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One of the Zermelo-Frankel axioms of set theory, also known as the subset axiom. Like the axiom of replacement, the axiom of separation is actually an infinite set of axioms, an 'axiom schema'. Specifically, for every formula P(x) of the language of set theory, the following is an axiom:

Axiom If A is a set, then there is a set B such that y is in B if and only if y is in A and P(y) is true.

The set B is normally denoted {y in A: P(y)}.

In other words, given a set A, we can separate out the elements of A satisfying a property P, and put them in a new set.

Notice that we have to pick the elements from some set A. In general, we can't take the collection of all sets satisfying some property, for if we could then we could take the set of all sets not members of themselves, giving Russell's paradox. The axiom of separation as I've given it above was formulated to avoid this problem.

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