@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-B4PHZ43K-K
  skos:prefLabel "théorème de Zermelo"@fr, "Zermelo's theorem"@en ;
  a skos:Concept ;
  skos:related psr:-RMP0BKNQ-Q .

psr:-JRSJ6RBM-L
  skos:prefLabel "lemme de Zorn"@fr, "Zorn's lemma"@en ;
  a skos:Concept ;
  skos:related psr:-RMP0BKNQ-Q .

psr:-RMP0BKNQ-Q
  a skos:Concept ;
  skos:prefLabel "axiom of choice"@en, "axiome du choix"@fr ;
  skos:definition """En mathématiques, l'axiome du choix, abrégé en « AC », est un axiome de la théorie des ensembles qui « affirme la possibilité de construire des ensembles en répétant une infinité de fois une action de choix, même non spécifiée explicitement. » 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Axiome_du_choix">https://fr.wikipedia.org/wiki/Axiome_du_choix</a>)"""@fr, """In mathematics, the <b>axiom of choice</b>, abbreviated <b>AC</b> or <b>AoC</b>, is an axiom of set theory equivalent to the statement that <i>a Cartesian product of a collection of non-empty sets is non-empty</i>. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to construct a new set by arbitrarily choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (S_{i})_{i\\\\in I}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>S</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>∈<!-- ∈ --></mo>
<br/>            <mi>I</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (S_{i})_{i\\\\in I}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/678af8cfde54ae12ee7322dcac4b969eae1b021a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.758ex; height:2.843ex;" alt="(S_{i})_{i\\\\in I}"></span> of nonempty sets, there exists an indexed set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (x_{i})_{i\\\\in I}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mo stretchy="false">)</mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>∈<!-- ∈ --></mo>
<br/>            <mi>I</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle (x_{i})_{i\\\\in I}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cbb47cf9bb3374016df9c9c71f54f5b28ff475" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.663ex; height:2.843ex;" alt="(x_{i})_{i\\\\in I}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle x_{i}\\\\in S_{i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>x</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <msub>
<br/>          <mi>S</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle x_{i}\\\\in S_{i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09ce730f6f736a42e32c24806f0731933fd82ba" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.195ex; height:2.509ex;" alt="x_{i}\\\\in S_{i}"></span> for every <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle i\\\\in I}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>i</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>I</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle i\\\\in I}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="i\\\\in I"></span>. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">https://en.wikipedia.org/wiki/Axiom_of_choice</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Axiom_of_choice>, <https://fr.wikipedia.org/wiki/Axiome_du_choix> ;
  skos:inScheme psr: ;
  dc:modified "2023-07-03"^^xsd:date ;
  dc:created "2023-07-03"^^xsd:date ;
  skos:related psr:-B4PHZ43K-K, psr:-JRSJ6RBM-L ;
  skos:broader psr:-T88XBMNP-M .

psr: a skos:ConceptScheme .
psr:-T88XBMNP-M
  skos:prefLabel "set theory"@en, "théorie des ensembles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RMP0BKNQ-Q .