of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation (Aussonderungsaxiom), subset axiom, axiom of...

an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. An axiom schema is a formula in the metalanguage of an axiomatic...

proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first...

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...

Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity...

use the Axiom of Infinity combined with the Axiom schema of specification. Let I {\displaystyle I} be an inductive set guaranteed by the Axiom of Infinity...

form the intersection ⋂ A {\displaystyle \bigcap A} using the axiom schema of specification as ⋂ A = { c ∈ E : ∀ D ( D ∈ A ⇒ c ∈ D ) } {\displaystyle \bigcap...

elements of the first infinite von Neumann ordinal ω {\displaystyle \omega } . And another application of the axiom (schema) of specification means ω {\displaystyle...

y]\rightarrow x=y].} The converse of this axiom follows from the substitution property of equality. 2) Axiom Schema of Specification (or Separation or Restricted...

of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible due to the axiom schema of specification. The theory of semisets...

The axiom of extensionality, also called the axiom of extent, is an axiom used in many forms of axiomatic set theory, such as Zermelo–Fraenkel set theory...

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...

of the natural numbers using the axiom of infinity and axiom schema of specification. One variation of the principle of complete induction can be generalized...

the axiom of union. Together with the axiom of empty set and the axiom of union, the axiom of pairing can be generalised to the following schema: ∀ A...

schema of unrestricted comprehension is weakened to the axiom schema of specification or axiom schema of separation, If P is a property, then for any set X...

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...

sets using the axiom schemas of specification and replacement, as well as the axiom of power set, introduces impredicativity, a type of circularity, into...

the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty...

containing precisely those elements x for which φ(x) holds. (This is an axiom schema.) Axiom of Δ0-collection: Given any Δ0 formula φ(x, y), if for every set x...

{\displaystyle B=\{x\in A\mid x\notin f(x)\}} exists via the axiom schema of specification, and B ∈ P ( A ) {\displaystyle B\in {\mathcal {P}}(A)} because...

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...

well-ordered, so the axiom of choice is not needed to well-order them. The following construction of the Vitali set shows one way that the axiom of choice can be...

the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory. It guarantees for every set x {\displaystyle x} the existence of a...

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...

\ldots ,x_{n})].} Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified...

first three of these characterizations can be proven equivalent in Zermelo–Fraenkel set theory without the axiom of choice, but the equivalence of the third...

notation represents the set of all values of x that belong to some given set E for which the predicate is true (see "Set existence axiom" below). If Φ ( x ) {\displaystyle...

0} . Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a...

foundation axiom of ZFC is replaced by axioms implying its negation. The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers...

In mathematics, the axiom of dependent choice, denoted by D C {\displaystyle {\mathsf {DC}}} , is a weak form of the axiom of choice ( A C {\displaystyle...