FOM: Re: class theory

[Harvey Friedman]

Shoenfield 3:50PM 1/29/00 wrote:

> I am not exactly sure what Friedman had in mind, but one of his
>aims seems to be to show that global choice is not provable from
>local choice in NBG.   This was proved by Solovay, I think in 1963,
>although it has never (so far as I know) been published or even
>announced.   This was the first independence proof by class forcing;
>the ideas eventually found their way into Easton's thesis.   I do
>not know exacly what forcing class Solovay used; I'm sure he'd be
>glad to tell you if you send him an e-mail.

I have been well aware of this extremely well known result for over 30
years, which you attribute to Solovay - and I have no reason to doubt that
this is a correct attribution. You just add a generic class by set
conditions.

That is only the starting part. I explicitly stated in my posting of 9:27AM
1/27/00, that the point was to answer the very specific question of Charles
Parsons stated in his posting of 6:04PM 1/26/00, that comes directly out of
a consideration of some of Frege's work.

In particular, ZF has a definable cardinality operator (Scott), but - and
here's the point - NBG, or even MK + AxC (axiom of choice for sets) does
not have a definable cardinality operator (for classes). Obviously, NBGC =
NBG + global choice has a definable cardinality operator (for classes), and
so my result immdiately implies the classical result that NBGC does not
follow from NBG + AxC, or even the stronger classical type result that MKC
does not follow from MK + AxC.

>    One thing is important to understand; each of N,B, and G
>intended a class to be not an arbitrary collection of sets but a
>collection which is definable (from set parameters) in the language
>of set theory.   Thus a model of NBG is best regarded as obtained
>by taking a model M of ZFC and defining the classes of the model to
>be the collections of individals of the model which are definable
>from parameters.

This rules out models of MK = Morse Kelley, which are also very important.
The results I stated are stronger when stated with MK. NBG is a fragment of
MK.

>This is the smallest extension of the notion of
>a class which will make M into a model of NBG (with local choice);
>and there is no reason to consider any other.   (Failure to under-
>stand this has led some to think that Morse-Kelley is a simpler or
>better axiomatic system.   However, not every model of ZFC can be
>extended to a model of MK; so there are many new problems in the
>semantics of MK.)

MK is obviously important in its own right, and is stronger. It has the
extremely natural models of the form

P(V(theta))

where theta is a strongly inaccessible cardinal, and P is the power set
operation.

>    Why did Godel use NBG?  .....

One reason to use NBG is that it is finitely axiomatized, and can be given
an axiomatization without the use of schemes. However, even in
presentations of NBG, the axioms are usually formulated in terms of
schemes, instead of formulating them as a small number of specific
mathematical assertions.

On the other hand, MK is not finitely axiomatizable.