Godel's Second Incompleteness Theorem says, officially, that given a
set of
axioms A and rules by which you can deduce (prove)
theorems from the axioms, if you can deduce all the laws of good old elementary arithmetic from the axioms A, then you
can't prove that the axioms are
consistent (i.e. that they aren't self-contradictory).
George Boolos has (i think) a great explanation of the second incompleteness theorem using only words of one syllable:
First of all, when I say "proved", what I will mean is "proved with the aid of the
whole of math". Now then:
two plus two is four, as you well know. And, of course,
it can be proved that two plus two is four (proved, that is, with the aid of the whole of math, as I said, though in the case of two plus two, of course we do not need the
whole of math to prove that it is four). And, as may not be quite so clear, it can be proved that it can be proved that two plus two is four, as well. And it can be proved that it can be proved that it can be proved that two plus two is four.
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.
So, if math is not a lot of bunk, then, though it can't be proved that two plus two is five, it can't be proved that it can't be proved that two plus two is five.
By the way, in case you'd like to know: 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.
If this was a little too convuluted, check out
Godel's Theorem; if you get it over there, I highly recommend coming back to read this--it's great fun.