CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> real number in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> real number </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4322/#Item_37" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="arithmetic">Arithmetic</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/arithmetic+topology">arithmetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+arithmetic+geometry">higher arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+arithmetic+geometry">E-∞ arithmetic geometry</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/number">number</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/integer+number">integer number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, <a class="existingWikiWord" href="/nlab/show/irrational+number">irrational number</a>, <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a>, <a class="existingWikiWord" href="/nlab/show/quaternion">quaternion</a>, <a class="existingWikiWord" href="/nlab/show/octonion">octonion</a>, <a class="existingWikiWord" href="/nlab/show/adic+number">adic number</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+number">cardinal number</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+number">ordinal number</a>, <a class="existingWikiWord" href="/nlab/show/surreal+number">surreal number</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic">arithmetic</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+arithmetic">Peano arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/second-order+arithmetic">second-order arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfinite+arithmetic">transfinite arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/cardinal+arithmetic">cardinal arithmetic</a>, <a class="existingWikiWord" href="/nlab/show/ordinal+arithmetic">ordinal arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+integer">p-adic integer</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+rational+number">p-adic rational number</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+complex+number">p-adic complex number</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></strong>, <a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+scheme">arithmetic scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+genus">arithmetic genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chern-Simons+theory">arithmetic Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Chow+group">arithmetic Chow group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil-%C3%A9tale+topology+for+arithmetic+schemes">Weil-étale topology for arithmetic schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absolute+cohomology">absolute cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+conjecture+on+Tamagawa+numbers">Weil conjecture on Tamagawa numbers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borger%27s+absolute+geometry">Borger's absolute geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Iwasawa-Tate+theory">Iwasawa-Tate theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adelic+integration">adelic integration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shtuka">shtuka</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenioid">Frobenioid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Arakelov+geometry">Arakelov geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+Riemann-Roch+theorem">arithmetic Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebraic+K-theory">differential algebraic K-theory</a></p> </li> </ul> </div></div> <h4 id="analysis">Analysis</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/analysis">analysis</a></strong> (<a class="existingWikiWord" href="/nlab/show/differential+calculus">differential</a>/<a class="existingWikiWord" href="/nlab/show/integral+calculus">integral</a> <a class="existingWikiWord" href="/nlab/show/calculus">calculus</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a>, <a class="existingWikiWord" href="/nlab/show/topology">topology</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+analysis">infinitesimal analysis</a></p> <p><a class="existingWikiWord" href="/nlab/show/computable+analysis">computable analysis</a></p> <p><em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">Introduction</a></em></p> <h2 id="basic_concepts">Basic concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a></p> <p><a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> <p><a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a></p> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a>, <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limit of a sequence</a></p> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compactness</a>, <a class="existingWikiWord" href="/nlab/show/sequentially+compact+space">sequential compactness</a></p> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/integration">integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a></p> <h2 id="basic_facts">Basic facts</h2> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> <p>…</p> <h2 id="theorems">Theorems</h2> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> <p>…</p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#history'>History</a></li> <li><a href='#definitions_and_characterizations'>Definitions and characterizations</a></li> <ul> <li><a href='#the_terminal_archimedean_ordered_field'>The terminal Archimedean ordered field</a></li> <li><a href='#Dedekind'>Dedekind cuts</a></li> <li><a href='#cauchy_sequences'>Cauchy sequences</a></li> <li><a href='#the_initial_sequentially_modulated_cauchy_complete_archimedean_field'>The initial sequentially modulated Cauchy complete archimedean field</a></li> <li><a href='#the_dedekind_complete_ordered_field'>The Dedekind complete ordered field</a></li> <li><a href='#the_cauchy_complete_ordered_field'>The Cauchy complete ordered field</a></li> <li><a href='#as_an_archimedean_tarski_group'>As an archimedean Tarski group</a></li> <li><a href='#the_locale_of_real_numbers'>The locale of real numbers</a></li> <li><a href='#unit_intervals'>Unit intervals</a></li> <li><a href='#_as_a_terminal_coalgebra'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+</annotation></semantics></math> as a terminal coalgebra</a></li> <li><a href='#smooth_real_numbers'>Smooth real numbers</a></li> <li><a href='#other_definitions'>Other definitions</a></li> </ul> <li><a href='#Topologies'>Topologies</a></li> <li><a href='#generalisations'>Generalisations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>real number</em> is a <a class="existingWikiWord" href="/nlab/show/number">number</a> that may be <a class="existingWikiWord" href="/nlab/show/Dedekind+completion">approximated</a> by <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>. Equipped with the operations of <a class="existingWikiWord" href="/nlab/show/addition">addition</a> and <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a> induced from the rational numbers, real numbers form a <em><a class="existingWikiWord" href="/nlab/show/field">field</a></em>, commonly denoted <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></em>. The underlying set is the <em><a class="existingWikiWord" href="/nlab/show/Dedekind+completion">completion</a></em> of the <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> of rational numbers: the result of adjoining to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/suprema">suprema</a> for every <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> <a class="existingWikiWord" href="/nlab/show/bounded+set">bounded subset</a> with respect to the natural <a class="existingWikiWord" href="/nlab/show/order">ordering</a> of rational numbers.