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However, because the extensional quotient map need not reflect the relation, there is still content to the axiom: if two sets would be identified in the extensional quotient, then they must be members of the same sets and have the same sets as members.</p> <p>If one models <a class="existingWikiWord" href="/nlab/show/pure+sets">pure sets</a> in structural <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, then this property may be made to hold by construction.</p> <h2 id="statements">Statements</h2> <h3 id="weak_extensionality">Weak extensionality</h3> <h4 id="in_unsorted_set_theories">In unsorted set theories</h4> <p>In any <a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a>, the <strong>axiom of <a class="existingWikiWord" href="/nlab/show/weak+extensionality">weak extensionality</a></strong> states that given a set <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 a set <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>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math> if and only if for all <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">C \in A</annotation></semantics></math> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C \in B</annotation></semantics></math>.</p> <p>If the set theory does not have <a class="existingWikiWord" href="/nlab/show/equality">equality</a> as a primitive, we could define equality as the predicate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mo>≔</mo><mo>∀</mo><mi>C</mi><mo>.</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = B \coloneqq \forall C.(C \in A) \iff (C \in B)</annotation></semantics></math></div> <p>The axiom of weak extensionality is a foundational axiom in most <a class="existingWikiWord" href="/nlab/show/material+set+theories">material set theories</a>, such as Zermelo set theory and Mac Lane set theory, both which do not have the <a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a>, as well as <a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a> which does have the <a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a>, where the basic objects of the theory are functions, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> represents both the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> and the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on the empty set, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> represents the <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a>, the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> of the singleton, and the sole <a class="existingWikiWord" href="/nlab/show/element">element</a> of the singleton all at the same time, there are three possible notions of sets:</p> <ul> <li> <p>as <a class="existingWikiWord" href="/nlab/show/identity+functions">identity functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">set</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mi mathvariant="normal">dom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\mathrm{set}(a) \coloneqq \mathrm{dom}(a) = a</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">set</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mi mathvariant="normal">codom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\mathrm{set}(a) \coloneqq \mathrm{codom}(a) = a</annotation></semantics></math></p> </li> <li> <p>as <a class="existingWikiWord" href="/nlab/show/functions">functions</a> into the <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">set</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">codom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{set}(a) \coloneqq (\mathrm{codom}(a) = 1)</annotation></semantics></math></p> </li> <li> <p>as functions from the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">set</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">dom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{set}(a) \coloneqq (\mathrm{dom}(a) = 0)</annotation></semantics></math></p> </li> </ul> <p>The elements are functions with domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">element</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi mathvariant="normal">dom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{element}(a) \coloneqq (\mathrm{dom}(a) = 1)</annotation></semantics></math></p> <p>When sets are defined as functions into the singleton, the membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \in b</annotation></semantics></math> is defined by the function <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> being an element, the function <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> being a set, and the codomain 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> being <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>b</mi><mo>≔</mo><mi mathvariant="normal">set</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∧</mo><mi mathvariant="normal">element</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>∧</mo><mi mathvariant="normal">codom</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \in b \coloneqq \mathrm{set}(a) \wedge \mathrm{element}(b) \wedge \mathrm{codom}(a) = b</annotation></semantics></math></div> <p>Weak extensionality is a theorem in this case. By the universal property of the singleton, any two 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> with the same domain are equal to each other, which means that any proposition <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> between <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> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math>, and thus that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>C</mi><mo>.</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>C</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall C.(C \in A) \iff (C \in B)</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math>. The converse follows from <a class="existingWikiWord" href="/nlab/show/indiscernibility+of+identicals">indiscernibility of identicals</a>.</p> <h4 id="in_twosorted_set_theories">In two-sorted set theories</h4> <p>In any <a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a>, the <strong>axiom of <a class="existingWikiWord" href="/nlab/show/weak+extensionality">weak extensionality</a></strong> states that given a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>:</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">A:Set</annotation></semantics></math> and a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">B:Set</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>=</mo> <mi>Set</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A =_{Set} B</annotation></semantics></math> if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>:</mo><mi>Element</mi></mrow><annotation encoding="application/x-tex">a:Element</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>A</mi></mrow><annotation encoding="application/x-tex">a \in A</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>B</mi></mrow><annotation encoding="application/x-tex">a \in B</annotation></semantics></math>.</p> <p>If the set theory does not have <a class="existingWikiWord" href="/nlab/show/equality">equality</a> of sets as a primitive, we could define equality of sets as the predicate</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>=</mo> <mi>Set</mi></msub><mi>B</mi><mo>≔</mo><mo>∀</mo><mi>a</mi><mo>:</mo><mi>Element</mi><mo>.</mo><mo stretchy="false">(</mo><mi>a</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⇔</mo><mo stretchy="false">(</mo><mi>a</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A =_{Set} B \coloneqq \forall a:Element.