>) must be equal to each other (i.e. <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 a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}"></span>). </p><p>For example, to show that the equation <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+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+2=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3387b0e2d7885c1fe21886cb6b2536f8dcb75b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle x+2=5}"></span> has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds: </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 3+2=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+2=5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4011f9c8012f01a64b37e67426926efd6a9790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.073ex; height:2.343ex;" alt="{\displaystyle 3+2=5.}"></span></dd></dl> <p>To establish the uniqueness of the solution, one would proceed by assuming that there are two solutions, namely <i><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span></i> and <i><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span></i>, satisfying <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+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+2=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3387b0e2d7885c1fe21886cb6b2536f8dcb75b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle x+2=5}"></span>. That is, </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 a+2=5{\text{ and }}b+2=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+2=5{\text{ and }}b+2=5.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4387791dd78f3748d5d7a015076a71938785850a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.311ex; height:2.343ex;" alt="{\displaystyle a+2=5{\text{ and }}b+2=5.}"></span></dd></dl> <p>Then since equality is a <a href="/wiki/Transitive_relation" title="Transitive relation">transitive relation</a>, </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 a+2=b+2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+2=b+2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ab99bf2b51dffef251a467686ea01f0fbc6cd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.978ex; height:2.343ex;" alt="{\displaystyle a+2=b+2.}"></span></dd></dl> <p>Subtracting 2 from both sides then yields </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 a=b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6053fe8070065a7d8818843b82d98ac9ff3708d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.973ex; height:2.176ex;" alt="{\displaystyle a=b.}"></span></dd></dl> <p>which completes the proof that 3 is the unique solution of <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+2=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+2=5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3387b0e2d7885c1fe21886cb6b2536f8dcb75b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.593ex; height:2.343ex;" alt="{\displaystyle x+2=5}"></span>. </p><p>In general, both existence (there exists <i>at least</i> one object) and uniqueness (there exists <i>at most</i> one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition. </p><p>An alternative way to prove uniqueness is to prove that there exists an object <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> satisfying the condition, and then to prove that every object satisfying the condition must be equal to <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Reduction_to_ordinary_existential_and_universal_quantification">Reduction to ordinary existential and universal quantification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniqueness_quantification&action=edit&section=2" title="Edit section: Reduction to ordinary existential and universal quantification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uniqueness quantification can be expressed in terms of the <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential</a> and <a href="/wiki/Universal_quantifier" class="mw-redirect" title="Universal quantifier">universal</a> quantifiers of <a href="/wiki/Predicate_logic" class="mw-redirect" title="Predicate logic">predicate logic</a>, by defining the formula <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 !xP(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mi>x</mi> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !xP(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd1dab26425ce45dbd9027ad51796ee6437f4d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.153ex; height:2.843ex;" alt="{\displaystyle \exists !xP(x)}"></span> to mean </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 x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>∧</mo> <mi mathvariant="normal">¬</mi> <mi mathvariant="normal">∃</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>y</mi> <mo>≠</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82d02778f1b80aeabb058e66ac520df8115cb645" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.391ex; height:2.843ex;" alt="{\displaystyle \exists x\,(P(x)\,\wedge \neg \exists y\,(P(y)\wedge y\neq x)),}"></span></dd></dl> <p>which is logically equivalent to </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 x\,(P(x)\wedge \forall y\,(P(y)\to y=x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,(P(x)\wedge \forall y\,(P(y)\to y=x)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7593031b2ccc8b3132ad08e4144a3c7fcee395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.485ex; height:2.843ex;" alt="{\displaystyle \exists x\,(P(x)\wedge \forall y\,(P(y)\to y=x)).}"></span></dd></dl> <p>An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is </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 x\,P(x)\wedge \forall y\,\forall z\,[(P(y)\wedge P(z))\to y=z].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>z</mi> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,P(x)\wedge \forall y\,\forall z\,[(P(y)\wedge P(z))\to y=z].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20527ebf09b76788d0f238861c10aeec61c3415c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.721ex; height:2.843ex;" alt="{\displaystyle \exists x\,P(x)\wedge \forall y\,\forall z\,[(P(y)\wedge P(z))\to y=z].}"></span></dd></dl> <p>Another equivalent definition, which has the advantage of brevity, is </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 x\,\forall y\,(P(y)\leftrightarrow y=x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists x\,\forall y\,(P(y)\leftrightarrow y=x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84314abb372c81eba7ece13248a163052702045f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.208ex; height:2.843ex;" alt="{\displaystyle \exists x\,\forall y\,(P(y)\leftrightarrow y=x).