(X¡ôN)¡ê(XX)

Now take X to be N, and we have the contradiction

 

 

(N¡ôN)¡ê(NN)

Known of these was given by Russell himself in 1919 and concerns the plight of the barber of a certain village who has enunciated the principle that he shaves all those persons ans only those persons of the village who do not shave themselves. The paradoxical nature of this situation is realized when we try to answer the question, "Does the barber shave himself?" If he does shave himself, then he shouldn't according to his principle; if he doesn't shave himself, then he should according to his principle.

Other attempts to solve the paradoxes of set theory look for the trouble in logic, and it must be admitted that the discovery of the paradoxes in the general theory of sets has brought about a thorough investigation of the foundations of logic

 

http://library.thinkquest.org/22584/emh1800.htm

 

*You may have to scroll a bit to get to it.

 

My question is basic I think. Has this been resolved yet and what are the implications?

You have symbols that my internet reader can't interpret. However, yes, though I am no expert, "naive set theory" has been replaced by "classes" which are essentially hierarchies of "sets". At the base, the things that are still called "sets", cannot have sets as members. Then we have a "class" that can contain sets, a class that can contain THOSE things, etc.