Bertrand Russell (1872-1970) constructed a famous paradox (an "antinomy") to persuade the mathematical world that in developing consistent systems (systems in which every statement is either true or false), familiarity and intuitive clarity are not solid bases.

The argument goes on like this:
 

1. There are sets than contain themselves (examples: "the set of all objects that can be described with exactly thirteen  words", "the set of all thinkable things")
2. Therefore, a set either contains itself or not. Let's call a set "non-normal" in the first case and "normal" in the second
3. Let N be the collection of all normal sets, which of course, is itself a set
4. Question: is N normal?
5. If N is normal, then by definition of "normality" it does not contain itself. But N contains by construction all normal sets therefore itself too (contradiction)
6. If N is not normal, then by definition of "non-normality" N is itself a member of N. But by construction, any member of N is a normal set (contradiction too)
7. Conclusion: the statement "N is normal" is neither true nor false

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