TITLE: George Boolos and Gödel's Second Incompleteness Theorem
QUESTION [7 upvotes]: In Mind, Vol. 103, January 1994, pp. 1-3, George Boolos wrote:

And so on. In fact, if a claim can be proved, then it can be proved
  that the claim can be proved. And that too can be proved.
Now, two plus two is not five. And it can be proved that two plus two
  is not five. And it can be proved that it can be proved that two plus
  two is not five, and so on.
Thus: it can be proved that two plus two is not five. Can it be proved
  as well that two plus two is five? It would be a real blow to math, to
  say the least, if it could. If it could be proved that two plus two is
  five, then it could be proved that five is not five, and then there
  would be no claim that could not be proved, and math would be a lot of
  bunk.
So, we now want to ask, can it be proved that it can't be proved that two plus two is five? Here's the shock: no, it can't. Or, to
  hedge a bit: if it can be proved that it can't be proved that two plus
  two is five, then it can be proved as well that two plus two is five,
  and math is a lot of bunk. In fact, if math is not a lot of bunk, then
  no claim of the form "claim X can't be proved" can be proved.

...and...

...yes, it can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five. [i.e. what's asserted above can be proved]

[parentheticals added by me]
Please excuse the potentially ignorant questions:

Why proving that two plus two is four does not constitute as a proof that two plus two cannot be five (thus cannot be proved as such)?
Should I understand that the only way to prove that it cannot be proved that two plus two cannot make five is to prove that the system within we work is consistent, and that there is no other way? (If so, what is the proof that there is no other way?)

REPLY [12 votes]: While $2+2=4$ seems like an easy target for a way to prove $2+2\neq 5$, you're implicitly assuming that $4\neq 5$ and that $4=5$ cannot be proved. But if arithmetic is inconsistent, you can prove that $4=5$, in which case $2+2=5$ as well.
So stating that $2+2=5$ is not provable is a way of saying that $0=1$ is not provable. But now recall that the standard formulation of $\operatorname{Con}(T)$ is "$T$ does not prove $0=1$".
To your second question, the answer is more subtle than that. You need to assume that the system you work in is consistent, but that's why we have a meta-theory. Set theory proves that arithmetic is consistent, so working inside set theory you can prove that $2+2=5$ is not provable from the axioms of arithmetic. Of course, now you might want to ask, could set theory prove that $2+2=5$ is impossible? And Gödel snarks in the corner and says "try again". So you have to move to a stronger theory, and so on and so forth. 
It's turtles all the way down.