Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an element of M if and only if A is not an element of A. In the sense of Cantor, M is a well-defined set. Does it contain itself? If we assume that it does, it is not a member of M according to the definition. On the other hand, if we assume that M does not contain itself, than it has to be a member of M, again according to the very definition of M. In both cases we get a contradiction but one of them must be true. So this must be a contradiction in the underlying theory.
There are some versions of this paradox which are closer to real-life situations and maybe easier to understand for non-logicians: For example, the story of the barber who shaves everyone who does not shave himself. When you start to think about whether he should shave himself or not you will get puzzled...
After this paradox came up, set theory was formulated axiomatically in a way that avoided this paradox. The most common version of axiomatic set theory in mathematics nowadays is perhaps Zermelo-Fraenkel set theory, with some version of the axiom of choice.
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