I'm with Steve on the 2 + 2 = 4 question. It's not a fact, but simply something we agree about because we use a particular mathematical system. The axioms of that system are unproven assumptions. The idea that no mathematical system can ever be complete in this sense is down to this guy: Kurt Godel.
Umm, Godel says that an axiomatic system cannot be shown to be complete. Wouldn't that imply that the system has to be sufficiently complex to involve axioms? 2+2=4 does not contain any axioms. Rather the arithmetic operations are basically underlying the entire system. They are too basic for inclusion. I was taught Godel's theorem using the example of a geometry system where in fact parallel lines do meet at infinity. I don't recall having to assume 1+1=2 but then I was not a very good student.
Even if I fail at the first hurdle, I would go on to claim that whatever axioms cannot be proven, 2+2=4 does not rest on any axioms. It is a simple and observable fact that 2+2=4 and they do so in any consistent axiomatic system. Let me stress that word consistent. Assume a mathematical system where 2+2=5 and think about the consequences.