</p> <p>The <a class="existingWikiWord" href="/nlab/show/set">set</a> of real numbers also carries naturally the <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and as such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/real+line">real line</a></em> also known as <em><a class="existingWikiWord" href="/nlab/show/continuum">the continuum</a></em>. Equipped with both the topology and the field structure, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/topological+field">topological field</a> and as such is the <a class="existingWikiWord" href="/nlab/show/complete+space">uniform completion</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> <a class="existingWikiWord" href="/nlab/show/metric+space">metric</a>.</p> <p>Together with its <a class="existingWikiWord" href="/nlab/show/cartesian+products">cartesian products</a> – the <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> – the real line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is a standard formalization of the idea of <em>continuous space</em>. The more general concept of (<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a>) <em><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></em> is modeled on these Cartesian spaces. These, in turn are standard models for the notion of <a class="existingWikiWord" href="/nlab/show/space">space</a> in particular in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (see <em><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></em>), or at least in <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a>. See at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em> for more on this.</p> <h2 id="history">History</h2> <p>The original idea of a real number came from <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>; one thinks of a real number as specifying a <em>point on a line</em>, with <em><a class="existingWikiWord" href="/nlab/show/line">line</a></em> understood as the abstract idea of the object that a pencil and a ruler draw on a piece of paper. (More precisely, given two distinct points on the line, called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, you get a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> between the points and the real numbers.)</p> <p><a class="existingWikiWord" href="/nlab/show/Euclid">Euclid</a> (citing <a class="existingWikiWord" href="/nlab/show/Eudoxus">Eudoxus</a>) dealt with ratios of geometric magnitudes, which give positive real numbers; an arbitrary real number is then a difference of ratios of magnitudes. However, the Greeks did not think of such ratios as <a class="existingWikiWord" href="/nlab/show/number">number</a>s; that appears to have been an insight of the Arabs. See more at <a class="existingWikiWord" href="/nlab/show/Eudoxus+real+number">Eudoxus real number</a>.</p> <p>A big project of the 19th century (at least in hindsight) was the ‘arithmetisation of analysis’: showing how real numbers could be defined completely in terms of rational numbers (and the desired classes of functions on them could be defined in terms of the general point-set notion of <a class="existingWikiWord" href="/nlab/show/function">function</a>). Two successful approaches were developed in 1872, <a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Richard Dedekind</a>'s definition of real numbers as certain sets of rational numbers (called <em><a class="existingWikiWord" href="/nlab/show/Dedekind+cuts">Dedekind cuts</a></em>) and <a class="existingWikiWord" href="/nlab/show/Georg+Cantor">Georg Cantor</a>'s definition as certain sequences of rational numbers (called <em><a class="existingWikiWord" href="/nlab/show/Cauchy+sequences">Cauchy sequences</a></em>).</p> <p>A more modern approach is instead to characterise the properties that the set of real numbers must have and to prove that this is <a class="existingWikiWord" href="/nlab/show/generalized+the">categorical</a> (unique up to a unique bijection preserving those properties). Then the important result of the 19th-century programme is simply that this is consistent (that there exists at least one such set). One can even use <a class="existingWikiWord" href="/nlab/show/David+Hilbert">Hilbert</a>'s or <a class="existingWikiWord" href="/nlab/show/Alfred+Tarski">Tarski</a>'s axioms for geometry to do this characterisation, coming full circle back to geometry.</p> <p>Exactly how to define or characterise real numbers is still important in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> and <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> with its <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>. For more on this, see <a class="existingWikiWord" href="/nlab/show/real+numbers+object">real numbers object</a> and the examples below.</p> <h2 id="definitions_and_characterizations">Definitions and characterizations</h2> <p>There are two basic approaches possible: to define what a <strong>real number</strong> is as a mathematical object, or to define the <strong>real line</strong> as a specific object in some previously known <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <h3 id="the_terminal_archimedean_ordered_field">The terminal Archimedean ordered field</h3> <p>There is a well-known algebraic (more or less) characterisation of the real line as the ‘terminal Archimedean ordered field’ (cf. <a href="#Richman08">Richman08</a>) This can be interpreted as follows:</p> <ul> <li>A <strong>field</strong> is well known in algebra; if it matters, we mean a <a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a>.</li> <li>An <strong>ordered field</strong> means a <em><a class="existingWikiWord" href="/nlab/show/strictly+totally+ordered">strictly totally ordered</a></em> field.</li> <li>An <strong><a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+field">Archimedean ordered field</a></strong> is an ordered field satisfying the <a class="existingWikiWord" href="/nlab/show/Archimedean+property">Archimedean property</a>.</li> <li>An Archimedean ordered field is <strong>terminal</strong> if it is a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> in the category of Archimedean ordered fields.</li> </ul> <p>We speak of <a class="existingWikiWord" href="/nlab/show/the">the</a> such field because it is unique up to unique <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> by definition of <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a>, one can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}</annotation></semantics></math> by fiat via the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the real numbers as the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+field">Archimedean ordered field</a>. From this characterization, the real numbers are a <a class="existingWikiWord" href="/nlab/show/terminal+coalgebra">terminal coalgebra</a> of the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity endofunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \mapsto X</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+fields">Archimedean ordered fields</a>.</p> <p>If <a class="existingWikiWord" href="/nlab/show/quotient+sets">quotient sets</a> exist and <a class="existingWikiWord" href="/nlab/show/sequence+algebras">sequence algebras</a> of Archimedean ordered fields exist, then it is provable that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}</annotation></semantics></math> is Cauchy complete. From the definition of terminal object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}</annotation></semantics></math> is an algebra of the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto \mathcal{C}(X)</annotation></semantics></math> in the category of Archimedean ordered fields which takes Archimedean ordered fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the Archimedean ordered field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X)</annotation></semantics></math> of equivalence classes of Cauchy sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Every algebra of the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto \mathcal{C}(X)</annotation></semantics></math> in the category of Archimedean ordered fields is a Cauchy complete Archimedean ordered field.