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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples of real closed fields</span> </div> </a> <ul id="toc-Examples_of_real_closed_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_closure" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_closure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Real closure</span> </div> </a> <ul id="toc-Real_closure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decidability_and_quantifier_elimination" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Decidability_and_quantifier_elimination"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Decidability and quantifier elimination</span> </div> </a> <button aria-controls="toc-Decidability_and_quantifier_elimination-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 Decidability and quantifier elimination subsection</span> </button> <ul id="toc-Decidability_and_quantifier_elimination-sublist" class="vector-toc-list"> <li id="toc-Complexity_of_deciding_𝘛rcf" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complexity_of_deciding_𝘛rcf"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Complexity of deciding 𝘛<sub>rcf</sub></span> </div> </a> <ul id="toc-Complexity_of_deciding_𝘛rcf-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Order_properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Order_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Order properties</span> </div> </a> <ul id="toc-Order_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_generalized_continuum_hypothesis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_generalized_continuum_hypothesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>The generalized continuum hypothesis</span> </div> </a> <ul id="toc-The_generalized_continuum_hypothesis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_Euclidean_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elementary_Euclidean_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Elementary Euclidean geometry</span> </div> </a> <ul id="toc-Elementary_Euclidean_geometry-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">8</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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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 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</div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Real closed field</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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.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">"Artin–Schreier theorem" redirects here. For the branch of Galois theory, see <a href="/wiki/Artin%E2%80%93Schreier_theory" title="Artin–Schreier theory">Artin–Schreier theory</a>.</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>real closed field</b> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>F</i> that has the same <a href="/wiki/First-order_logic" title="First-order logic">first-order</a> properties as the field of <a href="/wiki/Real_number" title="Real number">real numbers</a>. Some examples are the field of real numbers, the field of real <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, and the field of <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A real closed field is a field <i>F</i> in which any of the following equivalent conditions is true: </p> <ol><li><i>F</i> is <a href="/wiki/Elementarily_equivalent" class="mw-redirect" title="Elementarily equivalent">elementarily equivalent</a> to the real numbers. In other words, it has the same first-order properties as the reals: any <a href="/wiki/Sentence_(mathematical_logic)" title="Sentence (mathematical logic)">sentence</a> in the first-order language of fields is true in <i>F</i> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is true in the reals.</li> <li>There is a <a href="/wiki/Total_order" title="Total order">total order</a> on <i>F</i> making it an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> such that, in this ordering, every positive element of <i>F</i> has a <a href="/wiki/Square_root#In_integral_domains,_including_fields" title="Square root">square root</a> in <i>F</i> and any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of <a href="/wiki/Parity_(mathematics)" title="Parity (mathematics)">odd</a> <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> with <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in <i>F</i> has at least one <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> in <i>F</i>.</li> <li><i>F</i> is a <a href="/wiki/Formally_real_field" title="Formally real field">formally real field</a> such that every polynomial of odd degree with coefficients in <i>F</i> has at least one root in <i>F</i>, and for every element <i>a</i> of <i>F</i> there is <i>b</i> in <i>F</i> such that <i>a</i> = <i>b</i><sup>2</sup> or <i>a</i> = −<i>b</i><sup>2</sup>.</li> <li><i>F</i> is not <a href="/wiki/Algebraically_closed" class="mw-redirect" title="Algebraically closed">algebraically closed</a>, but its <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> is a <a href="/wiki/Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a>.</li> <li><i>F</i> is not algebraically closed but the <a href="/wiki/Field_extension" title="Field extension">field extension</a> <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 F({\sqrt {-1}}\,)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F({\sqrt {-1}}\,)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/825dd8f81e817c218b0190bf859b65df3420b054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.844ex; height:3.009ex;" alt="{\displaystyle F({\sqrt {-1}}\,)}"></span> is algebraically closed.</li> <li>There is an ordering on <i>F</i> that does not extend to an ordering on any proper <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic extension</a> of <i>F</i>.</li> <li><i>F</i> is a formally real field such that no proper algebraic extension of <i>F</i> is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.)