Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 26 Jul 2005 22:48 In article <1122419926.576155.170680(a)f14g2000cwb.googlegroups.com>, malbrain(a)yahoo.com wrote: > briggs(a)encompasserve.org wrote: > > In article <1122393788.077928.129590(a)o13g2000cwo.googlegroups.com>, > > malbrain(a)yahoo.com writes: > > > Daryl McCullough wrote: > > > > > >> Nothing in mathematics is excepted without question. Not by > > >> mathematicians, anyway. > > > > > > I think you meant ACCEPTED. See Barb's post for a discussion of > > > EXCEPTED. > > > > > >> Yes, it is certainly the case that *if* you can prove by > > >> induction "forall x, Phi(x)", *then* you can write a > > >> corresponding recursive function that given a number n, > > >> produces a proof of Phi(n). Nobody disputes that. What > > >> people are disputing is your bizarre belief that proving > > >> "forall x, Phi(x)" by induction means that you have proved > > >> Phi(0), Phi(1), Phi(2), ... It means that you *can* prove > > >> all those infinitely many statements, not that you *have*. > > > > > > Sorry, but the axiom states that you HAVE INDEED proved your assertion > > > for each and every n when its conditions are satisfied. > > > > The axiom makes no statement about what you *can* prove. > > The axiom makes no statement about what you HAVE INDEED proved. > > Well, it never hurts to look again. > > > The axiom states that the _PROPERTY HOLDS_ for each and every n > > given that the property holds for 0 and that whenever the property > > holds for i, the property holds for S(i). > > "Three years later Fermat identified an important property of the > positive integers, namely that it did not contain an infinite > descending sequence. He did this in introducing the method of infinite > descent 1659: > > ... in the cases where ordinary methods given in books prove > insufficient for handling such difficult propositions, I have at last > found an entirely singular way of dealing with them. I call this method > of proving infinite descent ... > The method was based on showing that if a proposition was true for some > positive integer value n, then it was also true for some positive > integer value less than n. Since no infinite descending chain existed > in the positive integers such a proof would yield a contradiction. > Fermat used his method to prove that there were no positive integer > solutions to > x4 + y4 = z4. " > > karl m May one restate that equation as x^4 + y^4 = z^4, as being rather more in the spirit that Fermat wrote it in?
From: Virgil on 26 Jul 2005 22:52 In article <1122420601.050240.249700(a)o13g2000cwo.googlegroups.com>, malbrain(a)yahoo.com wrote: > David Kastrup wrote: > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > > > When the only way to form a bijection is with a mapping function, > > > then that function needs to be taken into account. This nonsense > > > about an infinite set of finite whole numbers is pretty bad too, > > > > Well, it seemingly is not a concept accessible to everybody, > > surprising though this may appear. > > > > Core at this problem is the inability to differentiate between > > infinite (a property of a single element) and arbitrarily large (a > > property of available elements from an infinite set), in short, the > > inability to distinguish between > > How is this going to help? You're dealing with pre-axiom of infinity. > There are only the Peano axioms. Why don't you try describing the > difference between infinte, indetermate, and arbitrarily large??? > > >From the history of computers: 1/0 = infinite, 0/0 = indefinite and > 2^60-1 equals arbitrarily large. karl m For sets which are subsets of the reals, including the naturals, there is also *unbounded*.
