[FOM] Cardinality in weaker set theories

[Colin McLarty]

Hi,

The foundations of cohomological number theory lead me to work on
cardinals in Zermelo set theory and weaker theories and I want to
check that I've got this right.

Let ZC be the axioms of ZFC but with separation instead of
replacement.  So this includes foundation.  Are the following all
true?

A) We can define von Neumann ordinals in ZC as sets linearly ordered
by membership.

B) ZC proves for every natural number n there is a von Neumann ordinal
omega+n, and that is all it proves (if ZFC is consistent) since in ZFC
the set of all sets of finite rank over the naturals is a model of ZC.

C) So ZC does not prove every well ordered set is order-isomorphic to
a von Neumann ordinal.

D) ZC offers no canonical definition of the Alephs, that is no way of
selecting one representative of each cardinality.

Or have I missed something here?

thanks, Colin