# Definition:Zermelo Set Theory/Historical Note

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## 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 $\N$.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 9$ Zermelo set theory