# https://ncatlab.org/nlab/all\_pages https://ncatlab.org/nlab/all\_pages/reference https://ncatlab.org/

https://github.com/ncatlab/nlab-content

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

Mathematics history of mathematics

Structural Foundations logic internal language classical mathematics evil constructive mathematics predicative mathematics category:foundational axiom set theory

structural set theory category theory

Categories and Sheaves

topos theory

Sheaves in Geometry and Logic higher category theory

higher topos theory

(∞,1)-topos theory models for ∞-stack (∞,1)-toposes cohomology homotopy theory

stable homotopy theory rational homotopy theory Topology and Geometry

geometry (general list), topology (general list) general topology differential topology differential geometry synthetic differential geometry symplectic geometry algebraic geometry noncommutative algebraic geometry noncommutative geometry (general flavour) higher geometry Algebra

universal algebra

higher algebra

homological algebra

group theory, ring theory

representation theory algebraic approaches to differential calculus

counterexamples in algebra

analysis

nonstandard analysis

functional analysis

operator algebras Fourier transform

Lie theory

higher Lie theory probability theory

discrete mathematics

Edit this sidebar

Contents

1. Idea
2. Generalisations
3. Related concepts
4. 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.

1. 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;
2. 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.

EditBack in time (7 revisions)See changesHistory Cite Print TeX Source
