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) |

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.

The poet and grammarian Philitas of Cos is even said to have died prematurely from exhaustion, owing to his desperate efforts to solve the paradox.[9, page 493]

Philitas was concerned about a different paradox, but Russell's paradox is probably more deadly.