It seems like Hume's Principle is only a matter of semantics, not of mathematics. The concept of one-to-one correspondence is useful even if we don't refer to i

It seems like Hume's Principle is only a matter of semantics, not of mathematics. The concept of one-to-one correspondence is useful even if we don't refer to it as size. All existing proofs would be equally valid even if we didn't accept this definition of 'size', since it would still be true that there is or isn't a correspondence between two sets, even if we refer to it in a different way.