Believe it or not, you can prove absolutely anything you want.
For instance, you might wish to prove that the moon is made of green cheese. Take the statement you want to prove and insert it at number 3 in the box below:
1. There are three numbered statements in this box.
2. Exactly one of these statements is true.
3. The moon is made of green cheese.
Now we look at each statement in turn.
Statement 1 is true, as we can see from looking at the box.
Statement 2 cannot be true, for the following reasons: we know that 1 is true, and if 2 is also true, then there would be at least two true statements, not one, in the box. Therefore we mark statement 2 as false.
This means that there are the number of true statements in the box is either none, two or three. It’s not none, because 1 is true. It’s not three, because we’ve just decided that 2 is false.
Therefore there are exactly two true statements in the box. We already know that 1 is true, and 2 is false; thus 3 must be true regardless of what it says!). Thus we can prove that the moon is made of green cheese.
(I first ran across this problem many years ago, in a magazine advertisement for the logic training game Wff n Proof.)
This is an example of the type of paradox first attributed to Epimenides the Cretan (circa 500BC according to Plato, or 600BC according to Aristotle [Britannica]), who said "All Cretans are Liars". It surfaces later in the barber paradox propounded by Bertrand Russell, and arises from the fact that we are using statements to describe themselves.
In its purest form, the paradox resolves to "This statement is false".
There’s an interesting site describing most of the famous mathematical paradoxes at: http://eluzions.com/Puzzles/Logic/Paradox/