# Axiom:Axiom of Specification/Historical Note

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## Historical Note on Axiom of Specification

The **Axiom of Specification** was created by Ernst Zermelo 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, Ernst Zermelo found it necessary to create:

- the Axiom of the Empty Set, 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