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# A CLOSE SHAVE WITH SET THEORY

YESTERDAY’S ITEM on countability mentioned Russell’s Paradox, but, in the interests of set-theoretic sanity, side-stepped details. Now that we’re rested, what the hey, why not delve into it?

Bertrand Russell, 1872 – 1970, English philosopher, writer, mathematician, social critic, political activist. Nobel Laureate in Literature, 1950. Photo from 1938.

Among Bertrand Russell’s many achievements in philosophy, logic, literature and social action, in 1932 he was awarded the De Morgan Medal of the London Mathematical Society. The medal is named for Augustus De Morgan, 1806- 1871, British mathematician and first president of the society.

Principia Mathematica to *56, by A.N. Whitehead and Bertrand Russell, Cambridge University Press, 1962. See also the three-volume sets, Principia Mathematica – Volume One, etc., 2010

Russell is coauthor with Alfred North Whitehead of the three-volume Principia Mathematica, in various editions from 1910 to 1962. These are not to be confused with Isaac Newton’s classic work, full name Philosophiæ Naturalis Principia Mathematica. Russell and Whitehead’s work, by contrast, is on the foundations of modern mathematics, akin to the works of Phantom Mathematician Nicolas Bourbaki.

Researching the fundamentals of set theory in 1901, Russell discovered what came to be known as his paradox. It’s especially beguiling phrased as follows.

A town’s barber is required to shave only those who do not shave themselves. The paradox: Who shaves the barber?

If he doesn’t shave himself, then he must shave himself. If he shaves himself, then he mustn’t.

This can be stated as a set-theoretic quandary: Where to place the barber, in with the self-shaved or the barber-shaved?

Russell observed “… whether a class is or is not a member of itself is nonsense… because the whole form of words is just noise without meaning.”

On the other hand, one wit has suggested the barber is a woman. Presto, all is resolved. ds