I'm trying to understand Godel's second incompleteness theorem, which by my understanding is equivalent to, "An axiomatic system [with certain complexity and soundness properties] $F$ cannot prove its own consistency, i.e. $F⊬(F ⊬ 0=1)$" as a specific example following on from the first incompleteness theorem stating, "There is some syntatically valid sentence that $F$ cannot prove either true or false."
I understand that consistency is a ludicrously strong statement, since it's effectively an assertion over a set of completely general, arbitararily complex sentences and because of that, I wouldn't expect it to be provable to begin with. However, the 2IT says that this statement's not merely practically unreasonable, but directly implies a contradiction. (Or has some other self-defeating implication)
So, what contradiction can be derived from the assumption that a sound theory can prove that it itself is consistent, and how?