[FOM] Cardinality in weaker set theories

[Colin McLarty]

Thanks to everyone for helpful answers.

I am not sure how to apply the Scott trick to get a set of
representatives of each cardinality, without replacement.  I think the
key issue is whether you can prove that when all elements of a set S
have well defined ordinal rank then there is a supremum to their
ranks, so that S also has well defined rank.

Is there a way around that?  Or is it unnecessary?

thanks, Colin