The Information - James Gleick [85]
A cleaner formulation of Epimenides’ paradox—cleaner because one need not worry about Cretans and their attributes—is the liar’s paradox: This statement is false. The statement cannot be true, because then it is false. It cannot be false, because then it becomes true. It is neither true nor false, or it is both at once. But the discovery of this twisting, backfiring, mind-bending circularity does not bring life or language crashing to a halt—one grasps the idea and moves on—because life and language lack the perfection, the absolutes, that give them force. In real life, all Cretans cannot be liars. Even liars often tell the truth. The pain begins only with the attempt to build an airtight vessel. Russell and Whitehead aimed for perfection—for proof—otherwise the enterprise had little point. The more rigorously they built, the more paradoxes they found. “It was in the air,” Douglas Hofstadter has written, “that truly peculiar things could happen when modern cousins of various ancient paradoxes cropped up inside the rigorously logical world of numbers,… a pristine paradise in which no one had dreamt paradox might arise.”♦
One was Berry’s paradox, first suggested to Russell by G. G. Berry, a librarian at the Bodleian. It has to do with counting the syllables needed to specify each integer. Generally, of course, the larger the number the more syllables are required. In English, the smallest integer requiring two syllables is seven. The smallest requiring three syllables is eleven. The number 121 seems to require six syllables (“one hundred twenty-one”), but actually four will do the job, with some cleverness: “eleven squared.” Still, even with cleverness, there are only a finite number of possible syllables and therefore a finite number of names, and, as Russell put it, “Hence the names of some integers must consist of at least nineteen syllables, and among these there must be a least. Hence the least integer not nameable in fewer than nineteen syllables must denote a definite integer.”♦♦ Now comes the paradox. This phrase, the least integer not nameable in fewer than nineteen syllables, contains only eighteen syllables. So the least integer not nameable in fewer than nineteen syllables has just been named in fewer than nineteen syllables.
Another paradox of Russell’s is the Barber paradox. The barber is the man (let us say) who shaves all the men, and only those, who do not shave themselves. Does the barber shave himself? If he does he does not, and if he does not he does. Few people are troubled by such puzzles, because in real life the barber does as he likes and the world goes on. We tend to feel, as Russell put it, that “the whole form of words is just a noise without meaning.”♦ But the paradox cannot be dismissed so easily when a mathematician examines the subject known as set theory, or the theory of classes. Sets are groups of things—for example, integers. The set 0, 2, 4 has integers as its members. A set can also be a member of other sets. For example, the set 0, 2, 4 belongs to the set of sets of integers and the set of sets with three members but not the set of sets of prime numbers. So Russell defined a certain set this way:
S is the set of all sets that are not members of themselves.
This version is known as Russell’s paradox. It cannot be dismissed as noise.
To eliminate Russell’s paradox Russell took drastic measures. The enabling factor seemed to be the peculiar recursion within the offending statement: the idea of sets belonging to sets. Recursion was the oxygen feeding the flame. In the same way, the liar paradox relies on statements about statements. “This statement is false” is meta-language: language about language. Russell’s paradoxical set relies on a meta-set: a set of sets. So the problem was a crossing of levels, or, as Russell termed it, a mixing of types. His solution: declare it illegal, taboo, out of bounds. No mixing different levels of abstraction. No self-reference; no self-containment.