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.
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