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Foundations mathematical logic first-order logic type theory homotopy type theory set theory material set theory ZFC ZFA structural set theory ETCS SEAR universe
Foundational axioms
basic constructions: axiom of cartesian products axiom of disjoint unions axiom of the empty set axiom of fullness axiom of function sets axiom of power sets axiom of quotient sets
material axioms: axiom of extensionality axiom of foundation axiom of anti-foundation Mostowski's axiom axiom of pairing axiom of transitive closure axiom of union
structural axioms: axiom of materialization
axioms of choice: axiom of choice axiom of countable choice axiom of dependent choice axiom of excluded middle axiom of existence axiom of multiple choice Markov's axiom presentation axiom small cardinality selection axiom axiom of small violations of choice axiom of weakly initial sets of covers Whitehead's principle
large cardinal axioms: axiom of infinity axiom of universes regular extension axiom inaccessible cardinal measurable cardinal elementary embedding supercompact cardinal Vopěnka's principle
strong axioms axiom of separation axiom of replacement
reflection principle Removing axioms constructive mathematics predicative mathematics Edit this sidebar
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Contents
Idea
Generalisations
Related concepts
Idea An element of a set is a thing which “belongs to,” or “is an element of,” that set.
The circularity of this definition is unavoidable in foundational set theories in which “set” is an undefined term. In “definitional” set theories, where “set” is defined in terms of something else, elements are likewise defined in terms of the same “something else.”
If sets (or setoids) are regarded as the semantics of some type theory, then an element of a set is the interpretation of a term of some type.
Generalisations A term of some type. a global element of any object in any category or higher category with a terminal object. a generalised element of any object in any category or higher category; a morphism in a category, a different sort of element from an object; An object in a category; in general, k-morphisms or k-cells in an ∞-category;
Related concepts basic symbols used in logic
symbol meaning ∈ element relation : typing relation = equality ⊢ entailment / sequent ⊤ true / top ⊥ false / bottom ⇒ implication ⇔ logical equivalence ¬ negation ≠ negation of equality / apartness ∉ negation of element relation ¬¬ negation of negation ∃ existential quantification ∀ universal quantification ∧ logical conjunction ∨ logical disjunction ⊗ multiplicative conjunction ⊕ multiplicative disjunction Last revised on July 3, 2018 at 02:48:59. See the history of this page for a list of all contributions to it.
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