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Concept
Axiom schema of specification
Summary
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme 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 call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
One instance of the schema is included for each formula φ in the language of set theory with free variables among x, w1, ..., wn, A. So B does not occur free in φ. In the formal language of set theory, the axiom schema is:
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 holds for x.
Note that there is one axiom for every such predicate φ; 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 , we can find a subset B of A whose members are precisely the members of A that satisfy . By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as . 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 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.
Official source
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