Russell's Paradox

Probably, I heard the word paradox for the first time when I read about the twin paradox of Einstein. I read this atleast ten years ago. Then I was not matured enough to understand what the word paradox really mean. It was recently, when I was reading Terrence Tao's book on "Analysis", I was once again confronted with another paradox. This was Russell's paradox. I see what a paradox really means.

Russell's paradox: Let's make a statement.

Let x be any object and p(x) be any property of x (i.e., either p(x) is true or is false). Then there exists a set {x : p(x) is true for x}. -----(1)

This seems to be an innocent statement. But actually, it is a very dangerous and fallacious statement. Consider x be a set and p(x) be the statement :-

x does not contain itself ----(2).

Then according to the statement (1), there exists a set S which contains all sets for which property (2) holds (i.e., they do not contain themselves). Does S contain itself?
1) If S does not contain itself, then by its very definition, it must include itself.
2) If S contains itself, then it should not contain itself.
This is called Russell's paradox. And, it disproves the statement given in (1). In fact, until Betrand Russell discovered this paradox, (1) was considered as an axiom of set theory. Now, a set theory which considers (1) as an axiom is called naive set theory.

Even if one does not understand this, one can have a look at this webpage http://www.paradoxes.co.uk/index.htm, at whose end a few exciting, or rather mysterious pictures illustrating paradoxes are given. To know more about Russell's paradox look here: http://en.wikipedia.org/wiki/Russell's_paradox.