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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="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#bijection_of_the_integers_with_the_natural_numbers'>Bijection of the integers with the natural numbers</a></li> <li><a href='#sequential_cauchy_completeness'> Sequential Cauchy completeness</a></li> <li><a href='#cauchy_completeness'>Cauchy completeness</a></li> <li><a href='#dedekind_completeness'> Dedekind completeness</a></li> </ul> <li><a href='#terminology'>Terminology</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>An <strong>integer</strong> is a <a class="existingWikiWord" href="/nlab/show/number">number</a> that is a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> or the <a class="existingWikiWord" href="/nlab/show/negative+number">negative</a> of one.</p> <p>The <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of all integers may defined as the <a class="existingWikiWord" href="/nlab/show/free+group">free group</a> on one generator or as the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> <a class="existingWikiWord" href="/nlab/show/ring">ring</a>.</p> <p>In keeping with a historical point of view in which integers are natural numbers with a sign attached, one may write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>=</mo><mo stretchy="false">{</mo><mi>n</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">|</mo><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>,</mo><mn>0</mn><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>0</mn><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>…</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>3</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mi>…</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z} = \{n, -n | n \in \mathbb{N}, 0 = -0\} = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, \ldots \} \,. </annotation></semantics></math></div> <p>From an <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a>, one may consider this as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mover><mo>→</mo><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>ℕ</mi><mover><mo>→</mo><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>ℕ</mi><mover><mo>→</mo><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>…</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \mathbb{N} \stackrel{1 + (-)}{\to} \ldots</annotation></semantics></math></div> <p>whereby <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">-n \in \mathbb{Z}</annotation></semantics></math> is represented by the element <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> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>n</mi> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">n^{th}</annotation></semantics></math> copy of <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> appearing in this diagram (starting the count at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>0</mn> <mi>th</mi></msup></mrow><annotation encoding="application/x-tex">0^{th}</annotation></semantics></math> copy). The resulting induced map to the colimit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>×</mo><mi>ℕ</mi><mo>≅</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>m</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>ℕ</mi><mo>→</mo><mi>ℤ</mi><mo>:</mo><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>n</mi><mo>−</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">\mathbb{N} \times \mathbb{N} \cong \sum_{m \in \mathbb{N}} \mathbb{N} \to \mathbb{Z}: (m, n) \mapsto n-m</annotation></semantics></math></div> <p>imparts a monoid (in fact a group) structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> descended from the monoid structure on <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">\mathbb{N} \times \mathbb{N}</annotation></semantics></math>; compare double-entry bookkeeping in medieval mathematics (<em><a href="http://rfcwalters.blogspot.com/2011/02/mathematical-economics-double-entry.html">partita doppia</a></em>).</p> <p>As a group, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> and is the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> of the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (or <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</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> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> <p>The monoid of natural numbers is naturally even a <a class="existingWikiWord" href="/nlab/show/rig">rig</a> – in fact the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> rig – and this multiplicative structure extends to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> to make it a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> – in fact the initial ring.</p> <h2 id="properties">Properties</h2> <ul> <li> <p>The integers form a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>.</p> </li> <li> <p>The integers have <a class="existingWikiWord" href="/nlab/show/decidable+equality">decidable equality</a> and <a class="existingWikiWord" href="/nlab/show/decidable+relation">decidable order</a>.</p> </li> <li> <p>The trivial ring is a <a class="existingWikiWord" href="/nlab/show/tight+apartness+ring">tight apartness ring</a> and a <a class="existingWikiWord" href="/nlab/show/discrete+ring">discrete ring</a>.</p> </li> <li> <p>The integers are a <a class="existingWikiWord" href="/nlab/show/strictly+weakly+ordered+ring">strictly weakly ordered ring</a> and a <a class="existingWikiWord" href="/nlab/show/lattice-ordered+ring">lattice-ordered ring</a>.</p> </li> <li> <p>The integers are a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> and a <a class="existingWikiWord" href="/nlab/show/normed+space">normed space</a>.</p> </li> <li> <p>The integers are a <a class="existingWikiWord" href="/nlab/show/Euclidean+domain">Euclidean domain</a></p> </li> <li> <p>The integers satisfy the Archimedean property, making it into an <a class="existingWikiWord" href="/nlab/show/Archimedean+integral+domain">Archimedean integral domain</a>.</p> </li> </ul> <h3 id="bijection_of_the_integers_with_the_natural_numbers">Bijection of the integers with the natural numbers</h3> <p>The integers are in bijection with the natural numbers. Both the integers and the natural numbers are <span class="newWikiWord">submonoids<a href="/nlab/new/submonoids">?