[FOM] Re: ZFC with classes

[Ali Enayat]

Vladik Kreinovich has pointed out that the system of set theory suggested by
Victor Makarov's recent posting appears in Kelley's classical text on
General Topology.

This system is known in the literature as the Kelley-Morse theory of classes
(KM).

Roughly speaking, (KM/ZF)  ~  (PA/Second order arithmetic).

In particular, KM proves the consistency of ZF, and a lot more. This is in
contrast with the von Neumann-Goedel-Bernays system, which is a conservative
extension of ZF (and whose arithmetical analogue is ACA_0).

It is easy to see that if there is an inaccessible cardinal, then KM has a
model (indeed, a well-founded model).

There is a substantial body of work on KM, especially that  of A. Mostwoski
and his school in Warsaw, and R. Chuaqui and his school in Santiago.
However, relative to the volume of work on second order arithmetic, and its
subsystems, there is a great deal of work to be done.

One can find rudiments of the subject in the text "Foundation of Set
Theory", by Fraenkel et. al (North-Holland).

Ali Enayat