# Category: Axioms of set theory

Axiom of global choice
In mathematics, specifically in class theories, the axiom of global choice is a stronger variant of the axiom of choice that applies to proper classes of sets as well as sets of sets. Informally it st
Baumgartner's axiom
In mathematical set theory, Baumgartner's axiom (BA) can be one of three different axioms introduced by James Earl Baumgartner. A subset of the real line is said to be -dense if every two points are s
Axiom of limitation of size
In set theory, the axiom of limitation of size was proposed by John von Neumann in his 1925 axiom system for sets and classes. It formalizes the limitation of size principle, which avoids the paradoxe
Wholeness axiom
In mathematics, the wholeness axiom is a strong axiom of set theory introduced by in 2000.
Axiom schema of specification
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an
Axiom of empty set
In axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory th
Axiom of real determinacy
In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following: Axiom — Consider infinite two-person games with perfect information. Then, every
Axiom of projective determinacy
In mathematical logic, projective determinacy is the special case of the axiom of determinacy applying only to projective sets. The axiom of projective determinacy, abbreviated PD, states that for any
Aczel's anti-foundation axiom
In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel, as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every
Axiom of pairing
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a sp
Axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least o
Axiom of extensionality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theor
Ground axiom
In set theory, the ground axiom states that the universe of set theory is not a nontrivial set-forcing extension of an inner model. The axiom was introduced by and .
Axiom of constructibility
The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V and L denote the von Neuman
Open coloring axiom
In the field of mathematics called set theory, the open coloring axiom (abbreviated OCA) is an axiom about coloring edges of a graph whose vertices are a subset of the real numbers: two different vers
Axiom schema of replacement
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessa
Freiling's axiom of symmetry
Freiling's axiom of symmetry is a set-theoretic axiom proposed by Chris Freiling. It is based on intuition of Stuart Davidsonbut the mathematics behind it goes back to Wacław Sierpiński. Let denote th
In mathematical set theory, the axiom of adjunction states that for any two sets x, y there is a set w = x ∪ {y} given by "adjoining" the set y to the set x. Bernays introduced the axiom of adjunction
In set theory, AD+ is an extension, proposed by W. Hugh Woodin, to the axiom of determinacy. The axiom, which is to be understood in the context of ZF plus DCR (the axiom of dependent choice for real
Axiom of power set
In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: where y is the Power set o
Large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally ve
Proper forcing axiom
In the mathematical field of set theory, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper
Axiom of regularity
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint f
Axiom of finite choice
In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if is a family of non-empty finite sets, then (set-theoretic product). If every set can be linear
Axiom of union
In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo. The axiom states that for each set x there is a set y whose
Axiom schema of predicative separation
In axiomatic set theory, the axiom schema of predicative separation, or of restricted, or Δ0 separation, is a schema of axioms which is a restriction of the usual axiom schema of separation in Zermelo
Axiom of determinacy
In mathematics, the axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person topological game
Axiom of non-choice
In constructive set theory, the axiom of non-choice is a version of the axiom of choice limiting the choice to just one.
Well-ordering theorem
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X ha
Moschovakis coding lemma
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible with choice — that every two-player inte
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied b