David Peat

On Incompleteness 33while we try to understand what the words mean. Put another way, thenumber of a couple will be the class of all couples and the name of thisis “the number two.” Or as Russell puts it: “A number is anything whichis the number of some class.” 4 With his definition of “number” in termsof class, Frege felt that he had solved an important problem. Commonsense had no problem with numbers, but now Frege had been able toclarify the same concept at the very foundations of mathematics.Russell’s ParadoxThen Frege heard from Bertrand Russell that there was a fly in theointment! Frege had shown that you can put candies, apples, shoes,pigs, and so on each in their own class and match the members of oneclass with another and so determine what all of these different classeshave in common—that is, the number of objects in each of theseclasses. But Russell objected: the class of all candies is not itself a candy,neither is it an apple. In other words, since the class of all candies is nota candy, it is not a member of itself.There is nothing too shocking about this; it’s simple commonsense. Lots of classes are not members of themselves. The class of applesis not an apple; the class of shoes is not a shoe. So why not invent awhole new class called “the class of all classes that are not members ofthemselves”? So far so good. Now comes Russell’s turn of the screw: Isthis class a member of itself? or not? Trying to answer this questionexposed a major problem in the foundation of mathematics and mademathematicians and philosophers worry that certainty may not be assimple or obvious as they had hoped.Let us put Russell’s paradox in the following way. Within a biglibrary there is a room containing catalogs of books. Many of thesecatalogs contain references to their own titles, as well as to those ofother books. But some of these catalogs do not refer to themselves. Thelibrarian decides to make a new catalog called “The Grand Catalog that4Bertrand Russell. Introduction to Mathematical Philosophy (Fairlawn, N.J.:Macmillan, 1955).