(a \in A) \iff (a \in B)</annotation></semantics></math></div> <h3 id="material_strong_extensionality">Material strong extensionality</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/bisimulation">bisimulation</a>, a binary relation such that 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∼</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \sim B</annotation></semantics></math>, the following conditions hold:</p> <ul> <li>for all sets <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>A</mi></mrow><annotation encoding="application/x-tex">C \in A</annotation></semantics></math>, there exists a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">D \in B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∼</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C \sim D</annotation></semantics></math></li> <li>for all sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">D \in B</annotation></semantics></math>, there exists a set <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>A</mi></mrow><annotation encoding="application/x-tex">C \in A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∼</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">C \sim D</annotation></semantics></math></li> </ul> <p>The <strong>axiom of strong extensionality</strong> states that for every bisimulation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> and for every set <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>B</mi></mrow><annotation encoding="application/x-tex">A \sim B</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math>.</p> <p>If the set theory does not have <a class="existingWikiWord" href="/nlab/show/equality">equality</a> as a primitive, we could define equality as the <a class="existingWikiWord" href="/nlab/show/terminal">terminal</a> <a class="existingWikiWord" href="/nlab/show/bisimulation">bisimulation</a>, as the bisimulation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo></mrow><annotation encoding="application/x-tex">=</annotation></semantics></math> such that for every bisimulation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> and for every set <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>B</mi></mrow><annotation encoding="application/x-tex">A \sim B</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A = B</annotation></semantics></math>.</p> <p>In any set theory with the <a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a>, the axiom of weak extensionality implies the axiom of strong extensionality.</p> <h3 id="structural_strong_extensionality">Structural strong extensionality</h3> <p>In any <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a>, the axiom of strong extensionality states that 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> with an <a class="existingWikiWord" href="/nlab/show/injection">injection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i:A \hookrightarrow B</annotation></semantics></math>, the two definitions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> being a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> are logically equivalent to each other:</p> <ul> <li>there exists a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">i^{-1}:B \to A</annotation></semantics></math> such that for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a \in A</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>B</mi></mrow><annotation encoding="application/x-tex">b \in B</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">i^{-1}(i(a)) = a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">i(i^{-1}(b)) = b</annotation></semantics></math></li> <li>for every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">x \in B</annotation></semantics></math> there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">y \in A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">i(y) = x</annotation></semantics></math>.</li> </ul> <p>Similar to how in material set theory one can use the axiom of extensionality to define <a class="existingWikiWord" href="/nlab/show/equality">equality</a> of sets, in structural set theory one can use the axiom of strong extensionality to define <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of sets.</p> <h2 id="in_dependent_type_theory">In dependent type theory</h2> <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 extensionality</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">extensionality</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><mo stretchy="false">(</mo><mi>x</mi><msub><mo>=</mo> <mi>V</mi></msub><mi>y</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>z</mi><mo>:</mo><mi>V</mi></mrow></munder><mo stretchy="false">(</mo><mi>z</mi><mo>∈</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>z</mi><mo>∈</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{extensionality}_V:\prod_{x:V} \prod_{y:V} (x =_V y) \simeq \prod_{z:V} (z \in x) \simeq (z \in y)}</annotation></semantics></math></div> <h3 id="power_sets">Power sets</h3> <p>For <a class="existingWikiWord" href="/nlab/show/power+sets">power sets</a>, the <em><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></em> is a property of <a class="existingWikiWord" href="/nlab/show/power+sets">power sets</a>, and states that given a <a class="existingWikiWord" href="/nlab/show/type">type</a> <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 <a class="existingWikiWord" href="/nlab/show/subtypes">subtypes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B:\mathcal{P}(A)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C:\mathcal{P}(A)</annotation></semantics></math>, there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+types">equivalence of types</a> between the <a class="existingWikiWord" href="/nlab/show/identity+type">identity type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo>=</mo> <mrow><mi>𝒫</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mi>C</mi></mrow><annotation encoding="application/x-tex">B =_{\mathcal{P}(A)} C</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/dependent+function+type">dependent function type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod_{x:A} (x \in_A B) \simeq (x \in_A C)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mo>,</mo><msub><mi mathvariant="normal">El</mi> <mi>Ω</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega, \mathrm{El}_\Omega)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/type+of+all+propositions">type of all propositions</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mo>≔</mo><msub><mi mathvariant="normal">El</mi> <mi>Ω</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))</annotation></semantics></math> is the local membership relation between elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">x:A</annotation></semantics></math> and subtypes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B:\mathcal{P}(A)</annotation></semantics></math>. The axiom of extensionality holds in the <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> if and only if <a class="existingWikiWord" href="/nlab/show/function+extensionality">function extensionality</a> holds.</p> <h2 id="see_also">See also</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/extensionality">extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/identity+of+indiscernibles">identity of indiscernibles</a></li> </ul> <h2 id="references"> References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Shulman">Michael Shulman</a>, <em>Comparing material and structural set theories</em> (<a href="https://arxiv.org/abs/1808.05204">arXiv:1808.05204</a>)</p> </li> <li> <p><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>)</p> </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 02:57:04. 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