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Reply-To: cg@cs.umn.edu Date: Sun, 2 May 93 11:01:59 EDT From: sowa <sowa@turing.pacss.binghamton.edu> Message-id: <9305021501.AA01600@turing.pacss.binghamton.edu> To: cg@cs.umn.edu, interlingua@isi.edu, phayes@cs.uiuc.edu Subject: 20th Century Mathematics Cc: sowa@turing.pacss.binghamton.edu

Pat, I reread your note that had the 80+ character lines clipped, and most of the topics seem to have been addressed in our other notes. There is one issue that seems to keep coming up that I would like to clarify: You keep saying that I "reject all of 20th century mathematics." That is not true. There is only one aspect about which I am extremely skeptical, and that is Georg Cantor's so-called "proof" of the existence of uncountable sets. This so-called proof hangs on one slender thread, namely Cantor's diagonal construction. A lot of other people have also been skeptical about that "proof", including Alfred North Whitehead, who was by no means an intuitionist. He made the point that when you have a proof by contradiction, the most you can conclude is that somewhere among all your assumptions, there does indeed exist a contradiction. But the proof has no way of telling you which of those assumptions happened to be wrong. Whitehead was willing to use a proof by contradiction, but only in cases where the entire subject matter was well established, the consistency of the axioms was either proved (or as in the case of much of mathematics so well used and applied that its consistency was never in doubt), and there was exactly one assumption whose truth was doubtful. That is essentially my position. I am not an intuitionist, and I am willing to accept classical first-order logic, including proofs by contradiction -- but with all the caveats that Whitehead noted. I am also willing to accept all the techniques and results of 20th century mathematics, except for one: anything that critically depends for its existence on Cantor's diagonal "proof". To see why I don't accept that "proof", let me give a brief summary of the underlying assumptions: 1. If two infinite sets cannot be put in a one-to-one correspondence, one of them must be larger than the other. 2. If a set could be placed in a one-to-one correspondence with the integers, then it should be possible to list all pairs in that correspondence in a long table. 3. If such a table could be constructed for pairing the integers with the reals, then it should be possible to go along the diagonal of the expansion of the real numbers in some representation, say as unending decimal fractions, to select the first digit from the first one, the second digit from the second one, etc. 4. Then the sequence of digits so selected would correspond to the representation of some real number x, which may or may not be on the list. 5. But it would be possible to generate another real number x' by changing each digit of x to some other digit. That new number x' could not be on the list assumed by assumption #2, since for each integer n, x' would differ from the real number paired with n because its n-th digit would be different. 6. Therefore, x' would be a real number not on the list assumed by #2. I will certainly admit that this is a very clever construction, and I will also admit that somewhere among these six assumptions there lies a contradiction. But which assumption or assumptions should be changed? The first assumption sounds rather plausible, but there are so many counterintuitive cases in the whole field, that nothing should really be considered "intuitively obvious". The fact that you can find a one-to-one correspondence between the integers and the rationals is very odd. Even odder is the possibility of a one-to-one correspondence between the points on a line and the points on a plane. In measure theory (a 20th century development that I am willing to accept), a line on a plane has measure zero, yet Cantor claimed that they both have the same number of points. All these results are so counterintuitive that even assumption #1, as "obvious" as it sounds, is not above suspicion. Then the assumptions #2, #3, #4, and #5 are the same basic constructions that Turing used to show the existence of noncomputable functions. I am quite happy to believe Turing's result that there are many functions that cannot be computed -- even though they are countable (i.e. subsets of the pairs of integers). But this result makes me even more suspicious of Cantor's claim. I would be willing to believe that the real number x' in assumption #4 above or the list of pairs of integers & reals in assumption #3 are examples of noncomputable things. But there is a world of difference between Turing's noncomputable (but countable) and Cantor's claim of uncountable. One further point that casts further doubt on Cantor's claim is the Loewenheim-Skolem theorem, which says that for any finitely axiomatizable theory, there must exist a countable model. This says that even for Cantor's axioms, from which he supposedly "proved" the existence of uncountable sets, there must exist a countable model. This is one further point that makes me extremely suspicious of any claim that the diagonal proof shows the existence of uncountable sets. If Cantor had simply said that the whole field of infinite sets is rife with noncomputable functions and nonconstructive definitions, then I would certainly agree. What I don't accept is the conclusion that THEREFORE there must exist uncountable sets. As far as the Knowledge Sharing Effort or the ANSI standards, I am perfectly willing to let people use either KIF or CGs to state any kind of nonsense they please, including axioms for "uncountable" sets. But I don't want to let anything in the foundations of either KIF or CGs depend on any assumptions of their existence. So to allay your fears that I will prevent you from talking about uncountables, I assure you that you can translate all your dubious axioms into CGs or KIF and share them with anyone else who has your taste for such nonsense. John