</a></span> of the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, with <a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>ℤ</mi></msub><mo>:</mo><mi>ℤ</mi><mo>↪</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">i_\mathbb{Z}:\mathbb{Z} \hookrightarrow \mathbb{Q}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>ℕ</mi></msub><mo>:</mo><mi>ℕ</mi><mo>↪</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">i_\mathbb{N}:\mathbb{N} \hookrightarrow \mathbb{Q}</annotation></semantics></math> preserving addition, zero, and one. Let us take the relation on the rational numbers</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><mi>ℚ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>ℚ</mi><mo>⊢</mo><mrow><mo>|</mo><mi>x</mi><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>|</mo></mrow><msub><mo>=</mo> <mi>ℚ</mi></msub><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">x:\mathbb{Q}, y:\mathbb{Q} \vdash \left| x + x + \frac{1}{2} \right| =_\mathbb{Q} y + \frac{1}{2}</annotation></semantics></math></div> <p>The relation</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><mi>ℤ</mi><mo>,</mo><mi>y</mi><mo>:</mo><mi>ℕ</mi><mo>⊢</mo><mrow><mo>|</mo><msub><mi>i</mi> <mi>ℤ</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>|</mo></mrow><msub><mo>=</mo> <mi>ℚ</mi></msub><msub><mi>i</mi> <mi>ℕ</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow><annotation encoding="application/x-tex">x:\mathbb{Z}, y:\mathbb{N} \vdash \left| i_\mathbb{Z}(x + x) + \frac{1}{2} \right| =_\mathbb{Q} i_\mathbb{N}(y) + \frac{1}{2}</annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/one-to-one+correspondence">one-to-one correspondence</a>, meaning that the integers are in bijection with the natural numbers.</p> <h3 id="sequential_cauchy_completeness"> Sequential Cauchy completeness</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>ℕ</mi><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">x:\mathbb{N} \to \mathbb{Z}</annotation></semantics></math> be a sequence of integers, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</annotation></semantics></math> be an integer. Then, there is a limit relation defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isLimit</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>∀</mo><mi>ϵ</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>.</mo><mo>∃</mo><mi>N</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mo>∀</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mi>N</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>−</mo><mi>b</mi><mo stretchy="false">|</mo><mo><</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isLimit}(x, b) \coloneqq \forall \epsilon:\mathbb{Z}_+.\exists N:\mathbb{N}.\forall n:\mathbb{N}.(n \geq N) \to (\vert x(n) - b \vert \lt \epsilon)</annotation></semantics></math></div> <p>This relation is a <a class="existingWikiWord" href="/nlab/show/functional+relation">functional relation</a>, making the integers a <a class="existingWikiWord" href="/nlab/show/sequentially+Hausdorff+space">sequentially Hausdorff space</a>.</p> <p>A <strong>modulus of Cauchy convergence</strong> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">M:\mathbb{Z}_+ \to \mathbb{N}</annotation></semantics></math> with a witness</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo>∀</mo><mi>ϵ</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>.</mo><mo>∀</mo><mi>m</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mo>∀</mo><mi>n</mi><mo>:</mo><mi>ℕ</mi><mo>.</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>m</mi><mo>≥</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>−</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo><</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(M, x):\forall \epsilon:\mathbb{Z}_+.\forall m:\mathbb{N}.\forall n:\mathbb{N}.((m \geq M(\epsilon)) \wedge (n \geq M(\epsilon))) \to (\vert x(m) - x(n) \vert \lt \epsilon)</annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/sequentially+Cauchy+complete">sequentially Cauchy complete</a> if every sequence with a modulus of Cauchy convergence has a unique limit. But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is sequentially Cauchy complete, because the only sequences with a unique limit are those sequences for which there exists a natural number <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:\mathbb{N}</annotation></semantics></math> such that for all natural numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">m \geq N</annotation></semantics></math> and <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 \geq N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(m) = x(n)</annotation></semantics></math>. The sequence <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> has many moduli of Cauchy convergence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, where the natural number <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">N</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N = M(1)</annotation></semantics></math>.</p> <p>Since the integers are the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+integral+domain">Archimedean integral domain</a>, the integers are also the initial sequentially Cauchy complete Archimedean integral domain. Every other sequentially Cauchy complete Archimedean integral domain is provably an <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a> and has the <a class="existingWikiWord" href="/nlab/show/HoTT+book+real+numbers">HoTT book real numbers</a> <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 an integral subdomain. That means, in the context of the <a class="existingWikiWord" href="/nlab/show/limited+principle+of+omniscience">limited principle of omniscience</a>, the category of sequentially Cauchy complete Archimedean integral domains is equivalent to the <a class="existingWikiWord" href="/nlab/show/walking+arrow">walking arrow</a>, with objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> 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> and homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">h:\mathbb{Z} \to \mathbb{R}</annotation></semantics></math>.