Self-defeating axioms? By Godel's incompleteness theorem, no consistent r.e. theories $T$ that can encode PA can prove $Con(T)$. But is there is consistent r.e. theories $T$ that can encode PA that can actually prove $\neg Con(T)$?
Perhaps there is already a name for it, but I don't know, so I'm calling it self-defeating theories. Note that it still have to be consistent.
 A: Everything below assumes that PA is consistent.
Sure, there are lots of these (although I don't know a name for them) - by Goedel's incompleteness theorem, the theory PA + "PA is inconsistent" is consistent, assuming PA is! And clearly it proves its own inconsistency.
The issue here is one of soundness: PA can't convert a proof of "I prove $0=1$" to a proof of "$0=1$." In fact, this sort of conversion is very rare, in a precise sense: by Lob's theorem, PA proves "If I prove $\varphi$, then $\varphi$ is true" exactly when PA already proves $\varphi$! So there is actually very little new information PA can extract from a proof of "PA proves $\varphi$."

Let me say a little more about the details here, to address the worries of the commenter below.
First of all, note that if you believe the incompleteness theorem, you had better believe the statement I wrote above. If PA+"PA is inconsistent" proved "$0=1$", then by the deduction theorem we'd have that PA proves "If PA is inconsistent, the $0=1$." But PA proves "$0\not=1$," so by contrapositive PA would prove "PA is consistent" and so be inconsistent.
Alright, so what's the heart of the confusion expressed below? Note the sentence

"Starting from false statements, you can prove anything."

This is a common phrasing of the principle of explosion; however, it is not actually correct! There are plenty of false-but-consistent sets of axioms out there. 
Incidentally, there's already a nascent version of this conflation in the following two definitions of inconsistency used in the comment: "you can prove anything, and that's called inconsistency" (correct) and "theories that prove false statements are called inconsistent" (incorrect). Inconsistency is a purely syntactic property (do you prove every sentence?) whereas correctness is a semantic property (do you prove any sentences which are not true about the natural numbers?).
Accurately stating explosion is a bit subtle. One version is:

If $T$ proves $\neg\varphi$, then $T$ proves "$\varphi\implies \psi$" for every sentence $\psi$.

This is absolutely true. But note the hypothesis "$T$ proves $\neg\varphi$," not "$\varphi$ is false." That is, the principle of explosion is properly understood internally to a theory. Note that if this were not the case, then the true theory of arithmetic would be decidable: for every sentence $\varphi$ in the language of arithmetic, exactly one of PA+$\varphi$ or PA+$\neg\varphi$ would have to be inconsistent, and we would be able to search through possible proofs to determine which it was. But the true theory of arithmetic is extremely undecidable.
The takeaway is this: a theory which proves false statements need not be inconsistent; the principle of explosion is not as strong as it is often stated. While it is true that any theory proving its own inconsistency must have false axioms, it need not be able to prove every sentence, that is, it may not actually be inconsistent.
