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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/power+set'>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/unsorted+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' href='/nlab/show/dependently+sorted+set+theory'>dependently sorted set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/structurally+presented+set+theory'>structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class='existingWikiWord' href='/nlab/show/material+set+theory'>material set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/ZFC'>ZFC</a></li> <li><a class='existingWikiWord' href='/nlab/show/ZFA'>ZFA</a></li> <li><a class='existingWikiWord' href='/nlab/show/Mostowski+set+theory'>Mostowski set theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/New+Foundations'>New Foundations</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/structural+set+theory'>structural set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/categorical+set+theory'>categorical set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/ETCS'>ETCS</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/fully+formal+ETCS'>fully formal ETCS</a></li> <li><a class='existingWikiWord' href='/nlab/show/ETCS+with+elements'>ETCS with elements</a></li> <li><a class='existingWikiWord' href='/nlab/show/Trimble+on+ETCS+I'>Trimble on ETCS I</a></li> <li><a class='existingWikiWord' href='/nlab/show/Trimble+on+ETCS+II'>Trimble on ETCS II</a></li> <li><a class='existingWikiWord' href='/nlab/show/Trimble+on+ETCS+III'>Trimble on ETCS III</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/structural+ZFC'>structural ZFC</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/allegorical+set+theory'>allegorical set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/SEAR'>SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/class+theory'>class-set theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/class'>class</a>, <a class='existingWikiWord' href='/nlab/show/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> <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><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><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+set'>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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_2' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_5' xmlns='http://www.w3.org/1998/Math/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/axiom+of+separation'>bounded separation</a> is enough), we can prove the existence of a particular set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> are the <em>only</em> members of <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_9' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is unique; it is usually denoted <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_11' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>. Note that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_14' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_15' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> which takes of a material set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and returns a material set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_19' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> are (material) sets, then <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_22' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_23' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_25' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_27' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_28' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_30' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>z = x</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_32' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is unique; it is usually denoted <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_34' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>. Note that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_37' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_38' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_39' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and returns a material set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_42' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> are (material) sets, then <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_45' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_46' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi></mrow><annotation encoding='application/x-tex'>z</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_48' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>z = x</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_50' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_51' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and returns a material set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_54' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> are (material) sets, then <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_57' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_58' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_61' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>a = x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_63' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='block' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_64' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_65' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_66' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, as well as primitive quaternary relations <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_69' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_70' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_71' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_72' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_73' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_74' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> are sets, then there exists a set <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_78' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_81' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_82' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>P</mi></mrow><annotation encoding='application/x-tex'>c \in P</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_85' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_86' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is usually denoted <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_88' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, while <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math> is usually denoted <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_92' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_94' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_95' xmlns='http://www.w3.org/1998/Math/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+set'>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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_96' xmlns='http://www.w3.org/1998/Math/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/deductive+system'>inference rule</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_97' xmlns='http://www.w3.org/1998/Math/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/biased+definition'>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>\begin{theorem} If <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_98' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_100' xmlns='http://www.w3.org/1998/Math/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>. \end{theorem}</p> <p>Again, we can prove the existence of specific <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_102' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> and prove that this <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is unique; it is denoted <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_105' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_106' xmlns='http://www.w3.org/1998/Math/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+quantifier'>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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_107' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_108' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_109' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_110' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_111' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_112' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_113' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_114' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_115' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_116' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_117' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_118' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_119' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_120' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_121' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_122' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_123' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_124' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_125' xmlns='http://www.w3.org/1998/Math/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/logarithm'>logarithmic</a> in <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> rather than linear in <math class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_127' xmlns='http://www.w3.org/1998/Math/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/finite+set'>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 class='maruku-mathml' display='inline' id='mathml_43c5cb12b1adf8adce892d322c86f366c4358225_128' xmlns='http://www.w3.org/1998/Math/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+Stenholm'>Elisabeth Bonnevier</a>, <em>Non-wellfounded sets in HoTT</em> (<a href='https://arxiv.org/abs/2001.06696'>arXiv:2001.06696</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p><div class='property'> category: <a class='category_link' href='/nlab/list/foundational+axiom'>foundational axiom</a></div></p> </div>  </div>  </body> </html>