Surely You're Joking, Mr. Zeilberger?
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From: Aatu Koskensilta on 16 Feb 2010 05:37 Tonico <Tonicopm(a)yahoo.com> writes: > "Flaws"? Within ZFC? Name one, just for the fun of it...;) The main flaw is of course that the diagonal argument doesn't prove anything. > And it'd be interesting to get that paper(s) by Zenkin, if you have > some. From the abstract of Zenkin's _LOGIC OF ACTUAL INFINITY AND G. CANTOR'S DIAGONAL PROOF OF THE UNCOUNTABILITY OF THE CONTINUUM_ a little something to whet your appetite: Since Cantor first constructed his set theory, two independent approaches to infinity in mathematics have persisted: the Aristotle approach, based on the axiom that "all infinite sets are potential," and Cantor's approach, based on the axiom that "all infinite sets are actual." A detailed analysis of the "rule-governed" usage of 'actual infinity' reveals that Cantor's diagonal proof is based on two hidden, but nonetheless necessary conditions never explicitly mentioned but in fact algorithmically used both in Cantor's so called "naive" set theory as well as modern "nonnaive" axiomatic set the- ories. An examination of "rule-governed" usage of the first necessary condition opens the way for a rigorous proof that in reality Cantor's diagonal procedure proves nothing, and merely reduces one problem, that associated with the un- countability of real numbers (the continuum), to three new and additional problems. The second necessary condition is simply a teleological one possessing no real relation to mathematics. Further analysis reveals that Cantor's Diagonal Method (CDM), being the only procedure for distinguishing infinite sets on the basis of their cardinalities, does not distinguish infinite from finite sets just on the basis of the number of their elements (cardinality); the results of CDM depend fatally upon the order of real numbers in the sequences to which it is applied. Cantor's diagonal proof itself is formally а "half" of the well-known "Liar" paradox but which can be used to produce a new set-theoretical paradox of the "Liar" type. - - - The fact to be demonstrated is that ultimately Cantor's diagonal proof engages us in an endless, potentially infinite, and quite senseless paradoxical "game of two honest tricksters" (a new set-theoretical paradox) which, as Wittgenstein alleged, "has no relation to what is called a deduction in logic and mathematics." - - - Here it is argued that Cantor's proof does not in fact prove the uncountability of the continuum, but rather proves something else entirely, viz. Aristotle's Thesis (stated in its later canonized Latin form): "Infinitum Actu Non Datur." In other words, it proves that an actual infinity "is never permitted in mathematics" (Gauß), or alternatively speaking, that in the words of Poincaré "there is no actual infinity; Cantorians forgot that and fell into contradictions. [... ] Later generations will regard set theory as a disease from which one has recovered!" The fabricated quote from Poincaré is a particularly nice touch (although not an original contribution of Zenkin's). -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, darüber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: spudnik on 16 Feb 2010 23:26 why is it that implied infinities are so bad, if zeta(2) requires infinite terms to be 6/pi^2? thus: are the five eggs arrayed like a regular pentagon? http://sites.google.com/site/tommy1729/home/eggs-problem --Another Flower for Einstein: http://www.21stcenturysciencetech.com/articles/spring01/Electrodynamiics.html --les OEuvres! http://wlym.com --Stop Cheeny, Ricw & the ICC in Sudan; no more Anglo-american quagmires! http://larouchepub.com/pr/2010/100204rice
From: Tonico on 16 Feb 2010 07:09 On Feb 16, 12:37 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Tonico <Tonic...(a)yahoo.com> writes: > > "Flaws"? Within ZFC? Name one, just for the fun of it...;) > > The main flaw is of course that the diagonal argument doesn't prove > anything. > That, of course, is your personal opinion. The diagonal argument as used in Cantor Theorem's proof proves that the set of real numbers in the interval [0,1] cannot be equinumerous to the natural numbers. > > And it'd be interesting to get that paper(s) by Zenkin, if you have > > some. > > From the abstract of Zenkin's > >   _LOGIC OF ACTUAL INFINITY AND  G. CANTOR'S DIAGONAL PROOF OF THE >           UNCOUNTABILITY OF THE CONTINUUM_ > > a little something to whet your appetite: > >   Since Cantor first constructed his set theory, two independent >   approaches to infinity in mathematics have persisted: the Aristotle >   approach, based on the axiom that "all infinite sets are potential," >   and Cantor's approach, based on the axiom that "all infinite sets >   are actual." A detailed analysis of the "rule-governed" usage of >   'actual infinity' reveals that Cantor's diagonal proof is based on >   two hidden, but nonetheless necessary conditions never explicitly >   mentioned but in fact algorithmically used both in Cantor's so >   called "naive" set theory as well as modern "nonnaive" axiomatic set >   the- ories. An examination of "rule-governed" usage of the first >   necessary condition opens the way for a rigorous proof that in >   reality Cantor's diagonal procedure proves nothing, and merely >   reduces one problem, that associated with the un- countability of >   real numbers (the continuum), to three new and additional >   problems. The second necessary condition is simply a teleological >   one possessing no real relation to mathematics. > >   Further analysis reveals that Cantor's Diagonal Method (CDM), being >   the only procedure for distinguishing infinite sets on the basis of >   their cardinalities, does not distinguish infinite from finite sets >   just on the basis of the number of their elements (cardinality); the >   results of CDM depend fatally upon the order of real numbers in the >   sequences to which it is applied. Cantor's diagonal proof itself is >   formally а "half" of the well-known "Liar" paradox but which can be >   used to produce a new set-theoretical paradox of the "Liar" type. > >   - - - > >   The fact