</li> <li>There is an ordering on <i>F</i> making it an ordered field such that, in this ordering, the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> holds for all polynomials over <i>F</i> with degree <i>≥</i> 0.</li> <li><i>F</i> is a <a href="/wiki/Weakly_o-minimal_structure" title="Weakly o-minimal structure">weakly o-minimal</a> ordered field.<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></li></ol> <div class="mw-heading mw-heading2"><h2 id="Examples_of_real_closed_fields">Examples of real closed fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=2" title="Edit section: Examples of real closed fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>the field of real <a href="/wiki/Algebraic_numbers" class="mw-redirect" title="Algebraic numbers">algebraic numbers</a></li> <li>the field of <a href="/wiki/Computable_number" title="Computable number">computable numbers</a></li> <li>the field of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></li> <li>the field of <a href="/wiki/Real_number" title="Real number">real numbers</a></li> <li>the field of <a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a> with real coefficients</li> <li>the <a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li>the <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal number</a> fields</li> <li>the <a href="/wiki/Superreal_number" title="Superreal number">superreal number</a> fields</li> <li>the field of <a href="/wiki/Surreal_number" title="Surreal number">surreal numbers</a> (this is a <a href="/wiki/Proper_class" class="mw-redirect" title="Proper class">proper class</a>, not a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="Real_closure">Real closure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=3" title="Edit section: Real closure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>F</i> is an ordered field, the <b>Artin–Schreier theorem</b> states that <i>F</i> has an algebraic extension, called the <b>real closure</b> <i>K</i> of <i>F</i>, such that <i>K</i> is a real closed field whose ordering is an extension of the given ordering on <i>F</i>, and is unique <a href="/wiki/Up_to" title="Up to">up to</a> a unique <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> of fields identical on <i>F</i><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> (note that every <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> between real closed fields automatically is <a href="/wiki/Order_isomorphism" title="Order isomorphism">order preserving</a>, because <i>x</i> ≤ <i>y</i> if and only if ∃<i>z</i> : <i>y</i> = <i>x</i> + <i>z</i><sup>2</sup>). For example, the real closure of the ordered field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> is the field <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} _{\mathrm {alg} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\mathrm {alg} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3fce98358133606c4d3048f836464fad54909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.012ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{\mathrm {alg} }}"></span> of real algebraic numbers. The <a href="/wiki/Theorem" title="Theorem">theorem</a> is named for <a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> and <a href="/wiki/Otto_Schreier" title="Otto Schreier">Otto Schreier</a>, who <a href="/wiki/Mathematical_proof" title="Mathematical proof">proved</a> it in 1926. </p><p>If (<i>F</i>, <i>P</i>) is an ordered field, and <i>E</i> is a <a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a> of <i>F</i>, then by <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a> there is a maximal ordered field extension (<i>M</i>, <i>Q</i>) with <i>M</i> a <a href="/wiki/Field_extension" title="Field extension">subfield</a> of <i>E</i> containing <i>F</i> and the order on <i>M</i> extending <i>P</i>. This <i>M</i>, together with its ordering <i>Q</i>, is called the <b>relative real closure</b> of (<i>F</i>, <i>P</i>) in <i>E</i>. We call (<i>F</i>, <i>P</i>) <b> real closed relative to</b> <i>E</i> if <i>M</i> is just <i>F</i>. When <i>E</i> is the algebraic closure of <i>F</i> the relative real closure of <i>F</i> in <i>E</i> is actually the <b>real closure</b> of <i>F</i> described earlier.<sup id="cite_ref-Efr177_3-0" class="reference"><a href="#cite_note-Efr177-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>If <i>F</i> is a field (no ordering compatible with the field operations is assumed, nor is it assumed that <i>F</i> is orderable) then <i>F</i> still has a real closure, which may not be a field anymore, but just a <a href="/wiki/Real_closed_ring" title="Real closed ring">real closed ring</a>. For example, the real closure of the field <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 {Q} ({\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {2}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24f86d74924b1b3a2935e9fe1fddf3e32a567e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {2}})}"></span> is the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <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} _{\mathrm {alg} }\!\times \mathbb {R} _{\mathrm {alg} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\mathrm {alg} }\!\times \mathbb {R} _{\mathrm {alg} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44d7d15cf403e4d176075c5a2c7632f9c783b474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.477ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{\mathrm {alg} }\!