</p> <p>Similarly, if <a class="existingWikiWord" href="/nlab/show/power+sets">power sets</a> of Archimedean ordered fields exist, then it is provable that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}</annotation></semantics></math> is Dedekind complete. From the definition of terminal object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>R</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{R}</annotation></semantics></math> is an algebra of the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>𝒟</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto \mathcal{D}(X)</annotation></semantics></math> in the category of Archimedean ordered fields which takes Archimedean ordered fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the Archimedean ordered field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X)</annotation></semantics></math> of two-sided Dedekind cuts in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Every algebra of the endofunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mi>𝒟</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto \mathcal{D}(X)</annotation></semantics></math> in the category of Archimedean ordered fields is a Dedekind complete Archimedean ordered field.</p> <h3 id="Dedekind">Dedekind cuts</h3> <p>Consider two <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> subsets, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, of a <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> unbounded <a class="existingWikiWord" href="/nlab/show/dense+linear+order">dense linear order</a>, such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>10</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/10]</annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/decimal+fractions">decimal fractions</a>), such that:</p> <ul> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">a \in L</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">b \in L</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>></mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b \gt a</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">b \in U</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">a \in U</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math> are rational numbers, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">a \in L</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">b \in U</annotation></semantics></math>. (*)</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">a \in L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">b \in U</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math>.</li> </ul> <p>We may define a <strong>Dedekind real number</strong> to be such a pair, which is also called a <strong><a class="existingWikiWord" href="/nlab/show/Dedekind+cut">Dedekind cut</a></strong>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \coloneqq (L,U)</annotation></semantics></math> is a Dedekind cut, then we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">a \lt x</annotation></semantics></math> to mean that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">a \in L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">x \lt b</annotation></semantics></math> to mean that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">b \in U</annotation></semantics></math>.</p> <p>We may approximate a Dedekind cut <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> as closely as we like by applying (*) as often as necessary. This will be only finitely often, for any fixed positive level of approximation, given initial upper and lower bounds (which exist since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> are inhabited).</p> <p>See <a class="existingWikiWord" href="/nlab/show/Dedekind+completion">Dedekind completion</a> for more.</p> <h3 id="cauchy_sequences">Cauchy sequences</h3> <p>Classically, a real number can be given by an <a class="existingWikiWord" href="/nlab/show/infinite+sequence">infinite</a> <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a> of <a class="existingWikiWord" href="/nlab/show/decimal+fractions">decimal fractions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>10</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/10]</annotation></semantics></math>, each of which is a decimal fraction that approximates the real number to a given number of decimal places. However, many real numbers have several representations, i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>10</mn><mo>=</mo><mn>0.099999</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><mn>0.10000</mn><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">1/10 = 0.099999... = 0.10000...</annotation></semantics></math></div> <p>so we need to specify an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on the Cauchy sequences. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is constructed as a <a class="existingWikiWord" href="/nlab/show/subquotient">subquotient</a> of the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">/</mo><mn>10</mn><msup><mo stretchy="false">]</mo> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{Z}[1/10]^{\mathbb{N}}</annotation></semantics></math>.</p> <p>We can generalise this to any <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a> of rational numbers, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is constructed as a <a class="existingWikiWord" href="/nlab/show/subquotient">subquotient</a> of the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℚ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{Q}^{\mathbb{N}}</annotation></semantics></math>.</p> <p>This construction is equivalent to the construction by Dedekind cuts, at least assuming the <a class="existingWikiWord" href="/nlab/show/weak+countable+choice">weak countable choice</a> principle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">AC</mi> <mrow><mi>ℕ</mi><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathrm{AC}_{\mathbb{N}2}</annotation></semantics></math> (which also follows from <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>). Thus it is popular in both <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a> and traditional <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> (which accepts <a class="existingWikiWord" href="/nlab/show/countable+choice">countable choice</a>). However, in stricter forms of constructive mathematics, including those used as <a class="existingWikiWord" href="/nlab/show/internal+languages">internal languages</a> in <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>, the Cauchy reals and Dedekind reals are not equivalent. (On the other hand, by generalising to Cauchy <a class="existingWikiWord" href="/nlab/show/nets">nets</a>, we recover the Dedekind reals again.)</p> <p>See <a class="existingWikiWord" href="/nlab/show/Cauchy+real+number">Cauchy real number</a> and <a class="existingWikiWord" href="/nlab/show/generalized+Cauchy+real+number">generalized Cauchy real number</a> for more.</p> <h3 id="the_initial_sequentially_modulated_cauchy_complete_archimedean_field">The initial sequentially modulated Cauchy complete archimedean field</h3> <p>There is an algebraic (more or less) characterisation of the real line as the ‘initial sequentially modulated Cauchy complete archimedean field’. This can be interpreted as follows:</p> <ul> <li>A <strong>field</strong> is well known in algebra; if it matters, we mean a <a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a>.</li> <li>An <strong>ordered field</strong> means a <em><a class="existingWikiWord" href="/nlab/show/strict+total+order">strictly totally</a></em> ordered field.</li> <li>An <strong>archimedean field</strong> is an ordered field satisfying the <a class="existingWikiWord" href="/nlab/show/archimedean+property">archimedean property</a>.