# FOM: cardinals in ZFC fragments

Rupert McCallum rupertmccallum at yahoo.com
Tue Aug 6 18:21:41 EDT 2002

```--- Harvey Friedman <friedman at mbi.math.ohio-state.edu> wrote:
> I have the feeling that there are various results, positive and
> negative,
> scattered throughout the literature and folklore on the following
> problem.
> I would appreciate hearing about such results.
>
> Let T be a set theory. We say that T "handles cardinals as objects"
> if and
> only if there is a formula phi(x,y) of set theory, with at most x
> free,
> such that T proves the following.
>
> 1. (forall x)(therexistsunique y)(phi(x,y)).
> 2. phi(x,y) and phi(z,w) implies "x,z are equinumerous iff y = w".
>
> It is standard that ZFC handles cardinals as objects, by taking y =
> the von
> Neumann cardinal of x.
>
> Scott showed that ZF handles cardinals as objects.
>
> But what about other fragments of ZFC?
>

Scott's trick would work in Z, would it not? Call a set V a level if
there is a well-ordered sequence (with anything at all as its domain)
such that the next term after S, if there is one, is P(S), a term with
no immediate predecessor is the union of all preceding terms, and the
last term is V. Z with the axiom of foundation, I believe, can prove
that every set is in some level. Define [T]_V to be {U e V: U is
equipollent to T and there is no set equipollent to T in any proper
sublevel of V}. Take these sets as your cardinals. phi(x,y) is "(E a
level V) y = [x]_V".

>
>

__________________________________________________
Do You Yahoo!?
Yahoo! Health - Feel better, live better
http://health.yahoo.com

```