It should be noted that Godel gave two Incompleteness Theorems.
The first says that, in a system based on a list of axioms, there will always be some statements that aren't provable. (Remember Mr. Spock confusing a computer by telling it "I am lying"? "This statement can't be proved" behaves the same way -- it's true, but you can't prove it, because if you did then you couldn't prove it.) In other words, there is always truth the system can't account for.
The second theorem builds on the first. It says a system can't prove itself consistent, and that if a system includes the claim "this system is consistent" then it is necessarily inconsistent. (In essence, you can construct a more complex version of the "I am lying" statement in any system that includes a claim of "I am telling the truth".) In other words, the only way to prove something consistent is from outside of it.
Interestingly, Godel gave a presentation somewhere within 5 - 7th September at Koningsberg presenting the first incompleteness theorem. At this time, he did not have the second. [1]
Von Neumann was in attendance and he wrote a letter to Godel on November 20th announcing his discovery of the second incompleteness theorem. As it turns out, Godel had just sent a paper for publication on November 17th with the proof of the second incompleteness theorem. In reply to Godel, finding out he had just been scooped, von Neumann wrote:
"As you have established the theorem on the unprovability of consistency as a natural continuation and deepening of your earlier results, I clearly won't publish on this subject." [2]
This statement (and Godel's mathematical equivalent) does not assert anything to be proved, it is contentless, which is why logical, axiomatic systems choke on it, essentially due to recursion or self-reference. Logically there are only two fundamental ways to err; by contradiction and by circular reasoning.
> In other words, the only way to prove something consistent is from outside of it.
This is what I take from Godel's theorems but this is a poor formulation of the idea. A better way to say it is that proof presupposes consistency and more specifically the law of identity which is a metaphysical law that has to be validated not proved (since proof depends on it).
The first says that, in a system based on a list of axioms, there will always be some statements that aren't provable. (Remember Mr. Spock confusing a computer by telling it "I am lying"? "This statement can't be proved" behaves the same way -- it's true, but you can't prove it, because if you did then you couldn't prove it.) In other words, there is always truth the system can't account for.
The second theorem builds on the first. It says a system can't prove itself consistent, and that if a system includes the claim "this system is consistent" then it is necessarily inconsistent. (In essence, you can construct a more complex version of the "I am lying" statement in any system that includes a claim of "I am telling the truth".) In other words, the only way to prove something consistent is from outside of it.