Axiom schema of specification

"Axiom of separation" redirects here. For the separation axioms in topology, see separation axiom.

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.

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