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 <math>\varphi</math> in the language of set theory with free variables among <math>x,w_1,w_2,\ldots,w_n,A</math> . So <math>B</math> does not occur free in <math>\varphi</math>. In the formal language of set theory, the axiom schema is:
:<math>\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) ] )</math>
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 <math>\varphi</math> holds for x.
Note that there is one axiom for every such predicate <math>\varphi</math>; 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 <math>\varphi</math>, we can find a subset B of A whose members are precisely the members of A that satisfy <math>\varphi</math>. By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as <math>B = \{x\in A | \varphi(x) \}</math>. 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
The axiom schema of specification is implied by the axiom schema of replacement together with the axiom of empty set.