\times \mathbb {R} _{\mathrm {alg} }}"></span> (the two copies correspond to the two orderings 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 \mathbb {Q} ({\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {2}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24f86d74924b1b3a2935e9fe1fddf3e32a567e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {2}})}"></span>). On the other hand, if <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 {Q} ({\sqrt {2}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {2}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24f86d74924b1b3a2935e9fe1fddf3e32a567e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.716ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {2}})}"></span> is considered as an ordered subfield 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>, its real closure is again the field <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} _{\mathrm {alg} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{\mathrm {alg} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c3fce98358133606c4d3048f836464fad54909" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.012ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} _{\mathrm {alg} }}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Decidability_and_quantifier_elimination">Decidability and quantifier elimination</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=4" title="Edit section: Decidability and quantifier elimination"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Formal_language" title="Formal language">language</a> of real closed fields <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 {\mathcal {L}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d072ef411c420812eda7108cf7cd211947c5c8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.714ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{\text{rcf}}}"></span> includes symbols for the operations of addition and multiplication, the constants 0 and 1, and the order relation <span class="texhtml">≤</span> (as well as equality, if this is not considered a logical symbol). In this language, the (first-order) theory of real closed fields, <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 {\mathcal {T}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b501a04b7cd24cb39621647e71c24adb933c801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}_{\text{rcf}}}"></span>, consists of all sentences that follow from the following axioms: </p> <ul><li>the <a href="/wiki/Axiom" title="Axiom">axioms</a> of <a href="/wiki/Ordered_field" title="Ordered field">ordered fields</a>;</li> <li>the axiom asserting that every positive number has a square root;</li> <li>for every odd 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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, the axiom asserting that all polynomials of degree <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> have at least one root.</li></ul> <p>All of these axioms can be expressed in <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> (i.e. <a href="/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantification</a> ranges only over elements of the field). Note 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 {\mathcal {T}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b501a04b7cd24cb39621647e71c24adb933c801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}_{\text{rcf}}}"></span> is just the set of all first-order sentences that are true about the field of real numbers. </p><p><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Tarski</a> showed 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 {\mathcal {T}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b501a04b7cd24cb39621647e71c24adb933c801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}_{\text{rcf}}}"></span> is <a href="/wiki/Complete_theory" title="Complete theory">complete</a>, meaning that any <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 {\mathcal {L}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d072ef411c420812eda7108cf7cd211947c5c8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.714ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{\text{rcf}}}"></span>-sentence can be proven either true or false from the above axioms. Furthermore, <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 {\mathcal {T}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b501a04b7cd24cb39621647e71c24adb933c801" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.377ex; height:2.676ex;" alt="{\displaystyle {\mathcal {T}}_{\text{rcf}}}"></span> is <a href="/wiki/Decidability_(logic)" title="Decidability (logic)">decidable</a>, meaning that there is an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> to determine the truth or falsity of any such sentence. This was done by showing <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a>: there is an algorithm that, given any <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 {\mathcal {L}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d072ef411c420812eda7108cf7cd211947c5c8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.714ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{\text{rcf}}}"></span>-<a href="/wiki/Well-formed_formula" title="Well-formed formula">formula</a>, which may contain <a href="/wiki/Free_variable" class="mw-redirect" title="Free variable">free variables</a>, produces an equivalent quantifier-free formula in the same free variables, where <i>equivalent</i> means that the two formulas are true for exactly the same values of the variables. Tarski's proof uses a generalization of <a href="/wiki/Sturm%27s_theorem" title="Sturm's theorem">Sturm's theorem</a>. Since the truth of quantifier-free formulas without free variables can be easily checked, this yields the desired decision procedure. These results were obtained <abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1930</span> and published in 1948.