[FOM] Must every proper class contain an infinite subset?

[Harvey Friedman]

On 3/1/06 3:35 PM, "Johan Belinfante" <belinfan at math.gatech.edu> wrote:

> This only applies to theories with proper classes, so it has no
> relevance to ZFC, etc...
> To be definite, consider this to be a question about Goedel's 1939
> theory of classes.
> 
> Question:  Must every proper class contain an infinite subset?  If so,
> does the proof
> require the global axiom of choice and/or the axiom of regularity
> (Goedel's Axioms D, E)?
> 

This can be proved in NBG. NBG doesn't have any axiom of choice.

I will take "infinite" to mean "not finite", and finite to mean that it is
the image of a set function whose domain is a natural number. It is easy to
prove in NBG, using replacement, that a finite class forms a set.

Let X be the proper class. Let Y be the set of all ranks of elements of X.
Then Y is a proper class of ordinals. (We have used foundation in an
essential manner here). In NBG we can prove the existence of a unique set
function f:omega into Y such that f(0) is the least element of Y, and each
f(i+1) is the next element of Y after f(i). The range of f is as required.

However, this cannot be proved in NBG + AxC without foundation. Add formal
elements x1,x2,... where each xi = {xi}. Build the cumulative hierarchy of
sets starting with "proper class" V0* = {x1,x2,...}. The only "sets" that
are allowed are those whose "transitive closure" contains at most finitely
many of the x's. The cumulative hierarchy is done along all set ordinals,
and the elements of the various set ordinal stages form the sets of the
model. The classes of the model are all the subclasses of the family of all
sets of the model. Obviously {x1,x2,...} is a class in the model; and all
subsets of it, in the model, are finite.

There is a lot of symmetry in the structure so that one can verify all of
NBG except of course foundation. Of course, we also get MK without
foundation. Obviously we lean heavily on the fact that Replacement has
uniqueness in its hypothesis.

To be more rigorous, the above gives a procedure for converting any model of
NBG + AxC to a model of NBG\F + AxC + "there is a proper class all of whose
subsets are finite", and converting any model of MK + AxC to a model MK\F +
AxC + "there is a proper class all of whose subsets are finite."

Harvey Friedman