From dana.scott at cs dot cmu dot edu Sun Dec 18 10:47:52 2005 Greetings Doron: Concerning you recent Opinion 68, I have to say emphatically that you are overreacting again! Clearly the new results of Fox are very dramatic; however, a quick internet search turned up slides of a recent talk by Daniel J. Kleitman and Jacob Fox where it is pointed out that Paul Erdős and Shizuo Kakutani already proved in 1943 that the NEGATION of the Continuum Hypothesis is equivalent to the equation x1 + x2 - x3 - x4 = 0 being countably-regular (i.e., with respect to countable partitions of the reals). This is a result of the same type as those of Fox (if not as refined and not as dramatic -- as far as the linear equation is concerned), so if you are to be shocked, shocked by such things, you should have been in a state of shock for the last 60 years! Remember the old results of Hausdorff-Banach-Tarski, which in particular claim that a solid sphere can be "broken" into finitely many subsets and "reassembled" as two spheres of the same size as the original ball. This method can only be done with a well ordering of the real numbers, and a well ordering as a subset of the plane is always a Lebesgue non-measurable set. In fact, Sierpinski showed many years ago that a non-principal ultrafilter in the Boolean algebra of all sets of integers also implies the existence of a non-measurable set. The lesson here is that arbitrary sets of reals can be monstrously horrible. So? We have known this for a very long time. And don't forget that people once hated the idea that there could be a continuous function nowhere differentiable. But they learned to live with these things and pay more attention to the non- pathological functions, sets, and spaces. The area where you -- as a combinatorial personality -- should really be worried concerns the recent work of Harvey Friedman. He finds propositions of FINITE combinatorics which are "true" but can only be proved with the aid of very large cardinals. He is speaking of cardinal numbers so large they make the aleph_n cardinals look like baby toys. You can say these "theorems" are not "interesting", but I personally think you would be wrong. Ask Harvey for a quick lesson about his results. And best wishes for the New Year! Dana S. Scott

From dana.scott at cs dot cmu dot edu Mon Dec 19 15:31:36 2005 Hey Doron! Thanks for the reply. We all could use some peaceful sleep these days, but I really wish you would take a peek at Harvey's brief paper on his new incompleteness results. (It is a six-page abstract without proofs but with much attention given to putting the statement into context.) He says: > 'Beautiful' is a word used by mathematicians with a semi > rigorous meaning. We give an 'arguably beautiful' explicitly > Pi01-sentence independent of ZFC. See Proposition A from section 1. The PDF-file can be downloaded from: http://www.math.ohio-state.edu/%7Efriedman/pdf/Pi01120905.pdf He makes a clear distinctions between things that are provable and analogous statements which are independent. And I don't feel that excluding potential infinity is the right move to happiness here. As regards Fox, his statements are very, very nice (and he ties up many loose ends), but I don't agree that he is "more concrete" than the old Erdős-Kakutani result. (BTW, here is the reference: "On non-denumerable graphs", Bull. AMS, vol. 49 (1943), PP. 457-489. ) But we don't have to argue about this point considering all that Fox has done. I did some more searching and found a copy of his excellent paper (at a non-MIT site!) as follows: http://www.math.ucsd.edu/~sbutler/seminar/FoxRado1.pdf I notice he credits you (and others) "for helpful advice and comments concerning this subject matter" -- though he does not admit to doing the Devil's work! As regards measurability, I find it especially interesting that Fox and a collaborator have shown in ZF set theory that the equation x1 + 2 x2 + 4 x3 = x4 is NOT 4-regular if you assume the Axiom of Choice, but IT IS if you assume Dependent Choice and Every Set of Reals Lebesgue Measurable. (The latter assumptions were proved consistent with ZF -- assuming an inaccessible cardinal -- by Solovay back in 1970.) There is no question that measurable sets are better than non-measurable ones! (And Fox indicates other reasons to think so.) Indeed, the well known proof from Choice for finding a non-measurable set by picking one representative out of each coset of the quotient (R/Z)/(Q/Z) (where R/Z is the additive group of reals mod 1, and Q/Z is the subgroup of rationals mod 1), shows that choosing arbitrarily from a suite of uncountably many sets is highly non-constructive. So maybe the right advice here is simply: AVOID BAD SETS! All the best, -- Dana Scott P.S. And, yes, you can post my comments. I would like to hear other comments for other people as well.

