Russells Paradox 

The paradox deals with how Sets can be defined.
The formal statement of the paradox is given at the foot of this page but here is a more informal example.
Consider the word "pentasyllabic" (means a word with five syllables). It's interesting because it describes itself, that is, it has five syllables. We shall call words that describe themselves HOMODOXIC.
Now consider the word "peanut." It doesn't describe itself. We call words that don't describe themselves HETERODOXIC. All words are either HOMODOXIC or HETERODOXIC, and never both.
We now make two sets:
If it is in the first set, then the word HETERODOXIC has the property of being HOMODOXIC, and therefore it doesn't describe itself. It should be placed in the second set.
But if it is placed in the second set, the word HETERODOXIC has the property of being HETERODOXIC and therefore it does describe itself. It should be placed in the first set.
OUCH !!!
Another even more informal example.
Consider a small village where there are two sorts of men  those who always shave themself and those who are always shaved by the barber.
In which set is the barber?
(No, he doesn't have a beard as there are no nonshavers in the village.)
Enrichment Task
Can you think of any other Homodoxic words? Clue: Colour and Color.
In formal Mathematical notation, Russell's paradox is:
Suppose that for any coherent proposition P(x), we can construct a set $\{x:P(x)\}.$
Let $S=\{x:x\not\in~x\}.$
Suppose $S\in~S$ ; then, by definition, $S\not\in~S.$
Likewise, if $S\not\in{S},$ then by definition $S\in{S}.$ Therefore, we have a contradiction.