X-Message-Number: 10233 Subject: A math education for Bob. Date: Fri, 14 Aug 1998 11:15:17 -0400 From: "Perry E. Metzger" <> > From: > > In some town (not Seville) there is one barber, and the barber shaves all > those men who don't shave themselves. Who shaves the barber? > > Obviously, no problem arises except in the case of the barber himself, so the > statement essentially reduces to: "If the barber shaves himself, then he does > not shave himself; and if he does not shave himself, then he does shave > himself." This is merely self-contradiction, not paradox in any sense. > Childishly simple, with no shades of Goedel whatever. I'm afraid that is not correct. First of all, you have to understand the significance of the Barber paradox. The Barber paradox is the reason that early set theory got demolished. At one time, it was thought to be reasonable in set theory to speak of any set as though it could be reasoned about -- one could, for instance, postulate the "set of all sets". Russell's discovery of the Barber paradox was significant since it destroyed the notion of the universal set, and indeed almost any arbitrarily constructed set. From then on, set theory (especially naive set theory) depended on rules from which to construct new sets given old ones, and accepted that not all sets were constructible. This was a major revolution. (If the set theorists had been thinking, they might have seen the analogy to the fact that not all strings in any formal system can be derived from the axioms of the system, but that is leaping ahead decades to Goedel himself.) Second of all, you seem to believe that the Barber paradox and Goedel's string (which is approximated in English as "I am not provable in system X") are somehow different in that the first is a "simple contradiction" and the latter is not. However, this is not true. Both are nearly identical in so far as they have only one direction in which they can be interpreted. In the Barber paradox, one gains the knowledge that the original assumption was wrong -- translated into set theory, one may not construct the set of all self-containing sets (or the Universal set), and the initial statement at the beginning (that we had such a set) was false. Similarly, one may readily determine the truth of Goedel's string, because its disproof is contradicted. If Goedel's string *is* false, then the formal system of mathematics in which it is demonstrated false itself is inconsistent. (We know, of course, from Goedel, that we may not prove any consistent mathematics to be consistent, but that is very different from proving a mathematics inconsistent. One leaves uncertainty, the other leaves no doubt!) > Peter Merel (#10221) points out that questioning Goedel and other > eminences is presumptuous. Preposterous is more like it. > The full technical "reasoned argument" that he reasonably requests > is not yet in satisfactory form--it's very difficult to make it > clear enough If it isn't in a satisfactory form, what makes you think that it is formally correct? > Goedel himself said his theorem is analogous to the Liar (Epimenides > etc.) "paradox," Analogous, yes. > and the latter, while still in dispute, was deflated not only by > little old me, but by many others long ago, including Aristotle. The > sentence "This statement is false." is not a proposition, because it > is essentially meaningless--there is no root referent. You obviously have no understanding of formal systems, Bob. Mathematics performed in English is a vague approximation of the "true" mathematics, which is conducted entirely in formal systems. No one actually does math that way because it is too tedious, but in principle you have to be able to reduce all mathematical reasoning into formal systems in order to show that it is correct. Formal systems are rules for the mechanical transformations of initial strings (the "axioms") into new strings, the so-called "theorems" of the formal system. Goedel did his work entirely on formal systems. He showed that, provided the formal system was sufficiently "powerful" (that is, contained rules of inference known to be needed for doing any real mathematics) you could always construct a self referential string in such a formal system and get it to speak about itself. In such a system, there is no wiggle room for the sort of logical game you are trying to play. The rules of inference are mechanical. The game is not one of playing with English words, but playing with strings via mechanical methods. Interpreting strings of a formal system into English is something that one does in the same way that Muslims claim (speciously, possibly) that translating the Koran is accomplished -- at best, the translation is an approximation of the original, and is not the definitive text. The reasoning must be done with mechanical transformations on the strings, not with sophistry in the English. By the way, Goedel understood the problem of getting the self reference to work very clearly. A good deal of the complexity of the proof comes from the trick he uses in order to get the formal string to point back at itself. Hoffstadter playfully characterizes the process as "Quining", after the great logician Quine. Goedel's two great syntheses were, of course, in demonstrating that you could turn all formal systems into numbers so that any formal system in which you would care to do math could be forced to do mechanical reasoning about itself, and then to demonstrate that you could force a string into self reference. The Liar paradox and Barber paradoxes were just an inspiration to Goedel. His string, which may be constructed in any sufficiently powerful formal system, exists. It is not a theorem of any formal system (assuming the system is consistent) but it is true. You can pretend that this string doesn't exist, but that doesn't make it go away. You can try to reason it away with tricks in English, but as I noted, English isn't the language in which this game is played. (Even Ancient Greek won't do.) Claiming you don't "believe" it makes no difference. If you accept formal systems as the underlying basis for logic and mathematical reasoning, then you have to accept the existence of Goedel's string, and the implications thereof. > I know, there is no agreement on what constitutes a "proposition," > but surely anyone can see at least the possibility that Aristotle > and I are right, No, I'm afraid I can't. > which opens up at least the possibility that Goedel's conclusion was wrong. You don't even understand Goedel, from what I can tell. Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=10233