*a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth.*
Russell`s Paradox

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

This particular paradox is analogous to the contradictions found when examining infinite sets. Gödel's work led to the realization that such "antinomies" are unavoided and so the mathematical Holy Grail of complete consistent systems is an impossibility.