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class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></strong></p> <ul> <li>fundamentals of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic">propositional logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/typed+predicate+logic">typed predicate logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a></li> <li><a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a>, <a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, <a class="existingWikiWord" href="/nlab/show/large+set">large set</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/union+structure">union structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> <li><a class="existingWikiWord" href="/nlab/show/powerset+structure">powerset structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/natural+numbers+structure">natural numbers structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+set+theory">first-order set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/simply+sorted+set+theory">simply sorted set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/one-sorted+set+theory">one-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/three-sorted+set+theory">three-sorted set theory</a></li> </ul> </li> <li><a class="existingWikiWord" 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href="/nlab/show/proper+class">proper class</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+class">universal class</a>, <a class="existingWikiWord" href="/nlab/show/universe">universe</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+of+classes">category of classes</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/constructive+set+theory">constructive set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></li> </ul> </div></div> </div> </div> <h1 id="the_axiom_of_pairing">The axiom of pairing</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#statement'>Statement</a></li> <ul> <li><a href='#pairing'>Pairing</a></li> <li><a href='#unordered_pairing'>Unordered pairing</a></li> <li><a href='#ordered_pairing'>Ordered pairing</a></li> <li><a href='#with_sets_and_elements_different'>With sets and elements different</a></li> <li><a href='#in_dependent_type_theory'>In dependent type theory</a></li> </ul> <li><a href='#generalisation'>Generalisation</a></li> <li><a href='#related_notions'>Related notions</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> as a <a class="existingWikiWord" href="/nlab/show/foundation+of+mathematics">foundation of mathematics</a>, the axiom of pairing is an important axiom needed to get the foundations off the ground (to mix metaphors). It states that <a class="existingWikiWord" href="/nlab/show/pair+sets">pair sets</a> exist.</p> <h2 id="statement">Statement</h2> <h3 id="pairing">Pairing</h3> <p>The <strong>axiom of pairing</strong> (or <strong>axiom of pairs</strong>) states the following:</p> <p><strong>Axiom of pairing</strong>: <em>If <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are (material) sets, then there exists a set <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \in P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">y \in P</annotation></semantics></math>.</em></p> <p>Using the axiom of separation (<a class="existingWikiWord" href="/nlab/show/bounded+separation">bounded separation</a> is enough), we can prove the existence of a particular set <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> such that <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are the <em>only</em> members of <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>. Using the <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a>, we can then prove that this set <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> is unique; it is usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x,y\}</annotation></semantics></math> and called the <strong><a class="existingWikiWord" href="/nlab/show/pair+set">pair set</a></strong> 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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x,x\}</annotation></semantics></math> may also be denoted simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x\}</annotation></semantics></math>.</p> <p>One could also assume that the <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> has a primitive binary operation <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> which takes of a material set <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and returns a material set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x, y)</annotation></semantics></math>. Then the axiom of pairing becomes</p> <p><strong>Axiom of pairing</strong>: <em>If <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are (material) sets, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in P(x, y)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y \in P(x, y)</annotation></semantics></math>.</em></p> <h3 id="unordered_pairing">Unordered pairing</h3> <p>The <strong>axiom of unordered pairing</strong> (or <strong>axiom of unordered pairs</strong>) states the following:</p> <p><strong>Axiom of unordered pairing</strong>: If <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are (material) sets, then there exists a set <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x \in P</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">y \in P</annotation></semantics></math> and for all sets <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">z</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">z \in P</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">z = x</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">z = y</annotation></semantics></math>.</p> <p>Using the <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a>, we can then prove that this set <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> is unique; it is usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x,y\}</annotation></semantics></math> and called the <strong><a class="existingWikiWord" href="/nlab/show/pair+set">pair set</a></strong> 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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>. Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x,x\}</annotation></semantics></math> may also be denoted simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x\}</annotation></semantics></math>.