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?
Best Answer
But anyway, my point is this: just look at ON and totally forget that is the spine of V, and see it as a system of ordered numbers. Now axiomatize what you see, much in the same way as you axiomatize its initial segment N. – Mirco Mannucci Jul 8 at 22:21
for instance, one could add to the theory an equivalence relation, α≡β formalizing equinomerosity, and
talk about cardinals without the (ambient) set theory
Well, this construction [Gavrilovich and Hasson’s Exercices de Style, a homotopy theory of set theory] attempts to talk about cardinal invariants and use as little set theory as possible: instead it uses axioms of a model category to do that. and indeed, as you suggest, there equinomerosity (up to some fixed $\kappa$) is introduced, under the name of cofibrant. The construction does not work with the skeleton, though; but perhaps it would be closer to using the skeleton if you modify the defitinions and replace everywhere 'inclusion' aka 'subset' by 'injective map'; then you'd lose limits in your model category.
How powerful the language is, is unclear. But you can indeed say $\kappa$ is measurable: that's when the corresponding homotopy caetgory is not dense as a partial order.