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href="/nlab/show/natural+deduction">natural deduction</a>, <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a>, <a class="existingWikiWord" href="/nlab/show/lambda-calculus">lambda-calculus</a>, <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simple type theory</a>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/collection">collection</a>, <a class="existingWikiWord" href="/nlab/show/object">object</a>, <a class="existingWikiWord" href="/nlab/show/type">type</a>, <a class="existingWikiWord" href="/nlab/show/term">term</a>, <a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/size+issues">size issues</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a></p> </li> </ul> <h2 id="set_theory"> Set theory</h2> <div> <p><strong><a 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" 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-set+theory">class-set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/class">class</a>, <a class="existingWikiWord" 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> <h2 id="foundational_axioms">Foundational axioms</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></strong></p> <ul> <li> <p>basic constructions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+cartesian+products">axiom of cartesian products</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+disjoint+unions">axiom of disjoint unions</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+the+empty+set">axiom of the empty set</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+fullness">axiom of fullness</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+function+sets">axiom of function sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+quotient+sets">axiom of quotient sets</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/material+set+theory">material axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+anti-foundation">axiom of anti-foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski%27s+axiom">Mostowski's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+transitive+closure">axiom of transitive closure</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+materialization">axiom of materialization</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+theory">type theoretic axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axioms+of+set+truncation">axioms of set truncation</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a></li> <li><a class="existingWikiWord" href="/nlab/show/boundary+separation">boundary separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/equality+reflection">equality reflection</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+circle+type+localization">axiom of circle type localization</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+principle">Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axioms+of+choice">axioms of choice</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+excluded+middle">axiom of excluded middle</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+existence">axiom of existence</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+multiple+choice">axiom of multiple choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/Markov%27s+axiom">Markov's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/presentation+axiom">presentation axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/small+cardinality+selection+axiom">small cardinality selection axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+small+violations+of+choice">axiom of small violations of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+weakly+initial+sets+of+covers">axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/large+cardinal+axioms">large cardinal axioms</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+universes">axiom of universes</a></li> <li><a class="existingWikiWord" href="/nlab/show/regular+extension+axiom">regular extension axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/Vop%C4%9Bnka%27s+principle">Vopěnka's principle</a></li> </ul> </li> <li> <p>strong axioms</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+separation">axiom of separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">axiom of replacement</a></li> </ul> </li> <li> <p>further</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+principle">reflection principle</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axiom+of+tight+apartness">axiom of tight apartness</a></p> </li> </ul> </div> <h2 id="removing_axioms">Removing axioms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> </ul> <div> <p> <a href="/nlab/edit/foundations+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#homogeneous_and_heterogeneous_membership_relations'>Homogeneous and heterogeneous membership relations</a></li> <li><a href='#material_and_structural_membership_relations'>Material and structural membership relations</a></li> <ul> <li><a href='#structural_membership_relations_without_quine_atoms'>Structural membership relations without Quine atoms</a></li> <li><a href='#structural_membership_relations_with_quine_atoms'>Structural membership relations with Quine atoms</a></li> </ul> </ul> <li><a href='#see_also'> See also</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A membership relation is a <a class="existingWikiWord" href="/nlab/show/binary+relation">binary relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> found in <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simply sorted</a> <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> between terms which can be elements and terms which can be sets. Membership relations are found both <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> and <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a>.</p> <p>This is in contrast to <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependently sorted</a> set theories, where membership <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is a typing <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a> rather than a binary relation.</p> <h2 id="properties">Properties</h2> <p>Membership relations have different properties in different set theories which can be distinguished.