Russell’s Paradox & Frege’s Logicism

Bertrand Russell thought up a deep and perplexing paradox when reading about Gottlob Frege's system of logic. Frege thought that he could define all mathematical concepts and prove all mathematical truths solely from principles of logic. The view that mathematics can be reduced to logic in this way is called logicism. Had Frege demonstrated the truth of logicism, it would have been one of the greatest achievements in the history of philosophy. But his version of logicism was not successful. One of the logical principles used to prove the existence of numbers, functions, and other mathematical objects is: for every predicate, "is F (P)" there is a collection of things that are F. Two examples are: "is a prime number" determines the collection of numbers {2, 3, 5, 7, 11…. } and "is a collection" determines the collection of all collections. In 1903 Russell showed that (P) is self-contradictory with the following argument. Consider the predicate "is not a member of itself." With (P) there is a collection -call it R- of collections that are not members of themselves. Is R a member of itself? If it is then it isn't, and if it isn't then it is. A contradiction! This was a devastating blow to Frege and to logicism.

Remember: The collection of all collections that are not members of themselves is itself both a member of itself and not a member of itself.

Additionally: Here is a paradox involving reasoning similar to Russell's: "There is a barber who shaves all those and only those who don't shave themselves." If the barber shaves himself then he doesn't shave himself, and if he doesn't then he does. This paradox is easy to solve, simply by accepting that there cannot be such a barber. Frege couldn't accept the analogous way out for collections, since he used his principle to prove the existence of collections required by mathematics.

source:
30-Second Philosophies by Barry Loewer