FOM: cardinals in ZFC fragments

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

--- Harvey Friedman <friedman at> 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". 


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