The existence proof you describe is in terms of a yet more complex abstraction; we can imagine aliens might describe a more useful atomic theory than our own which does not involve nucleons or their discrete counts at all (ok, unlikely as I acknowledge that may be).
I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument.
I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up".
I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.
This may seem a bit extreme, but, in my opinion, while in mathematics (and especially in physics) there is a fair amount of scaffolding, what they do in mathematics is, actually, they discover things that do exist in reality in one form or another. (It is important to understand that few, if any, things in mathematics are pure fantasy; the objects and relations studied there are pretty much forced upon us - which, incidentally, is yet another evidence in support of their having real significance, i.e. as something lying outside our consciousness and acting there independently from it.)
Yes, but the reals (or the power set of the reals) are really a bit dodgy. The Nana h tarski paradox shows a big disconnect with reality. There is other axiom models without AOC, which allow for every subset to be measurable, but they seem to have their own problems.
The reals exist in nature in the same sense as, say, 𝜋 does (which occurs as the natural limit in a particular Monte Carlo experiment). The paradox that you mentioned is, rather, disconnected from our intuition, from which reality itself is, too, disconnected (if you think of quantum mechanics, for example).
I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument.
I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up".
I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.