# Definition:Morse-Kelley Set Theory

## Definition

**Morse-Kelley set theory** is a system of axiomatic set theory.

It is a stronger form of Zermelo-Fraenkel-Skolem set theory which allows not only for first order formulas to specify the existence of properties of sets, but also defines properties by quantifying over properties as well as over sets.

## Source of Name

This entry was named for Anthony Perry Morse and John Leroy Kelley.

## Historical Note

Morse-Kelley set theory was devised for the purpose of extending Zermelo-Fraenkel-Skolem set theory so as to allow properties themselves to be further specified.

While it is no bad idea to allow further formal axioms to extend the range of what can be defined, it is not possible to formally provide for *all* possible properties by means of an axiom schema.

This was proven by Gödel's Incompleteness Theorems.

## 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