@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-VWLLSCW9-R
  skos:related psr:-GWVJ4B59-7 ;
  skos:prefLabel "axiom of regularity"@en, "axiome de régularité"@fr ;
  skos:inScheme psr: ;
  skos:definition """In mathematics, the <b>axiom of regularity</b> (also known as the <b>axiom of foundation</b>) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set <i>A</i> contains an element that is disjoint from <i>A</i>. In first-order logic, the axiom reads:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\orall x\\\\,(x\\
eq \\\\varnothing \\
ightarrow \\\\exists y(y\\\\in x\\\\ \\\\land y\\\\cap x=\\\\varnothing )).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi mathvariant="normal">∀<!-- ∀ --></mi>
<br/>        <mi>x</mi>
<br/>        <mspace width="thinmathspace"></mspace>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>x</mi>
<br/>        <mo>≠<!-- ≠ --></mo>
<br/>        <mi class="MJX-variant">∅<!-- ∅ --></mi>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mi mathvariant="normal">∃<!-- ∃ --></mi>
<br/>        <mi>y</mi>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>y</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mi>x</mi>
<br/>        <mtext>&nbsp;</mtext>
<br/>        <mo>∧<!-- ∧ --></mo>
<br/>        <mi>y</mi>
<br/>        <mo>∩<!-- ∩ --></mo>
<br/>        <mi>x</mi>
<br/>        <mo>=</mo>
<br/>        <mi class="MJX-variant">∅<!-- ∅ --></mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\orall x\\\\,(x\\
eq \\\\varnothing \\
ightarrow \\\\exists y(y\\\\in x\\\\ \\\\land y\\\\cap x=\\\\varnothing )).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b186acc3628667ed14e8c169107244c8c5a1085c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:38.037ex; height:2.843ex;" alt="{\\\\displaystyle \\orall x\\\\,(x\\
eq \\\\varnothing \\
ightarrow \\\\exists y(y\\\\in x\\\\ \\\\land y\\\\cap x=\\\\varnothing )).}"></span></dd></dl>
<br/>The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (<i>a<sub>n</sub></i>) such that <i>a<sub>i+1</sub></i> is an element of <i>a<sub>i</sub></i> for all <i>i</i>. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_regularity">https://en.wikipedia.org/wiki/Axiom_of_regularity</a>)"""@en, """L'axiome de fondation, encore appelé axiome de régularité, est l'un des axiomes de la théorie des ensembles. Introduit par Abraham Fraenkel, Thoralf Skolem (1922) et John von Neumann (1925), il joue un grand rôle dans cette théorie, alors que les mathématiciens ne l'utilisent jamais ailleurs, même s'ils le considèrent souvent comme intuitivement vérifié. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Axiome_de_fondation">https://fr.wikipedia.org/wiki/Axiome_de_fondation</a>)"""@fr ;
  skos:altLabel "axiome de fondation"@fr, "axiom of foundation"@en ;
  skos:broader psr:-T88XBMNP-M ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Axiome_de_fondation>, <https://en.wikipedia.org/wiki/Axiom_of_regularity> ;
  a skos:Concept .

psr:-T88XBMNP-M
  skos:prefLabel "set theory"@en, "théorie des ensembles"@fr ;
  a skos:Concept ;
  skos:narrower psr:-VWLLSCW9-R .

psr:-GWVJ4B59-7
  skos:prefLabel "théorie des ensembles de Zermelo-Fraenkel"@fr, "Zermelo-Fraenkel set theory"@en ;
  a skos:Concept ;
  skos:related psr:-VWLLSCW9-R .