<sup id="cite_ref-:0_4-0" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Tarski%E2%80%93Seidenberg_theorem" title="Tarski–Seidenberg theorem">Tarski–Seidenberg theorem</a> extends this result to the following <i>projection theorem</i>. If <span class="texhtml"><b>R</b></span> is a real closed field, a formula with <span class="texhtml mvar" style="font-style:italic;">n</span> free variables defines a subset of <span class="texhtml"><b>R</b><sup><i>n</i></sup></span>, the set of the points that satisfy the formula. Such a subset is called a <a href="/wiki/Semialgebraic_set" title="Semialgebraic set">semialgebraic set</a>. Given a subset of <span class="texhtml mvar" style="font-style:italic;">k</span> variables, the <i>projection</i> from <span class="texhtml"><b>R</b><sup><i>n</i></sup></span> to <span class="texhtml"><b>R</b><sup><i>k</i></sup></span> is the <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> that maps every <span class="texhtml mvar" style="font-style:italic;">n</span>-tuple to the <span class="texhtml mvar" style="font-style:italic;">k</span>-tuple of the components corresponding to the subset of variables. The projection theorem asserts that a projection of a semialgebraic set is a semialgebraic set, and that there is an algorithm that, given a quantifier-free formula defining a semialgebraic set, produces a quantifier-free formula for its projection. </p><p>In fact, the projection theorem is equivalent to quantifier elimination, as the projection of a semialgebraic set defined by the formula <span class="texhtml"><i>p</i>(<i>x</i>, <i>y</i>)</span> is defined by </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,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∃</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\exists x)P(x,y),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca62774719a0fe1ac5a1baae608b09bb402524c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.152ex; height:2.843ex;" alt="{\displaystyle (\exists x)P(x,y),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> represent respectively the set of eliminated variables, and the set of kept variables. </p><p>The decidability of a first-order theory of the real numbers depends dramatically on the primitive operations and functions that are considered (here addition and multiplication). Adding other functions symbols, for example, the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> or the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a>, can provide undecidable theories; see <a href="/wiki/Richardson%27s_theorem" title="Richardson's theorem">Richardson's theorem</a> and <a href="/wiki/Decidability_of_first-order_theories_of_the_real_numbers" title="Decidability of first-order theories of the real numbers">Decidability of first-order theories of the real numbers</a>. </p><p>Furthermore, the completeness and decidability of the first-order theory of the real numbers (using addition and multiplication) contrasts sharply with <a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Gödel</a>'s and <a href="/wiki/Alan_Turing" title="Alan Turing">Turing</a>'s results about the incompleteness and undecidability of the first-order theory of the natural numbers (using addition and multiplication). There is no contradiction, since the statement "<i>x</i> is an integer" cannot be formulated as a first-order formula in the language <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 {\mathcal {L}}_{\text{rcf}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>rcf</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}_{\text{rcf}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d072ef411c420812eda7108cf7cd211947c5c8b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.714ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}_{\text{rcf}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Complexity_of_deciding_𝘛rcf"><span id="Complexity_of_deciding_.F0.9D.98.9Brcf"></span>Complexity of deciding 𝘛<sub>rcf</sub></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=5" title="Edit section: Complexity of deciding 𝘛rcf"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tarski's original algorithm for <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a> has <a href="/wiki/Nonelementary_problem" title="Nonelementary problem">nonelementary</a> <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a>, meaning that no tower </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 2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960c6481fd65c45d8e0e28aa548fb929b9b96898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.696ex; height:4.176ex;" alt="{\displaystyle 2^{2^{\cdot ^{\cdot ^{\cdot ^{n}}}}}}"></span></dd></dl> <p>can bound the execution time of the algorithm if <span class="texhtml mvar" style="font-style:italic;">n</span> is the size of the input formula. The <a href="/wiki/Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">cylindrical algebraic decomposition</a>, introduced by <a href="/wiki/George_E._Collins" title="George E. Collins">George E. Collins</a>, provides a much more practicable algorithm of complexity </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 d^{2^{O(n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2^{O(n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db3f9eb8e193641220f35716cb0dc2f91f92662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.294ex; height:3.176ex;" alt="{\displaystyle d^{2^{O(n)}}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">n</span> is the total number of variables (free and bound), <span class="texhtml mvar" style="font-style:italic;">d</span> is the product of the degrees of the polynomials occurring in the formula, and <span class="texhtml"><i>O</i>(<i>n</i>)</span> is <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>. </p><p>Davenport and Heintz (1988) proved that this <a href="/wiki/Worst-case_complexity" title="Worst-case complexity">worst-case complexity</a> is nearly optimal for quantifier elimination by producing a family <span class="texhtml">Φ<sub><i>n</i></sub></span> of formulas of length <span class="texhtml"><i>O</i>(<i>n</i>)</span>, with <span class="texhtml mvar" style="font-style:italic;">n</span> quantifiers, and involving polynomials of constant degree, such that any quantifier-free formula equivalent to <span class="texhtml">Φ<sub><i>n</i></sub></span> must involve polynomials of degree <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 2^{2^{\Omega (n)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{\Omega (n)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be01fb26016ef8e7a1a7951b95a6b3085823bb89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.184ex; height:3.176ex;" alt="{\displaystyle 2^{2^{\Omega (n)}}}"></span> and length <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 2^{2^{\Omega (n)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2^{\Omega (n)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05b04326beae993dfd47ff915e42dc061785e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.83ex; height:3.509ex;" alt="{\displaystyle 2^{2^{\Omega (n)}},}"></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 \Omega (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6becc31c61ad3420a1e4ee9e39c28baf73bda24d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle \Omega (n)}"></span> is <a href="/wiki/Big_Omega_notation" class="mw-redirect" title="Big Omega notation">big Omega notation</a>. This shows that both the time complexity and the space complexity of quantifier elimination are intrinsically <a href="/wiki/Double_exponential_time" class="mw-redirect" title="Double exponential time">double exponential</a>. </p><p>For the decision problem, Ben-Or, <a href="/wiki/Dexter_Kozen" title="Dexter Kozen">Kozen</a>, and <a href="/wiki/John_Reif" title="John Reif">Reif</a> (1986) claimed to have proved that the theory of real closed fields is decidable in <a href="/wiki/EXPSPACE" title="EXPSPACE">exponential space</a>, and therefore in double exponential time, but their argument (in the case of more than one variable) is generally held as flawed; see Renegar (1992) for a discussion. </p><p>For purely existential formulas, that is for formulas of the form </p> <dl><dd><span class="texhtml">∃<i>x</i><sub>1</sub>, ..., ∃<i>x</i><sub><i>k</i></sub> <i>P</i><sub>1</sub>(<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>k</i></sub>) ⋈ 0 ∧ ... ∧ <i>P</i><sub><i>s</i></sub>(<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>k</i></sub>) ⋈ 0,</span></dd></dl> <p>where <span class="texhtml">⋈</span> stands for either <span class="texhtml"><, ></span> or <span class="texhtml">=</span>, the complexity is lower. Basu and <a href="/wiki/Marie-Fran%C3%A7oise_Roy" title="Marie-Françoise Roy">Roy</a> (1996) provided a well-behaved algorithm to decide the truth of such an existential formula with complexity of <span class="texhtml"><i>s</i><sup><i>k</i>+1</sup><i>d</i><sup><i>O</i>(<i>k</i>)</sup></span> arithmetic operations and <a href="/wiki/PSPACE" title="PSPACE">polynomial space</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Order_properties">Order properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=6" title="Edit section: Order properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A crucially important property of the real numbers is that it is an <a href="/wiki/Archimedean_field" class="mw-redirect" title="Archimedean field">Archimedean field</a>, meaning it has the Archimedean property that for any real number, there is an <a href="/wiki/Integer" title="Integer">integer</a> larger than it in <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>. Note that this statement is not expressible in the first-order language of ordered fields, since it is not possible to quantify over integers in that language. </p><p>There are real-closed fields that are <a href="/wiki/Non-Archimedean_ordered_field" title="Non-Archimedean ordered field">non-Archimedean</a>; for example, any field of <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal numbers</a> is real closed and non-Archimedean. These fields contain infinitely large (larger than any integer) and infinitesimal (positive but smaller than any positive rational) elements. </p><p>The Archimedean property is related to the concept of <a href="/wiki/Cofinality" title="Cofinality">cofinality</a>. A set <i>X</i> contained in an ordered set <i>F</i> is cofinal in <i>F</i> if for every <i>y</i> in <i>F</i> there is an <i>x</i> in <i>X</i> such that <i>y</i> < <i>x</i>. In other words, <i>X</i> is an unbounded sequence in <i>F</i>. The cofinality of <i>F</i> is the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> of the smallest cofinal set, which is to say, the size of the smallest cardinality giving an unbounded sequence. For example, <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> are cofinal in the reals, and the cofinality of the reals is therefore <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 \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>. </p><p>We have therefore the following invariants defining the nature of a real closed field <i>F</i>: </p> <ul><li>The cardinality of <i>F</i>.</li> <li>The cofinality of <i>F</i>.</li></ul> <p>To this we may add </p> <ul><li>The weight of <i>F</i>, which is the minimum size of a dense subset of <i>F</i>.</li></ul> <p>These three cardinal numbers tell us much about the order properties of any real closed field, though it may be difficult to discover what they are, especially if we are not willing to invoke the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">generalized continuum hypothesis</a>. There are also particular properties that may or may not hold: </p> <ul><li>A field <i>F</i> is <b>complete</b> if there is no ordered field <i>K</i> properly containing <i>F</i> such that <i>F</i> is dense in <i>K</i>. If the cofinality of <i>F</i> is <i>κ</i>, this is equivalent to saying <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a> indexed by <i>κ</i> are <a href="/wiki/Convergent_sequence" class="mw-redirect" title="Convergent sequence">convergent</a> in <i>F</i>.</li> <li>An ordered field <i>F</i> has the <a href="/wiki/Eta_set" class="mw-redirect" title="Eta set">eta set</a> property η<sub><i>α</i></sub>, for the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a> <i>α</i>, if for any two subsets <i>L</i> and <i>U</i> of <i>F</i> of cardinality less than <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 \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span> such that every element of <i>L</i> is less than every element of <i>U</i>, there is an element <i>x</i> in <i>F</i> with <i>x</i> larger than every element of <i>L</i> and smaller than every element of <i>U</i>. This is closely related to the <a href="/wiki/Model-theoretic" class="mw-redirect" title="Model-theoretic">model-theoretic</a> property of being a <a href="/wiki/Saturated_model" title="Saturated model">saturated model</a>; any two real closed fields are η<sub><i>α</i></sub> if and only if they are <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 \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span>-saturated, and moreover two η<sub><i>α</i></sub> real closed fields both of cardinality <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 \aleph _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd1f42a5a41f25dc3689817aa40dca0ad1649bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.704ex; height:2.509ex;" alt="{\displaystyle \aleph _{\alpha }}"></span> are <a href="/wiki/Order_isomorphic" class="mw-redirect" title="Order isomorphic">order isomorphic</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="The_generalized_continuum_hypothesis">The generalized continuum hypothesis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=7" title="Edit section: The generalized continuum hypothesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The characteristics of real closed fields become much simpler if we are willing to assume the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">generalized continuum hypothesis</a>. If the continuum hypothesis holds, all real closed fields with <a href="/wiki/Cardinality_of_the_continuum" title="Cardinality of the continuum">cardinality of the continuum</a> and having the <i>η</i><sub>1</sub> property are order isomorphic. This unique field <i>Ϝ</i> can be defined by means of an <a href="/wiki/Ultraproduct" title="Ultraproduct">ultrapower</a>, as <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} ^{\mathbb {N} }/\mathbf {M} }"> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\mathbb {N} }/\mathbf {M} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eea02904b085b1af92724f063e88e7a8b75475f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.797ex; height:3.176ex;" alt="{\displaystyle \mathbb {R} ^{\mathbb {N} }/\mathbf {M} }"></span>, where <b>M</b> is a maximal ideal not leading to a field order-isomorphic 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>. This is the most commonly used <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal number field</a> in <a href="/wiki/Nonstandard_analysis" title="Nonstandard analysis">nonstandard analysis</a>, and its uniqueness is equivalent to the continuum hypothesis. (Even without the continuum hypothesis we have that if the cardinality of the continuum 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 \aleph _{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90b9e53511cd7409630a24b5412a3cead28b6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.594ex; height:2.843ex;" alt="{\displaystyle \aleph _{\beta }}"></span> then we have a unique <a href="/wiki/Eta_set" class="mw-redirect" title="Eta set"><i>η</i><sub><i>β</i></sub> field</a> of size <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 \aleph _{\beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>β</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{\beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90b9e53511cd7409630a24b5412a3cead28b6c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.594ex; height:2.843ex;" alt="{\displaystyle \aleph _{\beta }}"></span>.) </p><p>Moreover, we do not need ultrapowers to construct <i>Ϝ</i>, we can do so much more constructively as the subfield of series with a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countable</a> number of nonzero terms of the field <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} [[G]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [[G]]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d0c679a432a1b7895539cefa2d46c1e1bbde771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.092ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [[G]]}"></span> of <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> on a <a href="/wiki/Totally_ordered_group" class="mw-redirect" title="Totally ordered group">totally ordered</a> <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> <a href="/wiki/Divisible_group" title="Divisible group">divisible group</a> <i>G</i> that is an <a href="/wiki/Eta_set" class="mw-redirect" title="Eta set"><i>η</i><sub>1</sub> group</a> of cardinality <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 \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span> (<a href="#CITEREFAlling1962">Alling 1962</a>). </p><p><i>Ϝ</i> however is not a complete field; if we take its completion, we end up with a field <i>Κ</i> of larger cardinality. <i>Ϝ</i> has the cardinality of the continuum, which by hypothesis 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 \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>, <i>Κ</i> has cardinality <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 \aleph _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765bb468708b2eec66bf2dc6505a4c92959d697d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{2}}"></span>, and contains <i>Ϝ</i> as a dense subfield. It is not an ultrapower but it <i>is</i> a hyperreal field, and hence a suitable field for the usages of nonstandard analysis. It can be seen to be the higher-dimensional analogue of the real numbers; with cardinality <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 \aleph _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765bb468708b2eec66bf2dc6505a4c92959d697d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{2}}"></span> 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 \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span>, cofinality <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 \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span> 