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There are some logical paradoxes connected with the theory of sets. The book The Foundations of Mathematics by Evert Beth discusses 17 different paradoxes[9, pp. 481-492]. Here I discuss just one of these which was published by Bertrand Russell in 1903[43, ¶78,¶¶ 100-106].

Let be the set of all sets, let be the set of all infinite sets, and let be the set of all finite sets. Then we have

Here since there are infinitely many finite sets. since contains just one element, which is the set of all sets. Also since is not a set. Next, let

Then for all we have
 (1.59)

Thus

We now ask whether is in . According to (1.59),

i.e., is in if and only if it isn't!

I believe that this paradox has never been satisfactorily explained. A large branch of mathematics (axiomatic set theory) has been developed to get rid of the paradox, but the axiomatic approaches seem to build a fence covered with  keep out" signs around the paradox rather than explaining it. Observe that the discussion of Russell's paradox does not involve any complicated argument: it lies right on the surface of set theory, and it might cause one to wonder what other paradoxes are lurking in a mathematics based on set theory.