As the semester wraps up, and students are busily trying to write clearly about difficult stuff, they may be encouraged by this effort of George Boolos to explain Gödel’s second Incompleteness theorem in words of one syllable:

**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, 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.