1.a) take a look at this link: 2p2e4 - Metamath Proof Explorer
b) the wording of this question is a little too vague, for my liking. if the statement is that: it can be proved mathematics is (logically) consistent, it is untrue. this isn't to say mathematics is inconsistent, it's just that there are important statements in mathematics that cannot be proven. an example: the Axiom of Choice, is it true or false? it's been shown that including AC and denying AC both lead to consistent versions of set theory.
c) my guess is you are studying Gödel's theorems. i decline to answer this question, and the following.
3) why, Russell's set, of course.