</li> <li>An archimedean field is <strong><a class="existingWikiWord" href="/nlab/show/sequentially+modulated+Cauchy+complete">sequentially modulated Cauchy complete</a></strong> if all <a class="existingWikiWord" href="/nlab/show/Cauchy+sequences">Cauchy sequences</a> in the field with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modulus+of+convergence">modulus of convergence</a> <a class="existingWikiWord" href="/nlab/show/converge">converge</a>.</li> <li>A sequentially modulated Cauchy complete archimedean field is <strong>initial</strong> if it is an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the category of sequentially modulated Cauchy complete archimedean fields.</li> </ul> <p>The initial sequentially modulated Cauchy complete ordered field results in the <a class="existingWikiWord" href="/nlab/show/HoTT+book+real+numbers">HoTT book real numbers</a>.</p> <h3 id="the_dedekind_complete_ordered_field">The Dedekind complete ordered field</h3> <p>There is a well-known algebraic (more or less) characterisation of the real line as the ‘Dedekind complete ordered field’, or sometimes the ‘Dedekind complete archimedean field’. This can be interpreted as follows:</p> <ul> <li>A <strong>field</strong> is well known in algebra; if it matters, we mean a <a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a>.</li> <li>An <strong>ordered field</strong> means a <em><a class="existingWikiWord" href="/nlab/show/linear+order">linearly</a></em> ordered field.</li> <li>An <strong>archimedean field</strong> is an ordered field satisfying the <a class="existingWikiWord" href="/nlab/show/archimedean+property">archimedean property</a>.</li> <li>An ordered field is <strong>Dedekind complete</strong> if it is <a class="existingWikiWord" href="/nlab/show/Dedekind+completion">Dedekind-complete</a>:</li> </ul> <ol> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math>, the upwards unbounded <a class="existingWikiWord" href="/nlab/show/open+interval">open interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,\infty)</annotation></semantics></math> is inhabited.</li> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math>, the downwards unbounded <a class="existingWikiWord" href="/nlab/show/open+interval">open interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty,a)</annotation></semantics></math> is inhabited.</li> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">b \in F</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(b,\infty)</annotation></semantics></math> is a subinterval of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,\infty)</annotation></semantics></math></li> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">b \in F</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo><</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b \lt a</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty,b)</annotation></semantics></math> is a subinterval of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty,a)</annotation></semantics></math></li> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">b \in F</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a subinterval of the union of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, \infty)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty, b)</annotation></semantics></math></li> <li>For all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">a \in F</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">b \in F</annotation></semantics></math>, the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,\infty)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty,b)</annotation></semantics></math> is a subinterval of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,b)</annotation></semantics></math></li> </ol> <p>In impredicative mathematics, we speak of <a class="existingWikiWord" href="/nlab/show/the">the</a> such field because it is unique up to unique <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>Assuming impredicative foundations, there is an archimedean field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> which is Dedekind-complete, and into which every archimedean field embeds. Furthermore, every Dedekind-complete ordered field is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, and uniquely so.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> using, say, Dedekind cuts of rational numbers. Then it is well known how to prove these facts about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, so we omit the proof for now.</p> </div> <p>However, we note that the proof is valid in weak <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>, in particular internal to any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> with a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. One can actually work in even weaker foundations than that; see the constructions at <a class="existingWikiWord" href="/nlab/show/real+numbers+object">real numbers object</a>.</p> <p>Even weaker foundations are possible if one allows the <a class="existingWikiWord" href="/nlab/show/underlying+set">underlying set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> to be <a class="existingWikiWord" href="/nlab/show/proper+class">large</a>.</p> <p>However, if we are working in <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a> with multiple <a class="existingWikiWord" href="/nlab/show/universes">universes</a> in the foundation, the Dedekind real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Dedekind real numbers for each universe.</p> <h3 id="the_cauchy_complete_ordered_field">The Cauchy complete ordered field</h3> <p>There is a well-known algebraic (more or less) characterisation of the real line as the ‘Cauchy complete ordered field’, or sometimes the ‘Cauchy complete archimedean field’. This can be interpreted as follows:</p> <ul> <li>A <strong>field</strong> is well known in algebra; if it matters, we mean a <a class="existingWikiWord" href="/nlab/show/Heyting+field">Heyting field</a>.</li> <li>An <strong>ordered field</strong> means a <em><a class="existingWikiWord" href="/nlab/show/linear+order">linearly</a></em> ordered field.</li> <li>An <strong>archimedean field</strong> is an ordered field satisfying the <a class="existingWikiWord" href="/nlab/show/archimedean+property">archimedean property</a>.</li> <li>An ordered field is <strong><a class="existingWikiWord" href="/nlab/show/complete+space">Cauchy complete</a></strong> if every <a class="existingWikiWord" href="/nlab/show/Cauchy+net">Cauchy net</a> in the field <a class="existingWikiWord" href="/nlab/show/converges">converges</a>.</li> </ul> <p>In impredicative mathematics, we speak of <a class="existingWikiWord" href="/nlab/show/the">the</a> such field because it is unique up to unique <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <div class="un_theorem"> <h6 id="theorem_2">Theorem</h6> <p>Assuming impredicative foundations, there is an archimedean field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> which is Cauchy-complete, and into which every archimedean field embeds. Furthermore, every Cauchy-complete ordered field is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, and uniquely so.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> using, say, Cauchy nets of rational numbers. Then it is well known how to prove these facts about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, so we omit the proof for now.</p> </div> <p>However, we note that the proof is valid in weak <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>, in particular internal to any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> with a <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a>. One can actually work in even weaker foundations than that; see the constructions at <a class="existingWikiWord" href="/nlab/show/real+numbers+object">real numbers object</a>.</p> <p>Even weaker foundations are possible if one allows the <a class="existingWikiWord" href="/nlab/show/underlying+set">underlying set</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> to be <a class="existingWikiWord" href="/nlab/show/proper+class">large</a>.