# Definition:Frege Set Theory

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## Definition

The **Frege system of set theory** is a system of axiomatic set theory which has as its sole axiom the Axiom of Abstraction:

- Given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$:

- $\set {x: \map P x}$

In support of this, the various logical axioms supporting predicate logic also hold.

It is accepted that the above statement lacks clarity, but the field of symbolic logic was less well understood when this system was first defined.

## Also see

- Results about
**Frege set theory**can be found**here**.

## Source of Name

This entry was named for Gottlob Frege.

## Sources

- 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 7$ Frege set theory