Axiom schema of specification

The axiom comes from the Zermelo set theory of 1907 and is therefore also part of the extended, Zermelo -Fraenkel relevant today set theory ZF. It states informally that all sub-classes of sets are also quantities. In the predicate logic language, the axiom is specified as an axiom schema that includes infinitely many axioms; Therefore, it is now often referred to as a separation scheme.

Clarification

Axiom for each place predicate:

Verbalization and writing:

Importance

Zermelo set theory resulted in a the axiom of separation, because the usual in the naive set theory of the late 19th century comprehension, which classifies each class as the amount produced Russell's antinomy. Russell took over even this naive comprehension in his logic and was therefore forced to severely restrict the syntax of permissible predicates in his theory of types for the avoidance of contradiction. Unlike Russell Zermelo did not make any restrictions on the syntax, but pointed his axiom, which weakened the comprehension strongly that the contradictory in Russel 's antinomy class is no longer set. He reached a much simpler and more powerful set theory in this way.

Abraham Fraenkel, however, showed in 1921 that the Zermelo set theory was too weak with axiom to derive the set theory Georg Cantor, and supplemented this reason a stronger replacement axiom that filled the gap. This axiom integrated Zermelo 1930 in his IF system and remarked that for him the axiom is derivable, so that it is unnecessary in the IF system. Namely the separation amount obtained as apparently by substitution axiom.