</p> <p>However, if we are working in <a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a> with multiple <a class="existingWikiWord" href="/nlab/show/universes">universes</a> in the foundation, the Cauchy real numbers would no longer be unique up to unique isomorphism, but rather there would be a set of Cauchy real numbers for each universe.</p> <h3 id="as_an_archimedean_tarski_group">As an archimedean Tarski group</h3> <p>There is a characterisation of the real line as the ‘complete archimedean Tarski group’ due to Alfred Tarski. This can be interpreted as follows:</p> <ul> <li>An <strong>abelian group</strong> is well known in algebra.</li> <li>An <strong>linearly ordered abelian group</strong> is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> with a <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>.</li> <li>A <strong>densely linearly ordered abelian group</strong> is an linearly ordered abelian group that is also a <a class="existingWikiWord" href="/nlab/show/dense+linear+order">dense linear order</a>.</li> <li>A <strong>Tarski group</strong> is a densely linearly ordered abelian group with a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo><</mo><mn>1</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \lt 1 + 1</annotation></semantics></math>.</li> <li>An <strong>archimedean Tarski group</strong> is a <a class="existingWikiWord" href="/nlab/show/Tarski+group">Tarski group</a> satisfying the <a class="existingWikiWord" href="/nlab/show/archimedean+property">archimedean property</a>.</li> <li>An archimedean Tarski group is <strong>complete</strong> if it is <a class="existingWikiWord" href="/nlab/show/Dedekind+completion">Dedekind-complete</a>.</li> </ul> <p>See <a class="existingWikiWord" href="/nlab/show/Tarski%27s+axiomatization+of+the+real+numbers">Tarski's axiomatization of the real numbers</a> for more information.</p> <h3 id="the_locale_of_real_numbers">The locale of real numbers</h3> <p>Consider <a class="existingWikiWord" href="/nlab/show/binary+relations">binary relations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/countable">countable</a> <a class="existingWikiWord" href="/nlab/show/inhabited">inhabited</a> <a class="existingWikiWord" href="/nlab/show/dense+linear+order">dense linear order</a> without endpoints, such as the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, satisfying these four properties:</p> <ul> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≥</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \geq b</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∼</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \sim b</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>≥</mo><mi>b</mi><mo>∼</mo><mi>c</mi><mo>≥</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a \geq b \sim c \geq d</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∼</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a \sim d</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∼</mo><mi>b</mi><mo>></mo><mi>c</mi><mo>∼</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a \sim b \gt c \sim d</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∼</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a \sim d</annotation></semantics></math>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∼</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">b \sim c</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \lt b</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo><</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">c \lt d</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∼</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">a \sim d</annotation></semantics></math>.</li> </ul> <p>The collection of all such relations form a <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, which we may interpret (by definition) as the <strong><a class="existingWikiWord" href="/nlab/show/locale+of+real+numbers">locale of real numbers</a></strong>. It can also be defined as the <a class="existingWikiWord" href="/nlab/show/localic+completion">localic completion</a> of the rational numbers.</p> <p>We may then define a <strong>localic real number</strong> to be a <a class="existingWikiWord" href="/nlab/show/point+of+a+locale">point</a> of this locale.</p> <p>This agrees with the notion of Dedekind real number, even in very weak (<a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative</a> and <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a>) <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a>.</p> <p>See <a class="existingWikiWord" href="/nlab/show/locale+of+real+numbers">locale of real numbers</a> for more.</p> <h3 id="unit_intervals">Unit intervals</h3> <p>The <a class="existingWikiWord" href="/nlab/show/coalgebra+of+the+real+interval">unit interval</a> of the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> could be constructed as a <a class="existingWikiWord" href="/nlab/show/terminal+coalgebra+of+an+endofunctor">terminal coalgebra of an endofunctor</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/intervals">intervals</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℝ</mi><mo>,</mo><mn>0</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋅</mo><mo>,</mo><mo><</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{R},0,+,-,1,\cdot,\lt)</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0 \lt 1</annotation></semantics></math>, with a <a class="existingWikiWord" href="/nlab/show/monotone">monotone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:[0,1]\to \mathbb{R}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0) = 0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f(1) = 1</annotation></semantics></math>. The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> of real numbers is the initial such ordered field.</p> <p>See also: <a class="existingWikiWord" href="/nlab/show/dyadic+interval+coalgebra">dyadic interval coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/decimal+interval+coalgebra">decimal interval coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/rational+interval+coalgebra">rational interval coalgebra</a>.</p> <h3 id="_as_a_terminal_coalgebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+</annotation></semantics></math> as a terminal coalgebra</h3> <p>The positive real line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+</annotation></semantics></math> may be characterized as the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/coalgebra+for+an+endofunctor">coalgebra for an endofunctor</a></p> <p>Let <a class="existingWikiWord" href="/nlab/show/Pos">Pos</a> be the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/poset">poset</a>s with a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Pos</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> U\colon Pos \to Set </annotation></semantics></math></div> <p>Consider the endofunctor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Pos</mi><mo>→</mo><mi>Pos</mi></mrow><annotation encoding="application/x-tex"> F_1\colon Pos \to Pos </annotation></semantics></math></div> <p>defined as the <span class="newWikiWord">ordinal product<a href="/nlab/new/ordinal+product">?</a></span></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↦</mo><mi>ω</mi><mo>⋅</mo><mi>X</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> F_1\colon X \mapsto \omega \cdot X, </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><mi>Pos</mi></mrow><annotation encoding="application/x-tex">\omega \in Pos</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>⋅</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\omega \cdot X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\omega) \times U(X)</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/lexicographic+order">lexicographic order</a>.</p> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>The terminal coalgebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> is order isomorphic to the non-negative real line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+</annotation></semantics></math>, with its standard order.