Axiom schema of specification: Encyclopedia II - Axiom schema of specification - Relation to the axiom schema of replacement

Axiom schema of specification - Relation to the axiom schema of replacement

The axiom schema of separation can almost be derived from the axiom schema of replacement.

First, recall this axiom schema:

for any functional predicate F in one variable that doesn't use the symbols A, B, C or D. Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of A such that P(E) is true. Then the set B guaranteed by the axiom of replacement is precisely the set B required for the axiom of specification. The only problem is if no such E exists. But in this case, the set B required for the axiom of separation is the empty set, so the axiom of separation follows from the axiom of replacement together with the axiom of empty set.

For this reason, the axiom schema of separation is often left out of modern lists of the Zermelo-Fraenkel axioms. However, it's still important for historical considerations, and for comparison with alternative axiomatisations of set theory, as can be seen for example in the following sections.

Adapted from the Wikipedia article "Relation to the axiom schema of replacement", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki