Consider the theory ZFfin := ZF − axiom of infinity + “all ordinals are finite” (i.e., every ordinal is zero or successor).
Of course, if we add the axiom of choice, this is not very interesting: it's basically first-order arithmetic in disguise. Basically what I'd like to know is whether we can make something combinatorially interesting out of ZFfin (without Choice).
Question: Can we find some axiom (or perhaps axiom scheme) T of a combinatorial flavor which makes ZFfin + T arithmetically at least as strong as ZFC (meaning that all arithmetical statements which are theorems of ZFC should also be theorems of ZFfin+T; of course, we also demand consistency of ZFfin+T)?
I don't have a precise definition of “of combinatorial flavor” but an example of something that is definitely not of that nature is the axiom scheme “all arithmetical theorems of ZFC are true”. I want something which looks like a large cardinal axiom, or a reflection scheme, or something set-theoretic like that, except that you can't do that as such because all ordinals are finite.
(If the question turns out to be too difficult, I might settle for other axioms which make ZFfin much stronger than arithmetic, without going as far as ZFC.)
Motivation: there are various axioms that can be added to the theory NFU (Quine's New Foundations with Urelements) to make it strong in various ways, and can be compared to the large cardinal axioms of ZFC. My question is whether analogous axioms can be found for ZFfin instead of NFU.