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 Axiom of amalgamation Axiom of choice Axiom of collection Axiom of comprehension Axiom of constructibility Axiom of countability Axiom of countable choice Axiom of dependent choice Axiom of determinacy Axiom of elementary sets Axiom of empty set Axiom of extensionality Axiom of finite choice Axiom of foundation Axiom of global choice Axiom of heredity Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of power set Axiom of projective determinacy Axiom of real determinacy Axiom of regularity Axiom of replacement Axiom of subsets Axiom of substitution Axiom of union Axiom schema of predicative separation Axiom schema of replacement Axiom schema of separation Axiom schema of specification Aczel's anti-foundation axiom AD+ Axiom F Freiling's axiom of symmetry Martin's axiom The proper forcing axiom