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, 1…} 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.
Based on the book "30 Second Science of Thought"
