Russell's Paradox

All classes are either a member of themselves or not.

The class of all ideas is an idea. The class of all classes is a class. Both these classes are members of themselves.

The class of all men is not a man. The class of all illnesses is not an illness. Neither of these classes are members of themselves.

Let S be the class of all Self-membered classes. ie classes which are members of themselves.

Let N be the class of all Non-self-membered classes. ie. classes which are not members of themselves.

Consider N.

N is itself a class and must therefore be either a member of N or S.

Is N a member of itself? If it is not it must be a member of the class of non-self-members, which is N. But if N is a member of N, then it is a member of itself and therefore a member of S and not N. But if N is a member of S and not N, then it is not a member of its own class and must therefore be a member of N - which was where we began.

This paradox may be easier to follow in the following form:

If X is any class and N the class of all non-self-membered classes, then the following statement is true:

X is a member of N if and only if X is not a member of X.

Since X represents any class, and N is a class, we can substitute X for N:

N is a member of N if and only if N is not a member of N.