</p> <h3 id="cauchy_completeness">Cauchy completeness</h3> <p>Given a <a class="existingWikiWord" href="/nlab/show/Tarski+universe">Tarski universe</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U, T)</annotation></semantics></math> and a small type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> whose type reflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">x:T(A) \to \mathbb{Z}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/net">net</a> of integers, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</annotation></semantics></math> be an integer. Then, there is a limit relation defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isLimit</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mo>∀</mo><mi>ϵ</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>.</mo><mo>∃</mo><mi>N</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mo>∀</mo><mi>n</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mi>N</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>−</mo><mi>b</mi><mo stretchy="false">|</mo><mo><</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isLimit}(x, b) \coloneqq \forall \epsilon:\mathbb{Z}_+.\exists N:T(A).\forall n:T(A).(n \geq N) \to (\vert x(n) - b \vert \lt \epsilon)</annotation></semantics></math></div> <p>This relation is a <a class="existingWikiWord" href="/nlab/show/functional+relation">functional relation</a>, making the integers a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>.</p> <p>A <strong>modulus of <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>-Cauchy convergence</strong> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>→</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M:\mathbb{Z}_+ \to T(A)</annotation></semantics></math> with a witness</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>:</mo><mo>∀</mo><mi>ϵ</mi><mo>:</mo><msub><mi>ℤ</mi> <mo>+</mo></msub><mo>.</mo><mo>∀</mo><mi>m</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mo>∀</mo><mi>n</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>.</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>m</mi><mo>≥</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>n</mi><mo>≥</mo><mi>M</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>−</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mo><</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(M, x):\forall \epsilon:\mathbb{Z}_+.\forall m:T(A).\forall n:T(A).((m \geq M(\epsilon)) \wedge (n \geq M(\epsilon))) \to (\vert x(m) - x(n) \vert \lt \epsilon)</annotation></semantics></math></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is <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>-<a class="existingWikiWord" href="/nlab/show/Cauchy+complete">Cauchy complete</a> if every net with a modulus of <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>-Cauchy convergence has a unique limit. But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is <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>-Cauchy complete, because the only <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>-nets with a unique limit are those nets for which there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><mi>T</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N:T(A)</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">m \geq N</annotation></semantics></math> and <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 \geq N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>m</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x(m) = x(n)</annotation></semantics></math>. The net <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> has many moduli of <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>-Cauchy convergence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, where the element <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">N</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mi>M</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N = M(1)</annotation></semantics></math>.</p> <p>Since the integers are the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+integral+domain">Archimedean integral domain</a>, the integers are also the initial <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>-Cauchy complete Archimedean integral domain. Every other <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>-Cauchy complete Archimedean integral domain is provably an <a class="existingWikiWord" href="/nlab/show/ordered+field">ordered field</a> and has the <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>-<a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a> <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 an integral subdomain.</p> <h3 id="dedekind_completeness"> Dedekind completeness</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/type+of+all+propositions">type of all propositions</a>, so that the foundations is <a class="existingWikiWord" href="/nlab/show/impredicative">impredicative</a>. A <strong>Dedekind cut</strong> is an pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L, U)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/predicates">predicates</a> such that</p> <ul> <li>there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(a)</annotation></semantics></math></li> <li>there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(b)</annotation></semantics></math></li> <li>for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(a)</annotation></semantics></math> if and only if there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</annotation></semantics></math> such that <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>L</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(b)</annotation></semantics></math></li> <li>for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(b)</annotation></semantics></math> if and only if there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math> such that <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>U</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(a)</annotation></semantics></math></li> <li>for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math>, it is not true that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(a)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(a)</annotation></semantics></math></li> <li>for all integers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</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>ℤ</mi></mrow><annotation encoding="application/x-tex">b:\mathbb{Z}</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>L</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(a)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(b)</annotation></semantics></math>.