Virgil wrote: > In article <firstname.lastname@example.org>, > David Bernier <email@example.com> wrote: > > > Obviously, one wants ordinals of uncountable order type. > > In ZC without any additional axioms, it would seem that one > > can only prove the existence of von Neumann ordinals that are > > countable (either finite or infinite). > > > > So in plain ZC, what would be a good definition of an > > ordinal? > > Is there any reason why one cannot use the same definition as in ZF and > NBG? >
It doesn't work very well because you don't have the axiom of replacement, so you can't prove that omega times two exists as usually defined when we're working in ZF. It's better to define an ordinal to be a set which is equal to the set of all well-ordered sets of minimal rank order isomorphic to a given well-ordered set. You get more ordinals that way.
> The von Neumann definition of NBG is : > A set S is an ordinal if and only if S is totally ordered with respect > to set containment and every element of S is also a subset of S.