FOM: cardinals in ZFC fragments

[Harvey Friedman]

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?