Definition:Zermelo Set Theory/Historical Note

Historical Note on Zermelo Set Theory
The axiomatic system of Zermelo Set Theory was created by as way to circumvent the logical inconsistencies of Frege set theory.

The axiom of subsets was derived from the comprehension principle, with a domain strictly limited to the elements of a given pre-existing set.

Further axioms were then developed in order to allow the creation of such pre-existing sets:
 * the axiom of existence, allowing for the existence of $\O := \set {}$
 * the axiom of pairing, allowing for $\set {a, b}$ given the existence of $a$ and $b$
 * the axiom of union, allowing for $\bigcup a$ given the existence of a set $a$ of sets
 * the axiom of powers, allowing for the power set $\powerset a$ to be generated for any set $a$
 * the axiom of infinity, allowing for the creation of the set of natural numbers $\N$.