From: Virgil on 26 Jul 2005 23:01 In article <hglde1dk4s2v91pfpdb4b5fso8tfb87ho1(a)4ax.com>, Martin Shobe <mshobe(a)sbcglobal.net> wrote: > On Tue, 26 Jul 2005 16:51:05 +0200, David Kastrup <dak(a)gnu.org> wrote: > > >Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > > >> David Kastrup wrote: > >> > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > >>> > >>>>And this is precisely what anti-Cantorians find unacceptable. IMHO > >>>>an "infinite" set cannot consistently have a "size". > >> > >>> This is a _perfectly_ valid point of view (as opposed to the views > >>> of those you sympathize with which are wildly inconsistent). > >> > >> Uh, uh. How do you know with which I symphathize? Do you keep a record > >> of my responses to everybody? > > > >Well, you use the word "anti-Cantorians" above. If this was not > >intended to mean mostly a particular set of some outspoken people in > >this Usenet group, it would appear that I misinterpreted this. > > > >> [ .. rest deleted .. ] > >> > >> OK. It seems that we finally have arrived somewhere. I have one > >> final question, though. Is it "legal" (according to mainstream > >> mathematics) to call a set "countable" if it can be brought in 1:1 > >> correspondence with the naturals? > > > >That's the usual usage of the word. > > I believe that that is the definition of "countably infinite". A set > is "Countable" iff there exists a one-to-one correspondence to a > subset of the naturals. > > >> And uncountable otherwise? > > > >Well, in one direction: {1} can't be brought into 1:1 correspondence > >with the naturals, either. But if you can't map the naturals to cover > >the set exhaustively, than the usual term would be "uncountable". > > > >It may be that there is a bit of excluded middle: I think finite sets > >are usually classed neither as countable nor uncountable. Not sure > >about that, though. > > I'm pretty sure that finite sets are usually considered countable. > > Martin IIRC, countable usually includes both finite and countably infinite. If one wishes to restrict the meaning to countably infinite, the word "denumerable" is sometimes used in that sense.
From: Martin Shobe on 26 Jul 2005 23:40 On 26 Jul 2005 17:15:10 -0700, malbrain(a)yahoo.com wrote: >Chris Menzel wrote: >> On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6(a)cornell.edu> said: >> > ... >> > then that function needs to be taken into account. This nonsense >> > about an infinite set of finite whole numbers is pretty bad too, but >> > probably without any real consequences. >> >> You seem to agree that the set of whole numbers is infinite. But there >> was an inductive argument a few posts back that all the whole numbers >> are finite, and hence that the set of finite whole numbers is infinite. >> There was some real mathematics there. > >How does it follow that the count of finite whole numbers is infinite? >How is this established by the Peano axioms? A set, A, is infinite if, and only if, there exists a one-to-one function, f:A -> A, such that f(A) is a proper subset of A. Or equivalently, A set, A, is infinite if, and only if, there exists a function, f:A -> A, such that 1) for all x,y in A, f(y)=f(x) => x=y. 2) there exists an x in A such that for all y in A, f(y) =/= x. Since there are no sets, and we are interested only in the domain of PA, we have The domain of PA is infinite if, and only if, there exists a function, f, such that 1) for all x,y f(y)=f(x) => x=y. 2) there exists an x such that for all y, f(y) =/= x. The successor function meets those criteria. Therefore, the domain of PA is infinite. The only problem that I can see with this is that it's a theorem about PA instead of a theorem of PA. > You have Tony agreeing to >the axiom of infinity apriori, when this is not indicated. The axiom of infinity is not needed to prove that a set is infinite. The axiom of infinity is needed to prove that infinite sets exist. Martin
From: MoeBlee on 27 Jul 2005 00:32
Han de Bruijn wrote: > MoeBlee wrote: > > > georgie wrote: > > > >>>Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic > >>>theory. > >> > >>How can axioms be rigorous? Aren't they supposed to be self-evident? > >>Doesn't that sort of imply that you "believe" them to be true? Isn't > >>that faith? > > > > Axioms are rigorous in that there is an effective method by which to > > determine whether a formula is an axiom. Non-logical axioms are, by > > definition, true in some models and not true in others. Just to study a > > theory, one does not have to commit to a belief that a particular model > > is one of the real world, whatever one takes 'the real world' to mean. > > Huh? And again: huh? Flabbergasted ... > > Han de Bruijn You must be mocking me for my having stated the obvious. Meanwhile, I'm still fascinated by your inconsistent theory, posted at your web site, which is: Z, without axiom of infinity, bu with your axiom: Ax x = {x}. Would you say what we are to gain from this inconsistent theory? MoeBlee |