Originally posted by vistesd
What is a diagonalization proof?
Of anything pertaining to this discussion that you could have asked, that question is the most difficult, but one of the most interesting. I'll take a shot at it, but I bet royalchicken can provide a better explanation.
The abstract answer is that diagonalization proofs are used to demonstrate claims about sets of things being countable or listable.
One standard example of a diagonalization proof is that one which demonstrates that there exist more real numbers between 0 and 1 than there are integers. Intuitively, when considering the truth of this claim, you'd note that there are infinitely many integers, and infinitely many real numbers between 0 and 1, so you might suspect the claim to be false.
Suppose you then claim it is false.
You would then be claiming that there exist at least as many integers as there are real numbers between 0 and 1.
If I think the claim is true, then I would formulate this challenge: put those real numbers in a list to correspond with the list of integers.
Maybe you'd start like:
1 --- 0.340000...
2 --- 0.894420...
3 --- 0.600000...
4 --- 0.999300...
and so on, with each real on the right ending in an infinity of digits.
Since there are an infinite number of integers, our list will be infinite.
However, if I can demonstrate that there is a real number that cannot occur in the right-hand side of the list (that is, that for any way you order the reals in correspondence with the integers, there will be a real missing from your list), I have shown that there are in fact more reals between 0 and 1 than there are integers.
I will construct such a real number right now.
Let x be any real number 0.abcdefg..., where
a is not equal to the first digit of the first real in the list,
b is not equal to the second digit of the second real in the list,
c is not equal to the third digit in the third real in the list,
and so on ad infinitum.
It must be the case that x is nowhere in the list. (It can't be the first real in the list, since its first digit differs from that; it can't be the 2nd, 3rd, etc. one either, for the similar reason.) However, it is also the case that x is a real number. Thus, there must exist more real numbers between 0 and 1 than there are integers, since we couldn't fit at least one real number into a one-to-one correspondence with the integers.
So, why the name diagonalization? Look at how the a, b, c, ... are defined. They correspond to the diagonal of the digits of the reals aligned in a list; i.e., the first from the first, the second from the second, and so on.
This may be a boring explanation, but the proof technique is clever and has been used to demonstrate some difficult, profound and counter-intuitive theorems.