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 \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>, and weight <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 \aleph _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78c211ce8badf4ffbf9417ecceb0ef7ab0a8caed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{1}}"></span> 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 \aleph _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/721cd7f8c15a2e72ad162bdfa5baea8eef98aab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle \aleph _{0}}"></span>, and with the <i>η</i><sub>1</sub> property in place of the <i>η</i><sub>0</sub> property (which merely means between any two real numbers we can find another). </p> <div class="mw-heading mw-heading2"><h2 id="Elementary_Euclidean_geometry">Elementary Euclidean geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=8" title="Edit section: Elementary Euclidean geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Tarski%27s_axioms" title="Tarski's axioms">Tarski's axioms</a> are an axiom system for the first-order ("elementary") portion of <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. Using those axioms, one can show that the points on a line form a real closed field R, and one can introduce coordinates so that the Euclidean plane is identified with R<sup>2</sup> . Employing the decidability of the theory of real closed fields, Tarski then proved that the elementary theory of Euclidean geometry is complete and decidable.<sup id="cite_ref-:0_4-1" class="reference"><a href="#cite_note-:0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <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=Real_closed_field&action=edit&section=9" 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">D. Macpherson <i>et al.</i> (1998)</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">Rajwade (1993) pp. 222–223</span> </li> <li id="cite_note-Efr177-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Efr177_3-0">^</a></b></span> <span class="reference-text">Efrat (2006) p. 177</span> </li> <li id="cite_note-:0-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_4-1"><sup><i><b>b</b></i></sup></a></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): <span class="nowrap">91–</span>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=%3Cspan+class%3D%22nowrap%22%3E91-%3C%2Fspan%3E93&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%3AReal+closed+field" class="Z3988"></span></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=Real_closed_field&action=edit&section=10" 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="CITEREFAlling1962" class="citation cs2">Alling, Norman L. (1962), "On the existence of real-closed fields that are η<sub>α</sub>-sets of power ℵ<sub>α</sub>.", <i>Trans. Amer. Math. Soc.</i>, <b>103</b>: <span class="nowrap">341–</span>352, <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%2FS0002-9947-1962-0146089-X">10.1090/S0002-9947-1962-0146089-X</a></span>, <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=0146089">0146089</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Trans.+Amer.+Math.+Soc.&rft.atitle=On+the+existence+of+real-closed+fields+that+are+%CE%B7%3Csub%3E%CE%B1%3C%2Fsub%3E-sets+of+power+%E2%84%B5%3Csub%3E%CE%B1%3C%2Fsub%3E.&rft.volume=103&rft.pages=%3Cspan+class%3D%22nowrap%22%3E341-%3C%2Fspan%3E352&rft.date=1962&rft_id=info%3Adoi%2F10.1090%2FS0002-9947-1962-0146089-X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0146089%23id-name%3DMR&rft.aulast=Alling&rft.aufirst=Norman+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li>Basu, Saugata, <a href="/wiki/Richard_M._Pollack" title="Richard M. Pollack">Richard Pollack</a>, and <a href="/wiki/Marie-Fran%C3%A7oise_Roy" title="Marie-Françoise Roy">Marie-Françoise Roy</a> (2003) "Algorithms in real algebraic geometry" in <i>Algorithms and computation in mathematics</i>. Springer. <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/3-540-33098-4" title="Special:BookSources/3-540-33098-4">3-540-33098-4</a> (<a rel="nofollow" class="external text" href="http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html">online version</a>)</li> <li>Michael Ben-Or, Dexter Kozen, and John Reif, <i><a rel="nofollow" class="external text" href="http://www.cs.duke.edu/~reif/paper/benor/realclosed.pdf">The complexity of elementary algebra and geometry</a></i>, Journal of Computer and Systems Sciences 32 (1986), no. 2, pp. 251–264.</li> <li>Caviness, B F, and Jeremy R. Johnson, eds. (1998) <i>Quantifier elimination and cylindrical algebraic decomposition</i>. Springer. <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/3-211-82794-3" title="Special:BookSources/3-211-82794-3">3-211-82794-3</a></li> <li><a href="/wiki/Chen_Chung_Chang" title="Chen Chung Chang">Chen Chung Chang</a> and <a href="/wiki/Howard_Jerome_Keisler" title="Howard Jerome Keisler">Howard Jerome Keisler</a> (1989) <i>Model Theory</i>. North-Holland.</li> <li>Dales, H. G., and <a href="/wiki/W._Hugh_Woodin" title="W. Hugh Woodin">W. Hugh Woodin</a> (1996) <i>Super-Real Fields</i>. Oxford Univ. Press.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavenportHeintz1988" class="citation journal cs1"><a href="/wiki/James_H._Davenport" title="James H. Davenport">Davenport, James H.</a>; <a href="/wiki/Joos_Ulrich_Heintz" title="Joos Ulrich Heintz">Heintz, Joos</a> (1988). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0747-7171%2888%2980004-x">"Real quantifier elimination is doubly exponential"</a>. <i>J. Symb. Comput</i>. <b>5</b> (<span class="nowrap">1–</span>2): <span class="nowrap">29–</span>35. <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/10.1016%2Fs0747-7171%2888%2980004-x">10.1016/s0747-7171(88)80004-x</a></span>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0663.03015">0663.03015</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Symb.+Comput.&rft.atitle=Real+quantifier+elimination+is+doubly+exponential&rft.volume=5&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E29-%3C%2Fspan%3E35&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0663.03015%23id-name%3DZbl&rft_id=info%3Adoi%2F10.1016%2Fs0747-7171%2888%2980004-x&rft.aulast=Davenport&rft.aufirst=James+H.&rft.au=Heintz%2C+Joos&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fs0747-7171%252888%252980004-x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEfrat2006" class="citation book cs1">Efrat, Ido (2006). <i>Valuations, orderings, and Milnor </i>K<i>-theory</i>. Mathematical Surveys and Monographs. Vol. 124. Providence, RI: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-4041-X" title="Special:BookSources/0-8218-4041-X"><bdi>0-8218-4041-X</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1103.12002">1103.12002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Valuations%2C+orderings%2C+and+Milnor+K-theory&rft.place=Providence%2C+RI&rft.series=Mathematical+Surveys+and+Monographs&rft.pub=American+Mathematical+Society&rft.date=2006&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1103.12002%23id-name%3DZbl&rft.isbn=0-8218-4041-X&rft.aulast=Efrat&rft.aufirst=Ido&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li>Macpherson, D., Marker, D. and Steinhorn, C., <i>Weakly o-minimal structures and real closed fields</i>, Trans. of the American Math. Soc., Vol. 352, No. 12, 1998.</li> <li>Mishra, Bhubaneswar (1997) "<a rel="nofollow" class="external text" href="http://www.cs.nyu.edu/mishra/PUBLICATIONS/97.real-alg.ps">Computational Real Algebraic Geometry</a>," in <i>Handbook of Discrete and Computational Geometry</i>. CRC Press. 2004 edition, p. 743. <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/1-58488-301-4" title="Special:BookSources/1-58488-301-4">1-58488-301-4</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRajwade1993" class="citation book cs1">Rajwade, A. R. (1993). <i>Squares</i>. London Mathematical Society Lecture Note Series. Vol. 171. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-42668-5" title="Special:BookSources/0-521-42668-5"><bdi>0-521-42668-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0785.11022">0785.11022</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Squares&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1993&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0785.11022%23id-name%3DZbl&rft.isbn=0-521-42668-5&rft.aulast=Rajwade&rft.aufirst=A.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRenegar1992" class="citation journal cs1">Renegar, James (1992). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0747-7171%2810%2980003-3">"On the computational complexity and geometry of the first-order theory of the reals. Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals"</a>. <i>Journal of Symbolic Computation</i>. <b>13</b> (3): <span class="nowrap">255–</span>299. <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/10.1016%2FS0747-7171%2810%2980003-3">10.1016/S0747-7171(10)80003-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Computation&rft.atitle=On+the+computational+complexity+and+geometry+of+the+first-order+theory+of+the+reals.+Part+I%3A+Introduction.+Preliminaries.+The+geometry+of+semi-algebraic+sets.+The+decision+problem+for+the+existential+theory+of+the+reals&rft.volume=13&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E255-%3C%2Fspan%3E299&rft.date=1992&rft_id=info%3Adoi%2F10.1016%2FS0747-7171%2810%2980003-3&rft.aulast=Renegar&rft.aufirst=James&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252FS0747-7171%252810%252980003-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPassmore2011" class="citation thesis cs1"><a href="/wiki/Grant_Olney" title="Grant Olney">Passmore, Grant</a> (2011). <a rel="nofollow" class="external text" href="https://www.cl.cam.ac.uk/~gp351/passmore-phd-thesis.pdf"><i>Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex</i></a> <span class="cs1-format">(PDF)</span> (PhD). <a href="/wiki/University_of_Edinburgh" title="University of Edinburgh">University of Edinburgh</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Combined+Decision+Procedures+for+Nonlinear+Arithmetics%2C+Real+and+Complex&rft.inst=University+of+Edinburgh&rft.date=2011&rft.aulast=Passmore&rft.aufirst=Grant&rft_id=https%3A%2F%2Fwww.cl.cam.ac.uk%2F~gp351%2Fpassmore-phd-thesis.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li> <li><a href="/wiki/Alfred_Tarski" title="Alfred Tarski">Alfred Tarski</a> (1951) <i>A Decision Method for Elementary Algebra and Geometry</i>. Univ. of California Press.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErdösGillmanHenriksen1955" class="citation cs2">Erdös, P.; Gillman, L.; Henriksen, M. (1955), <a rel="nofollow" class="external text" href="https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=2016&context=hmc_fac_pub">"An isomorphism theorem for real-closed fields"</a>, <i>Ann. of Math.</i>, 2, <b>61</b> (3): <span class="nowrap">542–</span>554, <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%2F1969812">10.2307/1969812</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/1969812">1969812</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=0069161">0069161</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+of+Math.&rft.atitle=An+isomorphism+theorem+for+real-closed+fields&rft.volume=61&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E542-%3C%2Fspan%3E554&rft.date=1955&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0069161%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969812%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1969812&rft.aulast=Erd%C3%B6s&rft.aufirst=P.&rft.au=Gillman%2C+L.&rft.au=Henriksen%2C+M.&rft_id=https%3A%2F%2Fscholarship.claremont.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D2016%26context%3Dhmc_fac_pub&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReal+closed+field" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Real_closed_field&action=edit&section=11" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; 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