From alexzen at com2com dot ru Wed Dec 21 10:45:37 2005 Sent: Tuesday, December 20, 2005 10:00 PM To: Doron Zeilberger (zeilberg at math dot rutgers dot edu) Subject: As to 'Opinion 68' Dear Doron, Here are some 'people' who are your like-mindeds as to Actual Infinity THE LIST-1 of scientists who rejected Actual Infinity. - Aristotle, Euclid, Leibniz, Berkeley, Locke, Descartes, Kant, Spinoza ( d'Espinosa ), Lagrange, Gauss, Lobachevsky, Cauchy, Kronecker, Hermite, Felix Klein, Poincare, Bair, Borel, Lebesgue, Brouwer, Quine, Percy W.Bridgman (Nobel-1946), Wittgenstein, Weyl, Luzin, Errett Bishop, Solomon Feferman, Jaroslav Peregrin, Valentin Fedorovich Turchin, P.Vopenka and a lot of other outstanding CREATORS of the CLASSICAL (i.e., working really) LOGIC and the CLASSICAL (i.e., working really) MATHEMATICS AS A WHOLE !!! (see http://alexzen.by.ru/ ) And, of course, Alexander Zenkin. It should be accentuated here that beginning from Kronecker, i.e., from about 70s of XIX century, the protest against a use of the actual infinity in mathematics as a self-contradictory notion took the form of the distinctly negative attitude to the set theory by Georg Cantor that is based on the algorithmic usage of the actual infinity conception. Some fatal nonsenses of 'naive' Cantor's as well as 'non-naive' axiomatic set theory are considered in the recent papers. > A.A.Zenkin, Logic of Actual Infinity and G.Cantor's Diagonal > Proof of the Uncountability of the Continuum. - The Review of Modern > Logic, Vol. 9, Number 3&4, 27-82 (2004). > > A.A.Zenkin, Scientific Intuition of Genii Against Mytho-'Logic' > of Cantor's Transfinite 'Paradise'. - Philosophia Scientia, 9 (2), 145 > - 163 (2005). http://alexzen.by.ru/papers/2005/Zenkin-PhilSc-9-2-2005.pdf If one rejects actual infinity, then there is not "a continuous function nowhere differentiable". The actual infinity is a necessary condition of Cantor's diagonal proof of the power-set theorem. If one rejects actual infinity, then the theorem becomes unprovable and therefore there are not any cardinals greater than aleph_0. Best wishes for the New Year!

From alexzen at com2com dot ru Jan. 8, 2006 Dear Doron, In your Opinion 68 you wrote: "My mind was made up about a month ago, during a wonderful talk (in the INTEGERS 2005 conference in honor of Ron Graham's 70th birthday) by MIT (undergrad!) Jacob Fox (whom I am sure you would have a chance to hear about in years to come), that meta-proved that the answer to an extremely concrete question about coloring the points in the plane, has two completely different answers (I think it was 3 and 4) depending on the axiom system for Set Theory one uses. What is the right answer?, 3 or 4? Neither, of course! The question was meaningless to begin with, since it talked about the infinite plane, and infinite is just as fictional (in fact, much more so) than white unicorns. . ." Here is another example of the like stupidity matured in Cantor's 'paradise': a so-called cardinality of the continuum depends on an indexing of its elements within the framework of Cantor's 'diagonal' proof of his fundamental theorem on the uncountability of the continuum. DESIGNATIONS: X=[0,1], N={1,2,3,.}, |Z| is a cardinality of a set Z for any Z. CDM = Cantor's Diagonal Method, AST = Axiomatic Set Theory. RAA = Reductio ad ansurdum. Consider the traditional proof of Cantor's theorem. CANTOR'S THEOREM (1890). "The set X is uncountable", i.e., |X|>|N|. PROOF (by RAA-method). Assume that "the set X is countable", i.e., |X|=|N|. Then there exists a 1-1-correspondence between elements of X and N. Let x1 , x2 , x3 , . . . (1) be an enumeration (a list) of ALL real numbers from X. The CDM-application to the list (1) generates a new real, say, y, which, by its construction, differs from every real in the list (1). Consequently, the list (1) contains NOT ALL reals from X. Contradiction. Q.E.D. Now we shall prove a statement which clarifies one unknown logical peculiarity of the Cantor diagonal argument. THEOREM 3. The cardinality of the set X depends upon an indexing of its elements in the list (1). PROOF. Consider again traditional Cantor's proof. CANTOR'S THEOREM. "The set X is uncountable", i.e., |X|>|N|. PROOF. Assume that "X is countable", i.e., |X|=|N|. Then there is a list x1 , x2 , x3 , . . . (1) of ALL real numbers from X. However, from the standpoint of modern AST, the condition "X is countable" means that X is equivalent to any countable set, e.g., to the set Nev = {2,4,6,.