Definition:Zermelo Set Theory/Historical Note

From ProofWiki
Jump to navigation Jump to search

Historical Note on Zermelo Set Theory

The axiomatic system of Zermelo Set Theory was created by Ernst Friedrich Ferdinand Zermelo as way to circumvent the logical inconsistencies of Frege set theory.

The axiom of specification 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 unions, 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 $\N$.