</p> <h3 id="homogeneous_and_heterogeneous_membership_relations">Homogeneous and heterogeneous membership relations</h3> <p>Recall that a <a class="existingWikiWord" href="/nlab/show/binary+relation">binary relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> takes values in sorts <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>. A binary relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is <strong>homogeneous</strong> or an <strong>endorelation</strong> if it takes values in the same sort <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 binary relation is <strong>heterogeneous</strong> if it takes values in two distinct sorts <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>.</p> <p>The membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is homogeneous if both sets and elements are terms of the same sort, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is heterogeneous if sets and elements are terms of different sorts. For example, in <a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a>, <a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a>, and <a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a>, the membership relations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> are homogeneous, while in the traditional presentation of <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> as a two-sorted theory, the membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is heterogeneous.</p> <h3 id="material_and_structural_membership_relations">Material and structural membership relations</h3> <p>Another important difference is the distinction between the membership relations in material and structural set theories, in what we could call <strong>material membership relation</strong> and <strong>structural membership relation</strong>. This has been proposed to distinguish between <a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> and <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> in set theories with <a class="existingWikiWord" href="/nlab/show/homogeneous+membership+relations">homogeneous membership relations</a>:</p> <p>A set theory with a homogeneous membership relation is a <strong><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a></strong> if it has a material membership relations, and it is a <strong><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a></strong> if it has a structural membership relation. If the set theory has multiple homogeneous membership relations, the set theory is a <strong>structural set theory</strong> if there exists a structural membership relation, and the set theory is a <strong>material set theory</strong> if every homogeneous membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/material+membership+relation">material membership relation</a>.</p> <p>A few definitions have been proposed to distinguish between homogeneous membership relations in material and structural set theories:</p> <h4 id="structural_membership_relations_without_quine_atoms">Structural membership relations without Quine atoms</h4> <p>Both material set theories like <a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a> and <a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a> and structural set theories like <a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a> have a homogeneous membership relation on the theory (and fully formal ETCS has multiple homogeneous membership relations). This means that merely having a homogeneous membership relation is not sufficient for the membership relation to be a material membership relation.</p> <p>One proposed difference between material set theory and structural set theory is that in material set theory, elements have internal structure and/or sets are the internal structure of some other objects, while in structural set theory, elements do not have internal structure and sets are not the internal structure of other objects.</p> <p>Thus, an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> <strong>has internal structure</strong> if there exists an 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">a \in e</annotation></semantics></math>, and an set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> <strong>is an internal structure</strong> if there exists an 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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">S \in a</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 mathvariant="normal">hasInternalStructure</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>≔</mo><mi mathvariant="normal">isElement</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>∧</mo><mo>∃</mo><mi>a</mi><mo>.</mo><mi>a</mi><mo>∈</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\mathrm{hasInternalStructure}(e) \coloneqq \mathrm{isElement}(e) \wedge \exists a.a \in e</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isInternalStructure</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mi mathvariant="normal">isSet</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>∧</mo><mo>∃</mo><mi>a</mi><mo>.</mo><mi>S</mi><mo>∈</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\mathrm{isInternalStructure}(S) \coloneqq \mathrm{isSet}(S) \wedge \exists a.S \in a</annotation></semantics></math></div> <p>If an set theory only has one homogeneous membership relation, then the membership relation is a <strong>material membership relation</strong> if there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> which has internal structure or a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> which is an internal structure, and the membership relation is a <strong>structural membership relation</strong> if for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> does not have internal structure, and for all sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is not an internal structure.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isMaterial</mi><mo>≔</mo><mo>∃</mo><mi>a</mi><mo>.</mo><mo stretchy="false">(</mo><mi mathvariant="normal">hasInternalStructure</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∨</mo><mi mathvariant="normal">isInternalStructure</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isMaterial} \coloneqq \exists a.