} of all even natural numbers. It means that, by virtue of the transitivity of the equivalency relation between countable sets X, N and Nev, all elements of the initial Cantor's list (1) can be re-indexed with the elements of the set, Nev : x2, x4, x6, ... . (1.1) Since the list (1.1) saves the number and the order of reals in the initial list (1), the CDM-application to the list (1.1) generates the same new Cantor's real y which doesn't belong to the list (1.1) like the CDM-application to the initial list (1). But now we have a possibility to index any (even infinite) number of new Cantor's reals, generated by CDM-application to the initial list (1) (or to the list (1.1) what is the same) and not belonging to the list (1), with elements of the free set Nod = {1,3,5, ...}. It means that a number of elements in N is not less than a number of reals in X, i.e., |N| >= |X|. The last means that the RAA-assumption "X is countable" (|X|=|N|) becomes irrefutable, and consequently the statement "The set X is uncountable" (|X|>|N|) becomes unprovable. Thus, if the reals in the list (1) of Cantor's proof are indexed with ALL natural numbers 1,2,3, . . ., then, according to the traditional Cantor's proof, |X| > |N|. But if the same reals are indexed with NOT ALL natural numbers (e.g., with 2,4,6,. . .), then |N| >= |X|. It means that within the framework of Cantor's proof the cardinality of the set X (continuum) essentially depends upon an indexing of its elements in the list (1). Q.E.D. It's obvious that the dependence of the cardinality of the infinite set X upon an indexing of its elements is an absurdity from the point of view of Cantor's ('naive') set theory as well as from the point of view of modern ('non-naive') axiomatic set theory. REMARK 1. From the Theorem 3 it follows, in particular, that the infinite sets, X and N, may be equivalent, i.e., |X| = |N|, but there is not a rule (algorithm) to establish a 1-1-correspondence between elements of these sets. Here is a direct proof of such a non-trivial possibility. THEOREM 4. If X is equivalent to N, then there is not a rule (algorithm) to produce a 1-1-correspondence between elements of X and N. PROOF. Assume that X is equivalent to N, but there is a rule (algorithm) producing a 1-1-correspondence between elements of X and N, i.e., there is a list (1) containing ALL reals from X. Applying Cantor's diagonal method (algorithm) to the list (1), we shall construct a NEW real, say y, which doesn't belong to (1). Contradiction. Q.E.D. Applying the law of contraposition to the Theorem 4, we get the following quite strange statement. COROLLARY 1. If there is a 1-1-correspondence between elements of the sets, X and N, then these sets are non-equivalent. It's obvious that the Corollary 1 is an ordinary stupidity of Cantor's 'paradise' since that contradicts flatly to the definition itself of the paradigmatic notion of the non-equivalency of infinite sets of Cantor's set theory and modern AST. MY MAIN CONCLUSION. - Cantor's 'paradise' as well as all modern axiomatic set theory is based on the (self-contradictory) concept of actual infinity. Cantor emphasized plainly and constantly that all transfinite objects of his set theory are based on the actual infinity. Modern AST-people try to persuade us to believe that the AST does not use actual infinity. It is an intentional and blatant lie, since if infinite sets, X and N, are potential, then the uncountability of the continuum becomes unprovable, but without the notorious uncountablity of continuum the modern AST as a whole transforms into a long twaddle about nothing and really is "a pathological incident in history of mathematics from which future generations will be horrified" (Luitzen Egbertus Jan Brouwer). Best regards, Alexander Zenkin

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