Axiom of comprehension
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 solved 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 is not free in φ. In the formal language of set theory, the axiom schema is:
or in words:
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 P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as {C ∈ A : P(C)}. Thus the essence of the axiom is:
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.
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