Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.
One thing I never found satisfactory is that any axiom system like this has to define things in terms of decidability etc, so it's "more verbose" (or less axiomlike) than ZFC
I think the only axiom you need to rethink from ZF is powersets (since I think that's the only axiom that produces uncomputable sets from computable ones (ignoring the AC, briefly)). What you'd replace it with (some sort of one based on comprehension, presumably) I couldn't say though.
But, at most countably many of those sets will ever be individually thought about by any human being. So there are only countably many (and maybe more strongly finitely many) subsets which any human being could ever possibly think about, and an uncountable remainder which are unthinkable. In order to do mathematics, do we really need to posit the existence of an uncountable number of objects no human mind will ever be able to individually consider?
> Of course, you could throw out some axioms so that you create an axiom system in which I can't define such a thing as "a subset of the natural numbers", but such a world doesn't feel "real" (in a platonic sense) to me.
You could introduce a set theory where from a countable set you can only infer the existence of its computable subsets, or its first order definable subsets, or something like that. Not sure why that should feel any less "real" than an uncountable infinity of mathematical objects about which no one will ever individually think.
If your arguments are tied to such physical constraints, then there aren't even countably infinite natural numbers unless the Universe is infinite in space or in time (such that there can be infinite humans or "thinkers").