Gödel's Second Incompleteness Theorem

from: http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

                 Gödel's Second Incompleteness Theorem
                   Explained in 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, t hough
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
pr oved 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 tw o 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 pl us 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 fo rm "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.



George Boolos, Mind, Vol. 103, January 1994, pp. 1 - 3.