The material referenced above covers what Duy Bui calls "naive set theory". There is a complication with naive set theory, which was first published by Bertrand Russell in 1901, although the problem was known by some mathematicians before. Russell constructed the set of all sets that are not a member of themselves, i.e.
{ x | x ∉ x }
and then asked whether this set is a member of itself. In other words, is
{ x | x ∉ x } ∈ { x | x ∉ x }?
A little thought will show the problem. If this set, let's call it R, is a member of itself, then it must satisfy the condition of not being a member of itself. In other words,
R ∈ R → R ∉ R
On the other hand, if R is not a member of itself, then it satisfies the condition of of being a member of R. In other words
R ∉ R → R ∈ R.
We therefore have a paradox that can only be resolved by complicating naive set theory. You can read more about Russell's paradox and how mathematicians have dealt with it at http://en.wikipedia.org/wiki/Russell%27s_paradox.