One instance of the schema is included for each
formula \varphi in the language of set theory with
free variables among x,w_1,w_2,\ldots,w_n,A . So B does not occur free in \varphi. In the formal language of set theory, the axiom schema is: :\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 \varphi holds for
x. Note that there is one axiom for every such
predicate \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 \varphi, we can find a subset
B of
A whose members are precisely the members of
A that satisfy \varphi. By the
axiom of extensionality this set is unique. We usually denote this set using
set-builder notation as B = \{x\in A | \varphi(x) \}. Thus the essence of the axiom is: : Every
subclass of a set that is defined by a predicate is itself a set. The preceding form of separation was introduced in 1930 by
Thoralf Skolem as a refinement of a previous, non-first-order form by Zermelo. The axiom schema of specification is characteristic of systems of
axiomatic set theory related to the usual set theory
ZFC, but does not usually appear in radically different systems of
alternative set theory. For example,
New Foundations and
positive set theory use different restrictions of the
axiom of comprehension of
naive set theory. The
Alternative Set Theory of Vopenka makes a specific point of allowing proper subclasses of sets, called
semisets. Even in systems related to ZFC, this scheme is sometimes restricted to formulas with bounded quantifiers, as in
Kripke–Platek set theory with urelements. == Relation to the axiom schema of replacement ==