# Definition:Zermelo-Fraenkel-Skolem Set Theory

## Definition

**Zermelo-Fraenkel-Skolem set theory** is a system of axiomatic set theory.

It consists of Zermelo-Fraenkel set theory with the inclusion of a system of first order formulas to specify the existence of properties of sets.

## Source of Name

This entry was named for Ernst Friedrich Ferdinand Zermelo, Abraham Halevi Fraenkel and Thoralf Albert Skolem.

## Historical Note

It was Thoralf Skolem who proposed the amendment to Zermelo-Fraenkel set theory to sharpen the specification of the axiom of replacement.

He considered it necessary to express the axiom of specification as an infinite number of axioms: one for every first order formula.

Zermelo disagreed forcefully against Skolem's approach, preferring the philosophical position that a property should be allowed to be considered as all possible meaningful conditions, not just propositions in first order logic.

Skolem's position was that Zermelo's notion of property was too vague to be completely satisfactory.

## 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