Russell’s Barber Paradox

Bertrand Russell discovered a paradox in set theory that had important implications for mathematics, philosophy, and puzzles.

Russells Barber Paradox

Assume any collection of items is a set. “Things that are red” is a set. “Concepts” is a set. Assuming that a set is a concept, then the set called “Concepts” is also a concept, so it is a member of itself. The set called “Things that are red,” however, is not red, so it is not a member of itself.

Next imagine the set called “Sets that are not members of themselves.” This includes the set called “Things that are red,” of course, and many other sets. But does it include itself? If no, then it meets the definition of the set and should be included. But if yes, then it does not meet the definition and should not be included.

This paradox leads to a riddle: The village barber (who is a man) shaves all the men in the village who do not shave themselves, and only those men. So who shaves the barber?

A variation is the Grelling-Nelson paradox: Does the adjective “non-self-descriptive” describe itself?

LogicomixBertrand Russell’s discovery of this paradox was an important development in set theory. Russell described the paradox in a letter to Gottlob Frege, who was about to publish the second volume of his work, The Basic Laws of Arithmetic. The paradox undermined one of the Basic Laws that Frege was proposing. Frege only had time to add an appendix, which began with these words: “Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.” If you want to learn more about Bertrand Russell’s life and discoveries, check out the excellent graphic work Logicomix: An Epic Search for Truth.

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