</p> <p>One could also assume that the <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> has a primitive binary operation <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><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math> which takes of a material set <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and returns a material set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x, y\}</annotation></semantics></math>. Then the axiom of pairing becomes</p> <p><strong>Axiom of unordered pairing</strong>: If <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are (material) sets, then <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>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">x \in \{x, y\}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">y \in \{x, y\}</annotation></semantics></math>, and for all sets <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">z</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">z \in \{x, y\}</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">z = x</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">z = y</annotation></semantics></math>.</p> <h3 id="ordered_pairing">Ordered pairing</h3> <p>Let us assume that the <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> has a primitive binary operation <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><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-,-)</annotation></semantics></math> which takes of a material set <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> and returns a material set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math>.</p> <p>The <strong>axiom of ordered pairing</strong> (or <strong>axiom of ordered pairs</strong>) states the following:</p> <p><strong>Axiom of ordered pairing</strong>: <em>If <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>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> are (material) sets, then <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>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in (x, y)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y \in (x, y)</annotation></semantics></math>, and for all sets <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>, <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><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a, b) = (x, y)</annotation></semantics></math> if and only if <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 = x</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>y</mi></mrow><annotation encoding="application/x-tex">b = y</annotation></semantics></math>.</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>a</mi><mo>.</mo><mo>∀</mo><mi>b</mi><mo>.</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">}</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>=</mo><mi>x</mi><mo>∧</mo><mi>b</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall a.\forall b.\{a, b\} = \{x, y\} \iff (a = x \wedge b = y)</annotation></semantics></math></div> <h3 id="with_sets_and_elements_different">With sets and elements different</h3> <p>In set theories where sets and elements are not the same thing, pairing becomes an operation on both the sets and the elements. One has to add a primitive ternary relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(X, Y, P)</annotation></semantics></math> which says that <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> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> 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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, as well as primitive quaternary relations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X, P, c, a)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(Y, P, c, b)</annotation></semantics></math> which says that element <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 \in X</annotation></semantics></math> is the left element of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">c \in P</annotation></semantics></math> and element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">b \in Y</annotation></semantics></math> is the right element of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">c \in P</annotation></semantics></math>, and the following axiom:</p> <p><strong>Axiom of ordered pairing</strong>: <em>If <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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> are sets, then there exists a set <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(X, Y, P)</annotation></semantics></math> and for every object <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</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>X</mi></mrow><annotation encoding="application/x-tex">a \in X</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>Y</mi></mrow><annotation encoding="application/x-tex">b \in Y</annotation></semantics></math> implies that there exists an object <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">c</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">c \in P</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X, P, c, a)</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>P</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_2(Y, P, c, b)</annotation></semantics></math></em></p> <p><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> is usually denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math> and called the <strong><a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a></strong> 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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, while <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">c</annotation></semantics></math> is usually denoted <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> and called the <strong><a class="existingWikiWord" href="/nlab/show/ordered+pair">ordered pair</a> of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>.</strong></p> <h3 id="in_dependent_type_theory">In dependent type theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a>, it is possible to define 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>V</mi><mo>,</mo><mo>∈</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, \in)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pure+sets">pure sets</a> which behaves as a <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a>. The universal type family of the Tarski universe is given by the type family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>V</mi><mo>⊢</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>y</mi><mo>:</mo><mi>V</mi></mrow></msub><mi>y</mi><mo>∈</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x:V \vdash \sum_{y:V} y \in x</annotation></semantics></math>. The <strong>axiom of pairing</strong> is given by the following <a class="existingWikiWord" href="/nlab/show/inference+rule">inference rule</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mi mathvariant="normal">ctx</mi></mrow><mrow><mi>Γ</mi><mo>⊢</mo><msub><mi mathvariant="normal">pairing</mi> <mi>V</mi></msub><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>V</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>y</mi><mo>:</mo><mi>V</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>P</mi><mo>:</mo><mi>V</mi></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>y</mi><mo>∈</mo><mi>P</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{pairing}_V:\prod_{x:V} \prod_{y:V} \sum_{P:V} (x \in P) \times (y \in P)}</annotation></semantics></math></div> <h2 id="generalisation">Generalisation</h2> <p>The axiom of pairing is the binary part of a <a class="existingWikiWord" href="/nlab/show/binary%2Fnullary+pair">binary/nullary pair</a> whose nullary part is the axiom stating the existence of the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a>. We can use these axioms and the <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a> to prove every instance of the following <strong>axiom</strong> (or rather theorem) <strong>schema of finite sets</strong>:</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n</annotation></semantics></math> are sets, then there exists a set <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n \in P</annotation></semantics></math>.</p> </div> </p> <p>Again, we can prove the existence of specific <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n</annotation></semantics></math> are the <em>only</em> members of <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> and prove that this <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> is unique; it is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, \ldots, x_n\}</annotation></semantics></math> and is called the <strong><a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a></strong> consisting of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, \ldots, x_n</annotation></semantics></math>.</p> <p>Note that this is a <em>schema</em>, with one instance for every (metalogical) <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>. Within axiomatic set theory, this is very different from the single statement that begins with a <a class="existingWikiWord" href="/nlab/show/universal+quantification">universal quantification</a> over the (internal) set of natural numbers. In particular, each instance of this schema can be stated and proved without the <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a>. Of course, there is one proof for each natural number.</p> <ul> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>, this is simply the axiom of the empty set.</li> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>, we use the axiom of pairing with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≔</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x \coloneqq x_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≔</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">y \coloneqq x_1</annotation></semantics></math> to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1\}</annotation></semantics></math>.</li> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>, we use the axiom of pairing with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≔</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x \coloneqq x_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≔</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">y \coloneqq x_2</annotation></semantics></math> to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, x_2\}</annotation></semantics></math>.</li> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math>, we first use the axiom of pairing twice to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, x_2\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>3</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_3\}</annotation></semantics></math>, then use pairing again to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>3</mn></msub><mo stretchy="false">}</mo><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex">\big\{\{x_1, x_2\}, \{x_3\}\big\}</annotation></semantics></math>, then use the axiom of union to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, x_2, x_3\}</annotation></semantics></math>.</li> <li>In general, once we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, \ldots, x_{n-1}\}</annotation></semantics></math>, we use pairing to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_n\}</annotation></semantics></math>, use pairing again to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">{</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">}</mo><mo>,</mo><mo stretchy="false">{</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex">\big\{\{x_1, \ldots, x_{n-1}\}, \{x_n\}\big\}</annotation></semantics></math>, then use the axiom of union to construct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x_1, \ldots, x_n\}</annotation></semantics></math>. (A direct proof of a single statement for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n \gt 3</annotation></semantics></math> can actually go faster than this; the length of the shortest proof is <a class="existingWikiWord" href="/nlab/show/logarithmic">logarithmic</a> in <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> rather than linear in <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>.)</li> </ul> <p>Note that these ‘finite sets’ are precisely the <a class="existingWikiWord" href="/nlab/show/Kuratowski-finite+sets">Kuratowski-finite sets</a> in a <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive</a> treatment.</p> <h2 id="related_notions">Related notions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a></li> </ul> <p>In the <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>Lab, the term ‘<a class="existingWikiWord" href="/nlab/show/pairing">pairing</a>’ usually refers to <em><a class="existingWikiWord" href="/nlab/show/ordered+pair">ordered</a></em> pairs.</p> <h2 id="references">References</h2> <p>For the axiom of ordered pairing see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/H%C3%A5kon+Robbestad+Gylterud">Håkon Robbestad Gylterud</a>, <a class="existingWikiWord" href="/nlab/show/Elisabeth+Bonnevier">Elisabeth Bonnevier</a>, <em>Non-wellfounded sets in HoTT</em> (<a href="https://arxiv.org/abs/2001.06696">arXiv:2001.06696</a>)</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/foundational+axiom">foundational axiom</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on February 28, 2024 at 03:14:48. 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