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 (Aussonderungsaxiom), subset axiom, axiom of class construction, or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.

Some mathematicians refer to this axiom as the axiom schema of comprehension, although others reserve that term only for unrestricted comprehension; this axiom is a "restricted" version of unrestricted comprehension. Because restricting comprehension avoids Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.

Statement

One instance of the schema is included for each formula ⁠

φ

{\displaystyle \varphi }

⁠ in the language of set theory with free variables among ⁠

x

,

w

1

,

w

2

,

…

,

w

n

,

A

{\displaystyle x,w_{1},w_{2},\ldots ,w_{n},A}

⁠. So the set ⁠

B

{\displaystyle B}

⁠, whose existence is asserted by the axiom, does not occur free in ⁠

φ

{\displaystyle \varphi }

⁠. In the formal language of set theory, the axiom schema is:

∀

w

1

,

…

,

w

n

∀

A

∃

B

∀

x

(

x

∈

B

⇔

[

x

∈

A

∧

φ

(

x

,

w

1

,

…

,

w

n

,

A

)

]

)

{\displaystyle \forall w_{1},\ldots ,w_{n}\,\forall A\,\exists B\,\forall x\,(x\in B\Leftrightarrow [x\in A\land \varphi (x,w_{1},\ldots ,w_{n},A)])}

or in words:

Given any set A, there is a set B (a subset of A) such that, given any set x, x is a member of B if and only if x is a member of A and ⁠

φ

{\displaystyle \varphi }

⁠ holds for x.

Note that there is one axiom for every such predicate ⁠

φ

{\displaystyle \varphi }

⁠; thus, this is an axiom schema.

To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate ⁠

φ

{\displaystyle \varphi }

⁠, we can find a subset B of A whose members are precisely the members of A that satisfy ⁠

φ

{\displaystyle \varphi }

⁠. By the axiom of extensionality this set is unique.