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 attemps [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 sceletonskeleton, though; but perhaps it would be closer to using the sceleton skeleton if you modify the defitinions and replace everywhere 'inclusion' aka 'subset' by 'injective map'; then you'd loose 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.