</li> </ul> <p>There is a Dedekind cut for every integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">a:\mathbb{Z}</annotation></semantics></math>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>a</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>b</mi><mo><</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">L_a(b) \coloneqq b \lt a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>a</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>a</mi><mo><</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">U_a(b) \coloneqq a \lt b</annotation></semantics></math>. There are no other Dedekind cuts on the integers. Thus, the integers are Dedekind complete.</p> <p>Since the integers are the <a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> <a class="existingWikiWord" href="/nlab/show/Archimedean+integral+domain">Archimedean integral domain</a>, the integers are also the initial Dedekind complete Archimedean integral domain. The only other Dedekind complete Archimedean integral domain is the <a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a> <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>. That means, if there is a <a class="existingWikiWord" href="/nlab/show/type+of+all+propositions">type of all propositions</a>, the category of Dedekind complete Archimedean integral domains is equivalent to the <a class="existingWikiWord" href="/nlab/show/walking+arrow">walking arrow</a>, with objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> 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> and homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>ℤ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">h:\mathbb{Z} \to \mathbb{R}</annotation></semantics></math>.</p> <h2 id="terminology">Terminology</h2> <p>The underlying <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> and <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> are <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a>. Some subcultures of mathematics (and not only <a class="existingWikiWord" href="/nlab/show/set+theory">set theorists</a>) use the term ‘integer’ synonymously for a natural number. Computer scientists distinguish between ‘unsigned integers’ (natural numbers) and ‘signed integers’ (integers as described here). Translations can also cause confusion with the term ‘whole number’.</p> <p>In <a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a>, one generalises integers to <a class="existingWikiWord" href="/nlab/show/algebraic+integers">algebraic integers</a>, an instance of the <a class="existingWikiWord" href="/nlab/show/red+herring+principle">red herring principle</a>. Accordingly, some number theorists will call the integers ‘<a class="existingWikiWord" href="/nlab/show/rational+integers">rational integers</a>’ to clarify; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> in the <a class="existingWikiWord" href="/nlab/show/number+field">number 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 <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>. (Compare, for example, <a class="existingWikiWord" href="/nlab/show/Gaussian+integers">Gaussian integers</a> and <a class="existingWikiWord" href="/nlab/show/Gaussian+numbers">Gaussian numbers</a>.)</p> <p>The symbol ‘<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>’ derives from the German word ‘Zahlen’, which is a generic word for ‘numbers’. (Compare <a class="existingWikiWord" href="/nlab/show/Richard+Dedekind">Dedekind</a>'s use of that word in the title of his famous book on the <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+cyclic+group">infinite cyclic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/even+number">even number</a>, <a class="existingWikiWord" href="/nlab/show/odd+number">odd number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, <a class="existingWikiWord" href="/nlab/show/rational+number">rational number</a>, <a class="existingWikiWord" href="/nlab/show/real+number">real number</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+integer">algebraic integer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclotomic+integer">cyclotomic integer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> modeled on the <a class="existingWikiWord" href="/nlab/show/formal+duality">formal dual</a>, <a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a>, of the ring of integers is <a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integers+object">integers object</a> in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+zero+theorem">rational zero theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integers+type">integers type</a></p> </li> </ul> <h2 id="references">References</h2> <p>The first characterization of the integers as an <a class="existingWikiWord" href="/nlab/show/ordered+integral+domain">ordered integral domain</a> appeared in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hermann+Grassmann">Hermann Grassmann</a>, <em>Lehrbuch der Arithmetik für höhere Lehranstalten</em>, Berlin: Enslin, 1861. (<a href="https://books.google.com/books?id=jdQ2AAAAMAAJ">Google Books</a>)</li> </ul> <p>though the name “ordered integral domain” does not appear in the text.</p> <p>History:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Leo+Corry">Leo Corry</a>, <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>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Integer">Integer</a></em></li> </ul> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/univalent+foundations+of+mathematics">univalent foundations of mathematics</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> with the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a>):</p> <ul> <li id="UFP13"><a class="existingWikiWord" href="/nlab/show/Univalent+Foundations+Project">Univalent Foundations Project</a>, Rem. 6.10.7 <em><a class="existingWikiWord" href="/nlab/show/Homotopy+Type+Theory+--+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>and specifically in <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/UniMath+project">UniMath project</a>, <em><a href="https://unimath.github.io/agda-unimath/elementary-number-theory.integers.html#1032">agda-unimath/elementary-number-theory.integers</a></em></li> </ul> <p>and specifically in <a class="existingWikiWord" href="/nlab/show/cubical+Agda">cubical Agda</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>, <em><a href="https://1lab.dev/Data.Int.html">Data.Int</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 2, 2025 at 23:02:00. 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