(\mathrm{hasInternalStructure}(a) \vee \mathrm{isInternalStructure}(a))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isStructural</mi><mo>≔</mo><mo>∀</mo><mi>a</mi><mo>.</mo><mo stretchy="false">(</mo><mo>¬</mo><mi mathvariant="normal">hasInternalStructure</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∧</mo><mo>¬</mo><mi mathvariant="normal">isInternalStructure</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathrm{isStructural} \coloneqq \forall a.(\neg\mathrm{hasInternalStructure}(a) \wedge \neg\mathrm{isInternalStructure}(a))</annotation></semantics></math></div> <p>Thus, the membership relation in a theory like <a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a> is a material membership relation because sets and elements are the same, and thus all elements have internal structure and all sets are internal structure. The same is true of any theory of <a class="existingWikiWord" href="/nlab/show/pure+sets">pure sets</a>, such as <a class="existingWikiWord" href="/nlab/show/New+Foundations">New Foundations</a>. In a theory with non-set <a class="existingWikiWord" href="/nlab/show/urelements">urelements</a> such as <a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a>, the membership relation is still a material membership relation, because every object in the theory is an element, and while urelements do not have internal structure, there exist other elements which do have internal structure, namely those elements which are sets.</p> <p>In contrast, the homogeneous membership relation in <a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a>, defined by</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>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>∧</mo><mi>s</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>∧</mo><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \in b \coloneqq s(a) = 1 \wedge s(b) = 0 \wedge t(a) = t(b)</annotation></semantics></math></div> <p>where the constant primitive <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> is the identity morphism of the terminal object and the constant primitive <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> is the identity morphism of the initial object, and where sets and elements are defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isSet</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>s</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathrm{isSet}(a) \coloneqq s(b) = 0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">isElement</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>s</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\mathrm{isElement}(a) \coloneqq s(b) = 1</annotation></semantics></math></div> <p>is a structural membership relation, because it can be proven that every element does not have internal structure and every set is not internal structure.</p> <h4 id="structural_membership_relations_with_quine_atoms">Structural membership relations with Quine atoms</h4> <p>The above definition states that in a structural set theory with a homogeneous membership relation, <a class="existingWikiWord" href="/nlab/show/Quine+atoms">Quine atoms</a> cannot exist. However, this means that in <a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a> when sets are defined as identity functions or as functions with singleton codomain, the defined membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math> is a material membership relation, as in both cases the singleton is a <a class="existingWikiWord" href="/nlab/show/Quine+atom">Quine atom</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math>. From one point of view, one could argue that those given definitions of set are incorrect, and one should instead use a definition of set which results in a structural homogeneous membership relation.</p> <p>However, since the subset of <a class="existingWikiWord" href="/nlab/show/identity+functions">identity functions</a>, functions with <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>, and functions with <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> <a class="existingWikiWord" href="/nlab/show/domain">domain</a> are all in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with each other, it should not matter which subset of function to designate as the sets. This motivates an alternate definition of structural and material membership relations, which allow for Quine atoms to exist with respect to a structural membership relation.</p> <p>A membership relation is a <strong>structural membership relation</strong> if each connected component of the respective membership graph is either an edgeless vertex, a rooted <a class="existingWikiWord" href="/nlab/show/tree">tree</a> whose children are all leaves, or possibly a vertex with a single loop:</p> <ul> <li> <p>the edgeless vertex represents the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a>, as well as any non-set non-element objects in the set theory</p> </li> <li> <p>the root of the rooted tree represents a set with more than one element, as well as the <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> in presentations of unsorted structural set theories where the singleton is different from the element in the singleton. The children leaves of the root represent the elements of the set.</p> </li> <li> <p>the vertex with a single loop represents a <a class="existingWikiWord" href="/nlab/show/Quine+atom">Quine atom</a> with respect to the membership relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in</annotation></semantics></math>, which occur in some definitions of sets in structural set theories, such as defining sets as <a class="existingWikiWord" href="/nlab/show/identity+functions">identity functions</a> or as functions with <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>, where the <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> is always a <a class="existingWikiWord" href="/nlab/show/Quine+atom">Quine atom</a>.</p> </li> </ul> <p>A <strong>material membership relation</strong> is then a membership relation whose membership graphs have more complex structure than those of structural membership relations.</p> <h2 id="see_also"> See also</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 14, 2022 at 04:24:33. 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