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
Last updated