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Bertrand Russell
Gottlob Frege
In June 1902, Bertrand Russell, the great British mathematician and logician, sent the statement of a paradox to his friend Gottlob Frege, a German philosopher, logician and mathematician. Frege had been working for more than 10 years writing his monumental work “The Foundations of Arithmetic” and was finishing the final chapter of the second volume of this two-volume treatise. Russell and Frege had been friends for many years and Russell encouraged his friend to write a book about the mere foundations of arithmetic based on the set theory of Cantor. Frege obliged, but one day Russell found a contradiction of the work Frege was proposing in his book. This contradiction simply destroyed the very principle of his friend’s logic. Frege was devastated; his work of over ten years was simply irrelevant. Russell’s letter to Frege terminated the labor of more than ten years. Frege sank into a deep depression, while Russel tried to repair the damage by constructing a new theory of logic that would be immune to the paradox. He couldn’t. The paradox appeared again in his new theory.
The Russell paradox has been popularized in many ways. One of the best known of these was given by Russell in 1919 and concerns the plight of a barber of a certain village who has enunciated the principle that he shaves all those persons of the village who do not shave themselves. The paradoxical nature of the situation is realized when we try to answer the question: “Does the barber shave himself?” If he does shave himself, then he shouldn’t according to his principle; if he doesn’t shave himself, then he should according to his principle.
Since the discovery of the above contradictions, additional paradoxes have been produced in abundance. These modern paradoxes of set theory are related to several ancient paradoxes of logic, such as:
- Eubulides, of the fourth century B.C. is credited with making the remark, “The statement I am making is false.” If Eubulides statement is true, then, by what it says, the statement must be false. On the other hand, if Eubulides’ statement is false, it follows that the statement must be true. Then Eubulides’ statement can neither be true nor false without entailing a contradiction.
- Epimenides, who himself was a Cretan philosopher of the sixth century B.C., is claimed to have made the remark, “Cretans are always liars.” A simple analysis of this remark easily reveals that it, too, is self-contradictory.
The existence of paradoxes in set theory, like those described above, clearly indicates that something is wrong. Since their discovery, a great deal of literature on the subject has appeared, and numerous attempts at a solution have been offered. For some mathematicians there seems to be an easy way out. One has merely to reconstruct set theory on an axiomatic basis sufficiently restrictive to exclude these known antinomies. This is simply a procedure that avoids the paradoxes (putting your head in the sand.)
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