Definition:Morse-Kelley Set Theory/Historical Note

Historical Note on Morse-Kelley Set Theory

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