Although Frank Ramsey's proof isn't exactly a paradox, it is worth mentioning anyway.
Ramsey set out to prove that there were exactly two Londoners with exactly the same number of hairs on their heads.
How did he prove that?
Firstly, when Ramsey was writing he noted that there were more than a million Londoners. He also noted - though God knows how - that there were less than a million hairs on any one Londoner's head.
So how do we move from those two truths to the truth that there are at least two Londoners who have exactly the same number of hairs on their heads?
Well, for a start, there are more Londoners than hairs on any single Londoner’s head. That is, there are more than a million Londoners; though no person has more than a million hairs on his or her head. So what? This has been expressed in the following way:
This means that because there are more Londoners than there are hairs on any one individual's head, then at least two Londoners must share the same number of hairs. How does that prove what it claims to prove? And doesn't it depend on how many more (than a million) Londoners there are?“If there are more pigeons than pigeonholes, then at least two pigeons must share a hole.”
i) because there are over a million Londoners
ii) though only a maximum of a million hairs on any one person's head
iii) and since there are a million hairs to share between over a million people [eh? that's wrong]
iv) then two persons or more must share the same number of hairs.
Put it another way: if there were a million Londoners and a maximum number of a million hairs on a Londoner’s head, then each Londoner could have hairs ranging from one to one million.
In theory at least, every Londoner could have a different number of hairs on his and her head. But what happens when there are more than a million Londoners but still only a maximum of one million hairs on any one Londoner’s head? It can't be the case that every Londoner will have a different number of hairs – even in theory. This will mean, then, that at least two Londoners will have the same number of hairs.
But there's still something suspect about this proof.
In any case, this doesn't seem to be a paradox, or that complex, or even that interesting; though perhaps I've missed the profound or complex bit.