By definition, all axioms are not derivable within the axiom system they define. We can talk about sand but that does not mean our minds are sand or sand machi

By definition, all axioms are not derivable within the axiom system they define. We can talk about sand but that does not mean our minds are sand or sand machines. We can talk about infinity within a set theory that contains that concept. A very large majority of mathematicians use infinite sets and have no problem with them. There are mathematicians that don't like the existence of uncountable sets or of uncountable choice but almost all of them accept countably infinite sets. For instance, I've never heard of a mathematician who believes that there are finitely many primes.