Axiom:Axiom of Specification/Historical Note

Historical Note on
The ' was created by as a replacement for the ' of Frege set theory.

The latter had been demonstrated, via Russell's Paradox, to lead to the conclusion that Frege Set Theory is Logically Inconsistent.

Thus, rather than allowing a set to be constructed of any elements at all which satisfy a given property $P$, the elements in question are restricted to being elements of some pre-existing set.

This in turn leads to the further question of how to create such a pre-existing set in the first place.

Hence the need to develop further axioms in order to allow the creation of such sets.

As a result of this, found it necessary to create:
 * the, allowing for the existence of $\O := \set {}$
 * the, allowing for $\set {a, b}$ given the existence of $a$ and $b$
 * the, allowing for $\bigcup a$ given the existence of a set $a$ of sets
 * the, allowing for the power set $\powerset a$ to be generated for any set $a$
 * the, allowing for the creation of the set of natural numbers $\N$.