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Concept in axiomatic set theory

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.

In a typed language where we can quantify over predicates, the axiom schema of specification becomes a simple axiom. This is much the same trick as was used in the NBG axioms of the previous section, where the predicate was replaced by a class that was then quantified over. In second order logic and higher order logic with higher order semantics, the axiom of specification is a logical validity and does not need to be explicitly included in a theory.

In the New Foundations approach to set theory pioneered by W. V. O. Quine, the axiom of comprehension for a given predicate takes the unrestricted form, but the predicates that may be used in the schema are themselves restricted. The predicate (C is not in C) is forbidden, because the same symbol C appears on both sides of the membership symbol (and so at different "relative types"); thus, Russell's paradox is avoided. However, by taking to be , which is allowed, we can form a set of all sets. For details, see stratification.