Obections to Cantor's Theory (Wikipedia article)
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From: Jean-Claude Arbaut on 20 Jul 2005 08:46 Le 20/07/05 14:31, dans 1121862680.823992.164760(a)f14g2000cwb.googlegroups.com, ýýgeorgieýý <geo_cant(a)yahoo.com> a ýcritý: > > Stephen J. Herschkorn wrote: > >> Do anti-Cantorians accept that sum(i=1..infty, d_i / 10^i) "exists" >> for each collection (d_i) of decimal digits (as i ranges over the >> positive integers)? If so, how do they correctly justify the collection >> of these real number are countable? > > What do you mean i=1..infty? I understand i=1, i=2, i=3, and > so forth, but i=infty makes no sense. It looks to me like you > are trying to "slip in" infty as a natural number. > > I have seen mathematicians along with smart people use such > symbology when they mean to imply the sum of all terms as i > takes on the values of all the natural numbers. I assume thats > what you mean as well. (In this thread I think we should try to > avoid ambiguities.) It's a universal notation for the sum of a series, don't *create* ambiguities ! > Ususally the limit of the sum is then > discussed. That limit exists. > I don't think many people would > disagree. I don't see how limits of sums has anything to do > with justifying the countability of "these real number." He He... So little. Cantor's diagonal, for example ?
From: David C. Ullrich on 20 Jul 2005 09:05 On 19 Jul 2005 11:31:07 -0700, "Kevin Delaney" <kevind(a)y-intercept.com> wrote: >The classical method (Scholastic) method of education leaned primarily >on teaching grammar, logic, arithemetic and rhetoric. > >The modern era felt that these were all artificial edifaces. There was >a concerted effort to pull these subjects out of the schools. They were >there. They were not there when I went to school. > >You can verify this if you look at the standard school curriculum in >say 1900 and compare it to the curriculum in, say, the 1970s. I used to >do stupid things like look up the different books that were taught at >different times to try and see how different eras would see the world. > >Transfinite theory is not the only manifestation of modern thinking. >So, I am not saying that transfinite theory alone was the cause for >this transformation. No, you're saying it was the _primary_ cause: "Transfinite theory was the primary reason for removing the study of logic, grammar and arithematic in American public education and replacing it with new math." >The basic idea was that traditional logic, grammar >and arithematic were part of this horrible weight keeping people down, >and that we would transcend to a higher level of existence. > >Citing all the places where people attacked classical education would >take several years. Doubtless. So why not cite _one_ place where _one_ person gave "transfinite theory" as a justification for eliminating grammar from the curriculum? Hint: Justifying a ridiculous claim by saying that it would take too long to justify it is a little thin. Just give one example. >It is pretty much a fact that in the modern times >there was one curriculum replaced by another curriculum. Has anyone disputed that? Does it follow from that fact that transfinite theory was the primary reason for removing the study of logic, grammar and arithematic in American public education and replacing it with new math? >In some cases >it was a postive thing, curriculums always need improvement and >adjustment. The problem with a wholesale replacement is that you end up >losing the promising threads of the previous curriculum. ************************ David C. Ullrich
From: Michael Stemper on 20 Jul 2005 08:55 In article <42DD6183.5030705(a)netscape.net>, Stephen J. Herschkorn writes: >Should a reputable encyclopedia contain an entry devoted entirely to >people who think the earth is flat? The Flat Earth Society? Sure, why not? Of course, it should be listed under "sociological anomalies" rather than under "geography". I'm not sure of the relationship between "Wikipedia" and "reputable encyclopedia", however. -- Michael F. Stemper #include <Standard_Disclaimer> If it's "tourist season", where do I get my license?