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This is theorem 5.1 in (<a href="#PavlovicPratt">Pavlovic–Pratt 1999</a>).</p> <p>There are many ways of setting up this description of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+</annotation></semantics></math>, depending on the coalgebra structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup><mo>→</mo><mi>ω</mi><mo>⋅</mo><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^+ \to \omega \cdot \mathbb{R}^+</annotation></semantics></math> chosen. Here is one: there are evident poset isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup><mo>≅</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^+ \cong [1, \infty)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>≅</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msub><mo>=</mo><mo stretchy="false">{</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>:</mo><mi>n</mi><mo>≥</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\omega \cong \mathbb{N}_{\geq 2} = \{n \in \mathbb{N}: n \geq 2\}</annotation></semantics></math>. Define a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msub><mo>⋅</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\alpha, \beta): [1, \infty) \to \mathbb{N}_{\geq 2} \cdot [1, \infty)</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha(x)</annotation></semantics></math> is the smallest integer strictly greater than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta(x) = 1/(\alpha(x) - x)</annotation></semantics></math>. The stream of integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>n</mi></msub><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msup><mi>β</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_n = \alpha(\beta^n(x))</annotation></semantics></math> gives a <a class="existingWikiWord" href="/nlab/show/continued+fraction">continued fraction</a> representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>2</mn></msub><mo>−</mo><mi>…</mi></mrow></mfrac></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex">x = a_0 - \frac1{a_1 - \frac1{a_2 - \ldots}},</annotation></semantics></math></div> <p>and the resulting bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msub><mo>×</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msub><mo>×</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">[1, \infty) \to \mathbb{N}_{\geq 2} \times \mathbb{N}_{\geq 2} \times \ldots</annotation></semantics></math>, sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_0, a_1, \ldots)</annotation></semantics></math>, is in fact a poset isomorphism if we endow the right-hand side with the lexicographic order.</p> </div> <p>Another way, which circumvents the use of isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup><mo>≅</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^+ \cong [1, \infty)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>≅</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\omega \cong \mathbb{N}_{\geq 2}</annotation></semantics></math>, is to define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>ℝ</mi> <mo>+</mo></msup><mo>→</mo><mi>ω</mi><mo>⋅</mo><msup><mi>ℝ</mi> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">(\alpha, \beta): \mathbb{R}^+ \to \omega \cdot \mathbb{R}^+</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha(x)</annotation></semantics></math> is the floor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>+</mo><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta(x) = 1/(1 - x + \alpha(x)) - 1</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mi>n</mi></msub><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msup><mi>β</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a_n = \alpha(\beta^n(x))</annotation></semantics></math> gives a <a class="existingWikiWord" href="/nlab/show/continued+fraction">continued fraction</a> representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><msub><mi>a</mi> <mn>2</mn></msub><mo>+</mo><mi>…</mi></mrow></mfrac></mrow></mfrac></mrow></mfrac></mrow></mfrac><mo>,</mo></mrow><annotation encoding="application/x-tex">x = a_0 + \frac1{1 + \frac1{a_1 + \frac1{1 + \frac1{a_2 + \ldots}}}},</annotation></semantics></math></div> <p>and the resulting <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mo>+</mo></msub><mo>→</mo><mi>ω</mi><mo>×</mo><mi>ω</mi><mo>×</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}_+ \to \omega \times \omega \times \ldots</annotation></semantics></math>, sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>a</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a_0, a_1, \ldots)</annotation></semantics></math>, is again a poset isomorphism if we endow the right-hand side with the lexicographic order.</p> <p>There are more and similar characterizations along these lines. An interesting application of the coalgebraic views is that they allow defining morphisms between the reals. A coalgebaic version Dedekind cuts or Conway numbers makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> into a compact category where the addition arises as the biproduct. The real vector spaces are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-enriched bicompletions and linear operators arise as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-enriched Kan extensions. (<a href="#Pavlovic">Pavlovic 2020</a>)</p> <h3 id="smooth_real_numbers">Smooth real numbers</h3> <p>The <strong>smooth real numbers</strong> (say in a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>) are only an <a class="existingWikiWord" href="/nlab/show/ordered+local+ring">ordered local ring</a> rather than an <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a>, because there might be non-zero non-invertible <a class="existingWikiWord" href="/nlab/show/nilpotent">nilpotent</a> <a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a> in the smooth real numbers. Nevertheless, the quotient of the smooth real numbers by the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> of non-invertible elements is one of the ordered field of real numbers as defined above.</p> <h3 id="other_definitions">Other definitions</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eudoxus+real+number">Eudoxus real number</a></li> <li><a class="existingWikiWord" href="/nlab/show/prealgebra+real+number">prealgebra real number</a></li> </ul> <h2 id="Topologies">Topologies</h2> <p>There are alternative topologies on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> sometimes considered:</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a> (not very interesting),</li> <li>the <a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a> or <span class="newWikiWord">absolute-value topology<a href="/nlab/new/absolute-value+topology">?</a></span> (the usual topology of a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>),</li> <li>the <span class="newWikiWord">upper-interval topology<a href="/nlab/new/upper-interval+topology">?</a></span> or the <span class="newWikiWord">lower-interval topology<a href="/nlab/new/lower-interval+topology">?</a></span>,</li> <li>the <a class="existingWikiWord" href="/nlab/show/lower+semicontinuous+topology">lower semicontinuous topology</a> or the <a class="existingWikiWord" href="/nlab/show/upper+semicontinuous+topology">upper semicontinuous topology</a>,</li> <li>the <a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>.</li> </ul> <p>Another variant of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> as a topological space is the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/long+line">long line</a>.</li> </ul> <h2 id="generalisations">Generalisations</h2> <p>The term ‘real number’ was originally introduced to indicate that one is <em>not</em> considering the generalistion to <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> or other kinds of <a class="existingWikiWord" href="/nlab/show/hypercomplex+numbers">hypercomplex numbers</a>. Accordingly, that term ‘real’ may sometimes be used for another generalisation of real numbers to indicate again that one is not considering a complexification.</p> <p>The <a class="existingWikiWord" href="/nlab/show/extended+real+number">extended real number</a>s include <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\pm\infty</annotation></semantics></math> as well as the real numbers; one may speak of <em>finite numbers</em> or <em>bounded numbers</em> to indicate that one is not considering this extension. <a class="existingWikiWord" href="/nlab/show/lower+real">Lower reals</a>, <a class="existingWikiWord" href="/nlab/show/upper+reals">upper reals</a>, and <a class="existingWikiWord" href="/nlab/show/MacNeille+real+number">MacNeille reals</a> are related generalisations studied in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, although with <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> they are (at least if bounded) the same as ordinary real numbers; one may speak of <em>located numbers</em> to indicate that one is not considering such extensions.</p> <p><a class="existingWikiWord" href="/nlab/show/surreal+number">Surreal numbers</a> and the <a class="existingWikiWord" href="/nlab/show/hyperreal+number">hyperreal number</a>s of <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> are two ways to include <a class="existingWikiWord" href="/nlab/show/infinite+number">infinite</a> and <a class="existingWikiWord" href="/nlab/show/infinitesimal+number">infinitesimal</a> versions of real numbers (besides the trivial case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">\pm\infty</annotation></semantics></math>); one may speak of <em>standard numbers</em> to indicate that one is not considering such extensions (although the precise meaning of ‘standard’ depends on the <a class="existingWikiWord" href="/nlab/show/universe">universe</a> that one is working in).</p> <p>In <a class="existingWikiWord" href="/nlab/show/descriptive+set+theory">descriptive set theory</a>, one often says ‘real number’ for an element of <a class="existingWikiWord" href="/nlab/show/Baire+space+of+irrational+numbers">Baire space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math>. This is not really a generalisation; by the <a class="existingWikiWord" href="/nlab/show/Schroeder-Bernstein+theorem">Schroeder-Bernstein theorem</a>, the underlying sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/bijection">isomorphic</a>. Constructively, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℕ</mi> <mi>ℕ</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{N}^{\mathbb{N}}</annotation></semantics></math> can still be thought of as the set of <a class="existingWikiWord" href="/nlab/show/irrational+numbers">irrational numbers</a>, so this use of the term may actually be a <em>restriction</em>.</p> <p><span class="newWikiWord">Floating-point numbers<a href="/nlab/new/floating-point+number">?</a></span> are often used in computer programming to represent real numbers, but they do not behave very well; one may speak of <em>infinite-precision numbers</em> to indicate that one's programming environment models ‘<a href="http://math.fau.edu/richman/html/mm2.htm"><em>real</em> real numbers</a>’.</p> <p>As mentioned above, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adic+number">adic numbers</a> for various <a class="existingWikiWord" href="/nlab/show/prime+numbers">prime numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> are variations on the theme of real numbers; real numbers may be thought of as <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-adic numbers</em>. Similarly, the real numbers are <em>characteristic-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> numbers</em> since they are based on the <a class="existingWikiWord" href="/nlab/show/prime+field">prime field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>; one could also start the construction with a different <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> (although it makes more sense to get analogues of complex numbers than of real numbers).</p> <p>Finally, one can consider points on a <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative</a> line instead of the usual <em>commutative numbers</em>.</p> <p>So in summary, this page is about the <em>real, finite, located, standard, analytic, infinite-precision, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-adic, characteristic-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, commutative numbers</em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/real+numbers+object">real numbers object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/computable+real+number">computable real number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cauchy+real+number">Cauchy real number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dedekind+real+number">Dedekind real number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+cohomology">real cohomology</a>, <a class="existingWikiWord" href="/nlab/show/real+homotopy+theory">real homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a></p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a> one may use the <a class="existingWikiWord" href="/nlab/show/completion+monad">completion monad</a> for dealing with real numbers</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+real+computer+arithmetic">exact real computer arithmetic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real-cohesive+homotopy+type+theory">real-cohesive homotopy type theory</a></p> </li> </ul> <div> <p><strong><a href="spin+group#ExceptionalIsomorphisms">exceptional</a> <a class="existingWikiWord" href="/nlab/show/spin+representation">spinors</a> and <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lorentzian+spacetime">Lorentzian</a> <br /> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <br /> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> entry</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 = 2+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4 = 3+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2, \mathbb{C})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6 = 5+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">little string</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>10</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10 = 9+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%2C1%29">Spin(9,1)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">{\simeq}</annotation></semantics></math> “<a class="existingWikiWord" href="/nlab/show/SL%282%2CO%29">SL(2,O)</a>”</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a>/<a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/analysis">analysis</a></em>.</p> <ul> <li id="Bourbaki71"><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <em>Real Numbers</em>, Chapter IV in: <em>General topology</em>, Elements of Mathematics, Springer (1971, 1990, 1995) [<a href="https://doi.org/10.1007/978-3-642-61701-0">doi:10.1007/978-3-642-61701-0</a>]</li> </ul> <p>Review and history:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Leo+Corry">Leo Corry</a>, §10.5 in: <em>A Brief History of Numbers</em>, Oxford University Press (2015) [<a href="https://global.oup.com/academic/product/a-brief-history-of-numbers-9780198702597">ISBN:9780198702597</a>]</li> </ul> <p>As the <a class="existingWikiWord" href="/nlab/show/coalgebra+of+the+real+interval">coalgebra of the real interval</a>:</p> <ul> <li id="PavlovicPratt"><a class="existingWikiWord" href="/nlab/show/Dusko+Pavlovic">Dusko Pavlovic</a>, <a class="existingWikiWord" href="/nlab/show/Vaughan+Pratt">Vaughan Pratt</a>, <em>On coalgebra of real numbers</em>, Electronic Notes in Theoretical Computer Science <strong>19</strong> (1999) 103-117, doi:<a href="https://doi.org/10.1016/S1571-0661%2805%2980272-5">10.1016/S1571-0661(05)80272-5</a>, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204">CiteSeerx</a></li> </ul> <p>A definition of the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> as <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+field">Archimedean ordered field</a> and as a <a class="existingWikiWord" href="/nlab/show/complete+space">complete</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+ordered+field">Archimedean ordered field</a>:</p> <ul> <li id="Richman08"><a class="existingWikiWord" href="/nlab/show/Fred+Richman">Fred Richman</a>, <em>Real numbers and other completions</em>, Mathematical Logic Quarterly <strong>54</strong> 1 (2008) 98-108 [<a href="https://onlinelibrary.wiley.com/doi/10.1002/malq.200710024">doi:10.1002/malq.200710024</a>]</li> </ul> <p>A compact category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> making the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-linear operators into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-enriched Kan extensions:</p> <ul> <li id="Pavlovic"><a class="existingWikiWord" href="/nlab/show/Dusko+Pavlovic">Dusko Pavlovic</a>, <em>From process-propositions-as-types to categorified real numbers</em> <p>and monoidal computers_, Logic and Structure in Computer Science 1000-1070, doi:<a href="https://doi.org/10.1007/978-3-031-24117-8">10.1007/978-3-031-24117-8</a>, <a href="https://arxiv.org/abs/2007.10057v3">arXiv:2007.10057</a></p> </li> </ul> <p>The definition of the real numbers in <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a> as <a class="existingWikiWord" href="/nlab/show/Cauchy+real+numbers">Cauchy real numbers</a>, namely as <a class="existingWikiWord" href="/nlab/show/regular+sequences">regular</a> <a class="existingWikiWord" href="/nlab/show/Cauchy+sequences">Cauchy sequences</a> of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a> is due to:</p> <ul> <li id="Bishop67"> <p><a class="existingWikiWord" href="/nlab/show/Errett+Bishop">Errett Bishop</a>: <em><a class="existingWikiWord" href="/nlab/show/Foundations+of+Constructive+Analysis">Foundations of Constructive Analysis</a></em>, McGraw-Hill (1967)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Errett+Bishop">Errett Bishop</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Bridges">Douglas Bridges</a>: <em><a class="existingWikiWord" href="/nlab/show/Constructive+Analysis">Constructive Analysis</a></em>, Grundlehren der mathematischen Wissenschaften <strong>279</strong>, Springer (1985) [<a href="https://doi.org/10.1007/978-3-642-61667-9">doi:10.1007/978-3-642-61667-9</a>]</p> </li> </ul> <p>Direct formalization of the definition of <a class="existingWikiWord" href="/nlab/show/Cauchy+real+numbers">Cauchy real numbers</a> from <a href="#Bishop67">Bishop (1967)</a> in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Lundfall">Martin Lundfall</a>, <em>Formalizing real numbers in Agda</em> (2015) [<a href="https://wcl.cs.rpi.edu/pilots/library/papers/TAGGED/4211-Lundfall%20(2015)%20-%20Formalizing%20Real%20Numbers%20in%20Agda.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lundfall-RealNumbersInAgda.pdf" title="pdf">pdf</a>, <a href="https://github.com/MrChico/Reals-in-agda">github</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zachary+Murray">Zachary Murray</a>, <em>Constructive Analysis in the Agda Proof Assistant</em> [<a href="https://arxiv.org/abs/2205.08354">arXiv:2205.08354</a>, <a href="https://github.com/z-murray/honours-project-constructive-analysis-in-agda">github</a>]</p> </li> </ul> <p>review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zachary+Murray">Zachary Murray</a>, <em>Constructive Real Numbers in the Agda Proof Assistant</em>, <a href="Center+for+Quantum+and+Topological+Systems#MurrayFeb2023">talk at <em>CQTS</em></a>, (Feb 2023) [video:<a href="https://www.youtube.com/watch?v=7Q_sjfyPJqU">YT</a>]</li> </ul> <p>and closely related constructions in <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a> have been implemented in</p> <ul> <li id="KrebbersSpitters11"><a class="existingWikiWord" href="/nlab/show/Robbert+Krebbers">Robbert Krebbers</a>, <a class="existingWikiWord" href="/nlab/show/Bas+Spitters">Bas Spitters</a>, <em>Type classes for efficient exact real arithmetic in Coq</em>, Logical Methods in Computer Science, <strong>9</strong> 1 (2013) lmcs:958 [<a href="http://arxiv.org/abs/1106.3448/">arXiv:1106.3448</a>, <a href="https://doi.org/10.2168/LMCS-9(1:1)2013">doi:10.2168/LMCS-9(1:1)2013</a>]</li> </ul> <p>following a <a class="existingWikiWord" href="/nlab/show/monad+%28in+computer+science%29">monadic</a> re-formulation (via the <a class="existingWikiWord" href="/nlab/show/completion+monad">completion monad</a>) due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Russell+O%27Connor">Russell O'Connor</a>, <em>A Monadic, Functional Implementation of Real Numbers</em>, Math. Struc. Comp. Sci. <strong>17</strong> 1 (2007) 129-159 [<a href="https://arxiv.org/abs/cs/0605058">arXiv:cs/0605058</a>, <a href="https://doi.org/10.1017/S0960129506005871">doi:10.1017/S0960129506005871</a>]</li> </ul> <p>Overview of implementation of real numbers in <a class="existingWikiWord" href="/nlab/show/proof+assistants">proof assistants</a> (cf. <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive analysis</a>):</p> <ul> <li>Sylvie Boldo, Catherine Lelay, Guillaume Melquiond, <em>Formalization of Real Analysis: A Survey of Proof Assistants and Libraries</em>, Mathematical Structures in Computer Science <strong>26</strong> 7 (2016) 1196-1233 [<a href="https://hal.inria.fr/hal-00806920v1">hal:00806920</a>, <a href="https://doi.org/10.1017/S0960129514000437">doi:10.1017/S0960129514000437</a>]</li> </ul> <p>A novel construction principle of the type of real numbers, as a <a class="existingWikiWord" href="/nlab/show/higher+inductive-inductive+type">higher inductive-inductive type</a> in <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalent</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>, not reliant on representatives by sequences of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, and with provable <a class="existingWikiWord" href="/nlab/show/Cauchy+completeness">Cauchy completeness</a> even without extra <a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a> is laid out (cf. <em><a class="existingWikiWord" href="/nlab/show/HoTT+book+real+numbers">HoTT book real numbers</a></em>) in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>: Chapter 11 of: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+%E2%80%93+Univalent+Foundations+of+Mathematics">Homotopy Type Theory – Univalent Foundations of Mathematics</a></em> (2013) [<a href="http://homotopytypetheory.org/book/">web</a>, <a href="http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf">pdf</a>]</li> </ul> <p>Something like this has been implemented in <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ga%C3%ABtan+Gilbert">Gaëtan Gilbert</a>, <em>Formalising Real Numbers in Homotopy Type Theory</em>, in <em>CPP 2017: Proceedings of the 6th ACM SIGPLAN Conference on Certified Programs and Proofs</em> (2017) 112–124 [<a href="https://arxiv.org/abs/1610.05072">arXiv:1610.05072</a>, <a href="https://doi.org/10.1145/3018610.3018614">doi:10.1145/3018610.3018614</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 18, 2024 at 18:16:13. See the <a href="/nlab/history/real+number" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/real+number" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/4322/#Item_37">Discuss</a><span class="backintime"><a href="/nlab/revision/real+number/95" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/real+number" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/real+number" accesskey="S" class="navlink" id="history" rel="nofollow">History (95 revisions)</a> <a href="/nlab/show/real+number/cite" style="color: black">Cite</a> <a href="/nlab/print/real+number" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/real+number" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>