to be demonstrated is that ultimately Cantor's diagonal >   proof engages us in an endless, potentially infinite, and quite >   senseless paradoxical "game of two honest tricksters" (a new >   set-theoretical paradox) which, as Wittgenstein alleged, "has no >   relation to what is called a deduction in logic and mathematics." > >   - - - > >   Here it is argued that Cantor's proof does not in fact prove the >   uncountability of the continuum, but rather proves something else >   entirely, viz. Aristotle's Thesis (stated in its later canonized >   Latin form): "Infinitum Actu Non Datur." In other words, it proves >   that an actual infinity "is never permitted in mathematics" (GauÃ), >   or alternatively speaking, that in the words of Poincaré "there is >   no actual infinity; Cantorians forgot that and fell into >   contradictions. [... ] Later generations will regard set theory as a >   disease from which one has recovered!" > > The fabricated quote from Poincaré is a particularly nice touch > (although not an original contribution of Zenkin's). > Nice. I don't know who "the cantorians" are, specially nowadays, but the distinction between "potential and actual" mathematics as stressed above, and in much lamer and nonsensical fashion by many cranks around the house, isn't that clear to me: what do Zenkin, or you, or others, believe that people that has no problem with the diagonal argument of Cantor believe? I, for one, have no problem at all with that argument, and it looks to me a rather simple and elegant way to prove that [0,1] isn't countable. Fine, so what does that say about me with regards to "potential and/or actual" infinity? It'd be interesting as well to know to what "contradictions" does Poincare refered to above... Tonio > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" >  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Angus Rodgers on 16 Feb 2010 11:44 On Tue, 16 Feb 2010 07:38:54 -0500, David Bernier <david250(a)videotron.ca> wrote: >In the standard real analysis of today, the Intermediate Value Theorem and >the Mean Value Theorem are used a lot. For example, if f(x) is >a polynomial function of odd degree and real coefficients, >f: R -> R, then the IVT implies that f has at least one real root. >And the IVT rests upon the completeness of the reals. > >The proof at Wikipedia uses the least upper bound property for reals: >< http://en.wikipedia.org/wiki/Intermediate_value_theorem > > >Zenkin rejects cardinals above aleph_0, and also Cantor's Diagonal proof. > >I'm not sure how those who reject the orthodox treatment of real numbers >would want to prove that f in IR[x] of odd degree has at least one real root ... >Go back to geometrical intuition? Construct root-finding algorithms? I already seem to detect signs of the dreaded convergence towards a discussion of the usual suspects and their heresies. Of course I have no power to stop that, just because I started the thread, but, unless either there is something new being said about other mathematical heretics, or jokers (I haven't been around for a long time, and I never followed all such threads - only occasionally the JSH ones), or such a widened discussion sheds light on Zeilberger's serious or humorous beliefs about the natural number system (by the way, he seems to reject the entirety of mathematical analysis, root and branch, as meaningless!), I would beg for the discussion not to go too far in that direction. On a more positive note, what I would like is to be helped to see (a) how it is possible for anyone to think like Zeilberger at all, and (b) how thinking like Zeilberger is nevertheless impossible for me in particular, because it is incompatible with some principles I hold to and he doesn't. If I can understand how even a professional mathematician can believe something so fundamental, which I believe to be false, I might better understand why I believe it to be false. It might be a bit like coming to an understanding of the principle of conservation of energy, by seeing in detail a failed attempt to build a perpetual motion machine. That analogy might not work very well - because I can't imagine coming to a better understanding of field theory by following in detail some foolish person's attempt to duplicate the cube! - but I think that is because it omits one feature of this case, which is that what Zeilberger denies seems so 'obviously' true (unlike energy conservation, or the impossibility of duplicating the cube by Euclidean means), that being able to understand DZ's wacky point of view ought to help me to see beyond the 'obvious', which is always a good thing - so long as you don't completely lose sight of what is, in the end, still 'obvious'! At least, that's why I think I'm interested, but any account of reasons for being interested is less interesting than the thing itself. -- Angus Rodgers
From: Aatu Koskensilta on 16 Feb 2010 08:50
Tonico <Tonicopm(a)yahoo.com> writes: > That, of course, is your personal opinion. My dear Tonico, it was of course not my opinion but Zenkin's! My objection to the highly suspect diagonal argument -- which, in this context, we may take to be the very simple proof of the fact that |X| < |P(X)|, rather than any messy stuff with reals and decimal expansions |and whatnot -- is based on more abstruse and subtle considerations, involving the problematic business of normalising the diagonal proof (it gets a bit hairy, proof theoretically speaking, the sort of stuff that is beyond the logical ken of pretty much any average mathematician; hence the perfect spot to find something underhanded in the proof). But Cantor's ghost is upon us, in a cardboard cut-out version, weeping tears of despair and desolation. The ghost intones thusly: The crowd was caught in a murderous cross-fire; hundreds more died in the next few minutes, torn apart in a concentrated hail of cannon balls and grapeshot. Perhaps we'll find it in our hearts to mend our evil ways? Look into your hearts. You will find, as I did, an angel of mercy, an angel crying the unquenchable tears of Cantor's ghost, forever weeping over the evil bile we spit at each other in news. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |