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...

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...

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...

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

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...

members. Specifying sets by member properties is allowed by the axiom schema of specification. This is also known as set comprehension and set abstraction...

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

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

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...

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...

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...

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...

Axiom of choice Axiom of constructibility Axiom of extensionality Axiom of infinity Axiom of limitation of size Axiom of pairing Axiom of union Axiom...

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...

mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty...

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...

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...

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...

\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...

The axiom of countable choice or axiom of denumerable choice, denoted ACω, is an axiom of set theory that states that every countable 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...

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...

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...

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...

of axiom schema 1, and hence are axioms. It can be shown that with only these three axiom schemata and modus ponens, one can prove all tautologies of...

showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French...