Declension wrote:
Here is my favourite paradox. You need to know the idea of "proof by contradiction". If you're not sure, look it up on Wiki.
CLAIM: Each positive integer can be named in under fourteen English words.
PROOF: Suppose for contradiction that not every positive integer can be named in under fourteen English words. Then there is a smallest positive integer that cannot be named in under fourteen English words, call it x. But x can be named in under fourteen English words, since x is "the smallest positive integer that cannot be named in under fourteen English words", giving us our contradiction.
CLAIM: It is impossible that each positive integer can be named in under fourteen English words.
PROOF: There is a finite number of English words, call it N. So there are only N^1 + N^2 + ... + N^13 possible combinations of under fourteen English words. But there are an infinite number of positive integers!
This is the well know Richard's Paradox. The problem is that a name is not a description and vice versa.
ruveyn
24 Jul 2012, 1:32 am