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniqueness_quantification&action=edit&section=3" title="Edit section: Generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The uniqueness quantification can be generalized into <a href="/wiki/Counting_quantification" title="Counting quantification">counting quantification</a> (or numerical quantification<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>). This includes both quantification of the form "exactly <i>k</i> objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Uniqueness depends on a notion of <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality</a>. Loosening this to a coarser <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> yields quantification of uniqueness <a href="/wiki/Up_to" title="Up to">up to</a> that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in <a href="/wiki/Category_theory" title="Category theory">category theory</a> are defined to be unique up to <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. </p><p>The exclamation mark <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 !}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle !}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b552106cb3511c670d125b372d702e7cca7d630a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:0.647ex; height:2.009ex;" alt="{\displaystyle !}"></span> can be also used as a separate quantification symbol, so <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 !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mo>!</mo> <mi>x</mi> <mo>.</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">↔</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo>.</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>∧</mo> <mo stretchy="false">(</mo> <mo>!</mo> <mi>x</mi> <mo>.</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b005312dbbf571ef1d20b834034f0adc4abb1a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.057ex; height:2.843ex;" alt="{\displaystyle (\exists !x.P(x))\leftrightarrow ((\exists x.P(x))\land (!x.P(x)))}"></span>, 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 (!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>!</mo> <mi>x</mi> <mo>.</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>:=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mi mathvariant="normal">∀</mi> <mi>b</mi> <mo>.</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∧</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd01ce65c741c7285cf387f84174f52366bcf91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.964ex; height:2.843ex;" alt="{\displaystyle (!x.P(x)):=(\forall a\forall b.P(a)\land P(b)\rightarrow a=b)}"></span>. E.g. it can be safely used in the <a href="/wiki/Replacement_axiom" class="mw-redirect" title="Replacement axiom">replacement axiom</a>, instead of <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 !}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists !}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aab884c076b7332eab3860c2f32086012df840b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \exists !}"></span>. </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=Uniqueness_quantification&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Essentially_unique" title="Essentially unique">Essentially unique</a></li> <li><a href="/wiki/One-hot" title="One-hot">One-hot</a></li> <li><a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">Singleton (mathematics)</a></li> <li><a href="/wiki/Uniqueness_theorem" title="Uniqueness theorem">Uniqueness theorem</a></li></ul> <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=Uniqueness_quantification&action=edit&section=5" title="Edit section: References"><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"><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 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Retrieved <span class="nowrap">2019-12-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Uniqueness+Theorem&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FUniquenessTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniqueness+quantification" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.whitman.edu/mathematics/higher_math_online/section02.05.html">"2.5 Uniqueness Arguments"</a>. <i>www.whitman.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.whitman.edu&rft.atitle=2.5+Uniqueness+Arguments&rft_id=https%3A%2F%2Fwww.whitman.edu%2Fmathematics%2Fhigher_math_online%2Fsection02.05.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniqueness+quantification" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHelman2013" class="citation web cs1">Helman, Glen (August 1, 2013). <a rel="nofollow" class="external text" href="http://persweb.wabash.edu/facstaff/helmang/phi270-1314F/phi270PDF/phi270text/phi270txt8/phi270txt83/phi270txt83(4up).pdf">"Numerical quantification"</a> <span class="cs1-format">(PDF)</span>. <i>persweb.wabash.edu</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-14</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=persweb.wabash.edu&rft.atitle=Numerical+quantification&rft.date=2013-08-01&rft.aulast=Helman&rft.aufirst=Glen&rft_id=http%3A%2F%2Fpersweb.wabash.edu%2Ffacstaff%2Fhelmang%2Fphi270-1314F%2Fphi270PDF%2Fphi270text%2Fphi270txt8%2Fphi270txt83%2Fphi270txt83%284up%29.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniqueness+quantification" class="Z3988"></span></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">This is a consequence of the <a href="/wiki/Compactness_theorem" title="Compactness theorem">compactness theorem</a>.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Uniqueness_quantification&action=edit&section=6" title="Edit section: Bibliography"><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="CITEREFKleene1952" class="citation book cs1">Kleene, Stephen (1952). <i>Introduction to Metamathematics</i>. Ishi Press International. p. 199.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Metamathematics&rft.pages=199&rft.pub=Ishi+Press+International&rft.date=1952&rft.aulast=Kleene&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniqueness+quantification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndrews2002" class="citation book cs1">Andrews, Peter B. (2002). <i>An introduction to mathematical logic and type theory to truth through proof</i> (2. ed.). Dordrecht: Kluwer Acad. Publ. p. 233. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0763-9" title="Special:BookSources/1-4020-0763-9"><bdi>1-4020-0763-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+mathematical+logic+and+type+theory+to+truth+through+proof&rft.place=Dordrecht&rft.pages=233&rft.edition=2.&rft.pub=Kluwer+Acad.+Publ.&rft.date=2002&rft.isbn=1-4020-0763-9&rft.aulast=Andrews&rft.aufirst=Peter+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUniqueness+quantification" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output 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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 class="mw-selflink selflink">!</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 href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">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" style="width:1%">Related</th><td class="navbox-list-with-group 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