# List of axioms in set theory

* Axiom of choice: The product of any set of non-empty sets is non-empty
* Axiom of countable choice: The product of a countable number of non-empty sets is non-empty
* Axiom of dependent choice: A weak form of *the axiom of choice*
* Axiom of finite choice: Any product of non-empty finite sets is non-empty
* Axiom of global choice: There is a global choice function
* Axiom of adjunction: Adjoining a set to another set produces a set
* Axiom of comprehension: The class of all sets with a given property is a set (usually contradictory)
* Axiom of constructibility: Any set is constructible, often abbreviated as V=L
* Axiom of countability: Every set is hereditarily countable
* Axiom of elementary sets: describes the sets with 0, 1, or 2 elements
* Axiom of extensionality (extent)
* Axiom of heredity: Any member of a set is a set (used in Ackermann's system)
* Axiom of limitation of size: A class is a set iff it has smaller cardinality than the class of all sets
* Axiom of empty set: The empty set exists
* Axiom of infinity: The infinite set exists
* Axiom of pairing: Unordered pairs of sets are sets
* Axiom of power set (subsets): The powerset of any set is a set
* Axiom of regularity (foundation): sets are well founded
* Axiom of union (amalgamation): The union of all elements of a set is a set
* Axiom of substitution (replacement): The image of a set under a function is a set
* Axiom schema of replacement: The image of a set under a function is a set
* Axiom schema of separation (specification): The elements of a set with some property form a set
* Axiom schema of predicative separation: *Axiom of separation* for formulas whose quantifiers are bounded
* Axiom of collection: means either *the axiom of replacement* or *the axiom of separation*
* Axiom of determinacy: Certain games are determined, in other words one player has a winning strategy
* Axiom of projective determinacy: Certain games given by projective set are determined, in other words one player has a winning strategy
* Axiom of real determinacy: Certain games are determined, in other words one player has a winning strategy
* Aczel's anti-foundation axiom: Every accessible pointed directed graph corresponds to a unique set
* AD+: An extension of *the axiom of determinacy*
* Axiom F: The class of all ordinals is Mahlo (Mahlo cardinals) 
* Freiling's axiom of symmetry: Equivalent to the negation of the *continuum hypothesis*
* Martin's axiom: Cardinals, less than the cardinality of the continuum, behave like ℵ0
* The proper forcing axiom: Strengthening of *Martin's axiom*

[Axiom of adjunction](https://en.wikipedia.org/wiki/Axiom_of_adjunction) [Axiom of amalgamation](https://en.wikipedia.org/wiki/Axiom_of_amalgamation) [Axiom of choice](https://en.wikipedia.org/wiki/Axiom_of_choice) [Axiom of collection](https://en.wikipedia.org/wiki/Axiom_of_collection) [Axiom of comprehension](https://en.wikipedia.org/wiki/Axiom_of_comprehension) [Axiom of constructibility](https://en.wikipedia.org/wiki/Axiom_of_constructibility) [Axiom of countability](https://en.wikipedia.org/wiki/Axiom_of_countability) [Axiom of countable choice](https://en.wikipedia.org/wiki/Axiom_of_countable_choice) [Axiom of dependent choice](https://en.wikipedia.org/wiki/Axiom_of_dependent_choice) [Axiom of determinacy](https://en.wikipedia.org/wiki/Axiom_of_determinacy) [Axiom of elementary sets](https://en.wikipedia.org/wiki/Axiom_of_elementary_sets) [Axiom of empty set](https://en.wikipedia.org/wiki/Axiom_of_empty_set) [Axiom of extensionality](https://en.wikipedia.org/wiki/Axiom_of_extensionality) [Axiom of finite choice](https://en.wikipedia.org/wiki/Axiom_of_finite_choice) [Axiom of foundation](https://en.wikipedia.org/wiki/Axiom_of_foundation) [Axiom of global choice](https://en.wikipedia.org/wiki/Axiom_of_global_choice) [Axiom of heredity](https://en.wikipedia.org/wiki/Axiom_of_heredity) [Axiom of infinity](https://en.wikipedia.org/wiki/Axiom_of_infinity) [Axiom of limitation of size](https://en.wikipedia.org/wiki/Axiom_of_limitation_of_size) [Axiom of pairing](https://en.wikipedia.org/wiki/Axiom_of_pairing) [Axiom of power set](https://en.wikipedia.org/wiki/Axiom_of_power_set) [Axiom of projective determinacy](https://en.wikipedia.org/wiki/Axiom_of_projective_determinacy) [Axiom of real determinacy](https://en.wikipedia.org/wiki/Axiom_of_real_determinacy) [Axiom of regularity](https://en.wikipedia.org/wiki/Axiom_of_regularity) [Axiom of replacement](https://en.wikipedia.org/wiki/Axiom_of_replacement) [Axiom of subsets](https://en.wikipedia.org/wiki/Axiom_of_subsets) [Axiom of substitution](https://en.wikipedia.org/wiki/Axiom_of_substitution) [Axiom of union](https://en.wikipedia.org/wiki/Axiom_of_union) [Axiom schema of predicative separation](https://en.wikipedia.org/wiki/Axiom_schema_of_predicative_separation) [Axiom schema of replacement](https://en.wikipedia.org/wiki/Axiom_schema_of_replacement) [Axiom schema of separation](https://en.wikipedia.org/wiki/Axiom_schema_of_separation) [Axiom schema of specification](https://en.wikipedia.org/wiki/Axiom_schema_of_specification) [Aczel's anti-foundation axiom](https://en.wikipedia.org/wiki/Aczel's_anti-foundation_axiom) [AD+](https://en.wikipedia.org/wiki/AD%2B) [Axiom F](https://en.wikipedia.org/wiki/Axiom_F) [Freiling's axiom of symmetry](https://en.wikipedia.org/wiki/Freiling's_axiom_of_symmetry) [Martin's axiom](https://en.wikipedia.org/wiki/Martin's_axiom) [The proper forcing axiom](https://en.wikipedia.org/wiki/Proper_forcing_axiom)