From: Tony Orlow on 20 Jul 2005 09:08 Chris Menzel said: > On Tue, 19 Jul 2005 16:00:53 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > Chris Menzel said: > >> On Tue, 19 Jul 2005 14:50:33 -0400, Tony Orlow <aeo6(a)cornell.edu> > >> said: > >> > ... > >> > Is the above your 7-line proof? it makes no sense. There is no > >> > reason to expect the natural number corresponding to the subset to > >> > be a member of that subset. if this rests on the diagonal proof, > >> > there is a very clear flaw in that proof which you folks simply > >> > dismiss as irrelvant, but which is fatal. > >> > >> There is a simple, demonstrably valid proof of Cantor's Theorem in ZF > >> set theory. So you must think the proof is unsound. Which axiom of > >> ZF do you believe to be false? > > > > I was asked that before, and never got around to fully analyzing the > > axioms for lack of time, > > I suggest you do so, though your reluctance is understandable, as the > realization that all of the axioms used in the proof of the theorem are > intuitively true would of itself constitute a proof that your arguments > against diagonalization are flawed. Look at the positive side though -- > this may well be the cure for what ails you. > > > but the diagonal proof suffers from the fatal flaw of assuming that > > the diaginal traversal actually covers all the numbers in the list. > > Any complete list of digital numbers of a given length, even a given > > infinite length, is exponentially longer in members than wide in terms > > of the digits in each member. Therefore, the diagonal traversal only > > shows that the anti- diagonal does not exist in the first aleph_0 > > terms. Of course, the entire list is presumed to be aleph_1 long, > > being a list of the reals, > > To say that a list of the reals is aleph_1 in length assumes the > continuum hypothesis; is that what you intend? (And if so, why?) > Moreover, granted, aleph_1 is omega_1 in pure set theory, but one should > use ordinal numbers rather than cardinals when talking about such things > as the length of a list, as there are many lists of different ordinal > lengths that are the same size. I don't think it assumes any such thing. In fact, it is my position that there is an infinite spectrum of infinities between that of the naturals and that of the reals. If set theory considers the set of reals to have cardinality aleph_ 1, which is strictly larger than the number of naturals, aleph_0, and you have a list of reals each with a number of digits equal to the number of naturals, then you do not have a diagonally traversible square, but a rectangular grid of digits. > > Pretty clearly, you aren't terribly well-educated in set theory. Don't > you think you should understand a field before you try to point out its > flaws? And don't you see that it makes you look rather silly? Would > you consider attacking superstring theory without understanding basic > quantum mechanics? I have not immersed myself deeply in set theory, no. But, I can see clearly that some of the conclusions are wrong, and in arguing this, I have stumbled upon a few of the obvious flaws in the logic. Hopefully I will find time soon to decipher the specific axioms of ZF and see whether the roots of the problem lie there, or with subsequent assumptions. Now, if I came up with a theory that "proved" that 1=2, would you believe it, at first glance? Should I accept a theory that "proves" that we can take one ball, cut it into a finite jumber of pieces, and reassemble it into two solid balls, each the same size as the original? The Banach-Tarski "paradox" was offered as a disproof by contradiction of the axiom of choice, although I am not sure that axiom is directly at fault for what is obivously an incorrect result. Somehow, instead of being considered a proof of an inconsistency in set theory, this result has been accepted as a counter-intuitive "fact", which I simply cannot accept as having anything to do with reality. Yes, I am one of those that believes that real math reflects some actual reality, whether we have iscovered the application yet or not. > > > and the antidiagonal simply exists on the list, below the line of > > diagonal traversal. > > Aside from your tenuous grasp of basic transfinite arithmetic, the > critical terms in your argument -- "digital number", "length", > "exponentially longer" (cardinal or ordinal exponentiation?), "width", > "diagonal traversal", "below the line of diagonal traversal" -- are much > too vague for your argument to be evaluated. For all we know, you might > have some genuine insights. But currently, your argument is smoke and > mirrors; it hasn't been expressed as mathematics. I think you know exactly what i mean by each of those terms. They are all widely understood. It is typical for Cantorians to resort to claims of vagueness on the part of their opponents, while presenting such vague proofs as the diagonal argument, without defining anything themselves, and assuming unfounded postulates such as all infinities are the same, except when they're not. > > > Cantorians seem to think infinity is simply infinity, > > Hard to fault them on that score. The fact that you have doubts about > whether infinity is infinity (what does "simply" add here?) says quite a > lot -- though *perhaps* if you were to try to distinguish different > senses of "infinite" with any sort of mathematical precision there might > be a point there. That was a dishonest snip, really. Notice the comma? That means the sentence was not finished. The rest of it said that you do this, even in the midst of proving that it is not the case. The diagonal argument assumes that the list is essentially square, and can be traversed diagonally, but then proves that this is not the case, and yet, Cantorians still seem to insist that the diagonalization proves the reals cannot be listed, when what it proves is that there are more reals than naturals. Now, I did put forth, in another thread, some pretty precise ways of comparing infinities and ordering them, but i am not going into that now. I'll finish getting my web pages together, with graphics. > > Hit the books, Junior; you've got some work to do. Yes, I have a lot to do, Junior. > > Chris Menzel > > -- Smiles, Tony
From: David Kastrup on 20 Jul 2005 09:19
Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Chris Menzel said: > >> Pretty clearly, you aren't terribly well-educated in set theory. >> Don't you think you should understand a field before you try to >> point out its flaws? And don't you see that it makes you look >> rather silly? Would you consider attacking superstring theory >> without understanding basic quantum mechanics? > > I have not immersed myself deeply in set theory, no. But, I can see > clearly that some of the conclusions are wrong, and in arguing this, > I have stumbled upon a few of the obvious flaws in the > logic. Look, when _not_ immersing yourself into set theory, then you just get the superficial picture. What will strike you then will be the _unintuitive_ results, those that go against what you consider common sense. Now it is your contention that in hundreds of years, no mathematician has bothered double-checking the _unintuitive_ things and reconciling them with the rest of the system. Such that it needs somebody without an actual idea of set theory to get this corrected. Can't you see how _bloody_ unlikely this is? If anything, it is the _unintuitive_ stuff that gets the closest scrutiny by mathematicians. Oversights are _much_ more likely to occur in areas that are "obvious". > Hopefully I will find time soon to decipher the specific axioms of > ZF and see whether the roots of the problem lie there, or with > subsequent assumptions. If there is a problem there, do you really think a rank amateur has a chance to let something resurface that mathematicians have worked on letting it disappear for centuries? If that whole web of axioms and theorems contains wrinkles somewhere, they'll not be apparent in the middle of the carpet, but somewhere behind a strange cabinet in an inaccessible corner. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |