>>16
Things are a bit unclear with set theory - some axioms are independent. However, things are much more clearer with arithmetic - which you seem to deny as well because all models of it are infinite (countable or not, although there's only one standard model which is countably infinite, the rest are non-standard and involve higher ordinals). If the ontology is arithmetically realist (which it has to be if the processes in our brain responsible for our mind happen to be computable) then physics is completlely dependent on an infinity of computations (getting awfuly close to uncomputable stuff here if you want to talk at the meta-level). I've also seen some physicists try to see what the physical consequences of a set theoretical ontology would be like and some of their results are pretty interesting as well, although I'm somewhat skeptical of them.
Either way, either physics is how some math looks from the inside or there's magical matter and experience which obeys no mechanistic laws (whatever that means, I can't conceive it) we'll never understand and we might as well give up all human endeavours and become hippies or religious zealots.