Actually a = a is an axiom of number theory, if I remember my Godel, Escher, Bach correctly.
eh... having read GEB does not you a number theorist make.
It is well possible to have many different axiomatizations of the very same theory; thus talk about some given formula being an axiom or not of a theory is sloppy talk for the formula in question being an axiom in a particular axiomatization of that theory. That is, "a=a" does not have to be an axiom of number theory; you can set up things differently.
And indeed, I think one could get away with the following pair of axioms to replace it: "0=0", "a+1=a+1". These allow for any number "a" to prove "a=a" in a+1 steps.
Of course, in any self-respecting logical language with identity, "a=a" is taken to be axiomatic, since "=" is taken to be an equivalnce relation (i.e. a=a, a=b<->b=a, a=b&b=c->a=c)
If it were possible to prove a thing in mathematics that was not actually true, then at least one of the axioms would necessarily have to be false.
Outside of having an inconsistent theory, how could you show that the conclusion of a proof was not true?
Deciding whether an axiom is true or false is another whole can of worms entirely--most would say that the statement makes no sense at all, and that you can only say whether given set of axioms is consistent or inconsistent with itself.
This is a tough question of philosophy. Take the axiom of continuity in number theory, which is independent of the other axioms: both the theory with the axiom and that with its negation are consistent. If take a very strong platonist view of mathematics, and you assume that there are mathematical objects, that mathematical statements are statements about mathematical objects, and that all mathematical statements are either true or false, then you have to conclude that one of these theories is true, while the other one is false, but that you can't know which one is which.
On the other hand, if you take a completely different approach, and deny that there are mathematical objects, and thus that statements about mathematical objects are either trivially false or lack a truth value, you then take on the burden of explaining how come theories which seem make reference to mathematical objects and/or use mathematical statements as premises (e.g. pretty much all of modern science) can be true or false.
So, this *is* a mess, indeed.
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