Axiom:Axiom of Specification/Historical Note

Historical Note on
The  was created by as a replacement for the axiom of abstraction 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 $\N$.