In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the (ambient) set theory
the class ON is first-order axiomatizable, and thus it looks like I can carve out of ON the subclass CARD (for instance, one could add to the theory an equivalence relation, $\alpha \equiv \beta$ formalizing equinomerosity, and then define a cardinal in the usual way as the min ordinal in the equivalence class).
Once I have my definable predicate $CARD(\alpha)$, I can proceed to introduce cardinal arithmetics. For instance, I can define successor as the minimal cardinal greater than the given cardinal.
Obviously, I need to make some assumptions as to the basic cardinal arithmetics, so that it looks like the standard one in $ZFC$ + (possibly) generalized continuum hypothesis.
Now, assuming one has done all of the above, it appears that the "small" large cardinals, such as weak inaccessible, Mahlo, etc are definable in this theory (even in standard presentations, such as Drake, their definition is arithmetic ).
But what about the others, the heavy-weight ones? Do I necessarily have to resort to the ambient set theory ( stationary points, elementary embeddings, etc ) to talk about very large cardinals, or there is always a direct (algebraic/arithmetical/topological) way to provide their definition?
Prima facie, it looks like the answer is no, but maybe there is a clever path to answer in the affirmative. Or perhaps, there is some kind of intrinsic boundary, beyond which you need to think of cardinals within the context of set theory
Any thought, refs, or known fact?