Cantor Confusion
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From: Albrecht on 3 Oct 2006 11:13 William Hughes wrote: > Albrecht wrote: > > the_wign(a)yahoo.com schrieb: > > > > > Cantor's proof is one of the most popular topics on this NG. It > > > seems that people are confused or uncomfortable with it, so > > > I've tried to summarize it to the simplest terms: > > > > > > 1. Assume there is a list containing all the reals. > > > 2. Show that a real can be defined/constructed from that list. > > > 3. Show why the real from step 2 is not on the list. > > > 4. Conclude that the premise is wrong because of the contradiction. > > > > > > The steps are simple except for a possible debate about defined / > > > constructed. I don't think anyone believes the proof is invalid > > > because of that debate however. > > > > > > There seems to be another area that seems to be a problem > > > though. The problem is that step #2 doesn't seem valid. If we > > > assume the list contains all the real numbers, then defining or > > > constructing a real number in terms of that list would be > > > self-referential. The number from step #2, that is normally defined > > > digit-by-digit along the diagonal, must have its digits (or at least > > > one of them) defined as not equal to itself, if we are to assume the > > > list contains all the real numbers. Certainly the conclusion in that > > > case is that the premise is wrong or that the construction is not > > > valid, but the conclusion can't be simply that the premise is wrong. > > > > > > This same problem appears in the "power-set" theorem, where we > > > have a definition of a set, S, which is a subset of N, defined in > > > terms of the image of a function, f, whose image is assumed to be > > > the power-set of N. If the image of f is assumed to be the power-set > > > of N and S is defined in terms of f, then S must necessarily be > > > defined in terms of itself. Again, if we assume that the image of > > > f is P(N), then defining S as a set whose elements are defined > > > to be elements not in the image of f is a self-referential definition > > > of S because S is also a subset of N, making it meaningless. > > > Certainly a meaningless definition can't be used to prove a > > > contradiction. > > > > > > I'm guessing that the "discussions" that occur stem from the fact > > > that mathematicians disagree that the seemingly self-referential > > > definitions are a problem but it's not intuitively obvious why that is > > > so, therefore many people feel the need to try to refute the proofs. > > > The problem is that it really isn't clear why mathematicians seem to > > > accept the self-referential definitions. > > > > > > Cantor's argument shows just one only thing: > > there is no list of all reals - as there is no list of all naturals - > > as there isn't anything which is infinite and completed. > > A set does not have to be infinite to be countable. > Cantor's proof works just fine for finite sets. So > Cantor's proof can be used to show that the set > of real numbers is not finite. [If this is not cracking a walnut > with a sledgehammer, I don't know what is!] Oh really it isn't. > > If you do not allow "completed infinite sets", then the meat > of Cantor's proof (the cardinality of the reals is greater than > that of the integers) cannot be done > So, everyone who finds your "contradictions" to be convincing > will find Cantor rather trivial. Any proof which shows that infinity is incomprehensible isn't trivial. It must be a proof, which is working on the limit of the knowable. > > However, as far as I can see, 'everyone who finds your > "contradictions" to be convincing', is a set with at most > one element. > Maybe. Best regards Albrecht S. Storz
From: Albrecht on 3 Oct 2006 11:36 Dave L. Renfro wrote: > Dave L. Renfro wrote (in part): > > >> Each sequence of real numbers omits at least one real > >> number. (Sequence being a 1-1 and onto function from the > >> positive integers to the real numbers.) > > Jesse F. Hughes wrote: > > > Er, are you sure about that definition of sequence? Seems to > > me that sequence is any function from the positive integers > > to the reals. > > > > With your definition, the conclusion would be: every onto > > function N->R is not onto. This is true, of course, but is > > more likely to confuse the "Anti-Cantorians" rather than > > enlighten them. > > Ooops, you're correct ... I messed up. I knew I should have > stayed out of one of these kinds of threads. I was curious > how mueckenh would respond to my bringing up other situations > (in "real life") where one argues that something can't happen > by showing that something contradictory (or at least, something > not desirable) would arise if we assumed it did happen. He (she?) > seemed to have ignored my examples, though. > > And what's with this other thing I saw in the thread, where > someone argued that the diagonal argument is not complete > because the assumption that the list of real numbers containing > all the real numbers wasn't allowed for? That's just bizarre. > If this person is _really_ concerned about that issue (rather > than trolling, as I strongly suspect), why isn't he raising > the same issue for _every_ argument? Your consideration is really strange to me. I will try to explain my claim again: There is only a diagonal number which is proveable not in the list if there is a real number which is build up out of all the real numbers in the list. But for an infinite list you can't end the diagonal number. You may have a sequence which converge. But you never have a limit. Cantor argues that you must not have the limit. But with the same idea you can e.g. construct a kind of an diagonal (natural) number of any list of natural numbers (this idea is from Russell Easterly). In consequence the set of the natural numbers is uncountable... You can proof a lot of strange things with the idea of Cantor. But these things are not very useful. Best regards Albrecht S. Storz > For example, in proving > that 4 + 3 = 7 (using three applications of the successor > function), he should complain that we didn't consider the > case where 4 + 3 isn't 7, and when we consider this possibility, > the proof fails. More generally, he should find fault with > every proof of statements of the form "If P, then Q", including > the arguments that he is using to support his position. > > Now that I think about it, don't a lot of these anit-Cantor > arguments take the same form as what they're criticizing? > They argue that something about diagonalizing is incorrect, > because if it were correct, then [insert their attempt to > obtain a contradiction]. > > Dave L. Renfro
From: Albrecht on 3 Oct 2006 12:12 Albrecht wrote: > Dave L. Renfro wrote: > > Dave L. Renfro wrote (in part): > > > > >> Each sequence of real numbers omits at least one real > > >> number. (Sequence being a 1-1 and onto function from the > > >> positive integers to the real numbers.) > > > > Jesse F. Hughes wrote: > > > > > Er, are you sure about that definition of sequence? Seems to > > > me that sequence is any function from the positive integers > > > to the reals. > > > > > > With your definition, the conclusion would be: every onto > > > function N->R is not onto. This is true, of course, but is > > > more likely to confuse the "Anti-Cantorians" rather than > > > enlighten them. > > > > Ooops, you're correct ... I messed up. I knew I should have > > stayed out of one of these kinds of threads. I was curious > > how mueckenh would respond to my bringing up other situations > > (in "real life") where one argues that something can't happen > > by showing that something contradictory (or at least, something > > not desirable) would arise if we assumed it did happen. He (she?) > > seemed to have ignored my examples, though. > > > > And what's with this other thing I saw in the thread, where > > someone argued that the diagonal argument is not complete > > because the assumption that the list of real numbers containing > > all the real numbers wasn't allowed for? That's just bizarre. > > If this person is _really_ concerned about that issue (rather > > than trolling, as I strongly suspect), why isn't he raising > > the same issue for _every_ argument? > > > Your consideration is really strange to me. > I will try to explain my claim again: There is only a diagonal number (better: "anti-diagonal number", in German, we speak just about "Diagonalzahl", that's a little bit confusing). Albrecht S. Storz > which is proveable not in the list if there is a real number which is > build up out of all the real numbers in the list. But for an infinite > list you can't end the diagonal number. You may have a sequence which > converge. But you never have a limit. > Cantor argues that you must not have the limit. But with the same idea > you can e.g. construct a kind of an diagonal (natural) number of any > list of natural numbers (this idea is from Russell Easterly). In > consequence the set of the natural numbers is uncountable... > You can proof a lot of strange things with the idea of Cantor. But > these things are not very useful. > > > Best regards > Albrecht S. Storz > > > > For example, in proving > > that 4 + 3 = 7 (using three applications of the successor > > function), he should complain that we didn't consider the > > case where 4 + 3 isn't 7, and when we consider this possibility, > > the proof fails. More generally, he should find fault with > > every proof of statements of the form "If P, then Q", including > > the arguments that he is using to support his position. > > > > Now that I think about it, don't a lot of these anit-Cantor > > arguments take the same form as what they're criticizing? > > They argue that something about diagonalizing is incorrect, > > because if it were correct, then [insert their attempt to > > obtain a contradiction]. > > > > Dave L. Renfro
From: MoeBlee on 3 Oct 2006 13:42 Albrecht wrote: > I will try to explain my claim again: There is only a[n] [anti-]diagonal number > which is proveable not in the list if there is a real number which is > build up out of all the real numbers in the list. But for an infinite > list you can't end the diagonal number. You may have a sequence which > converge. But you never have a limit. Then you're not talking about the reals. We prove that every convergent sequence of rationals converges to a unique real number, which is the limit of the sequence. If it converges, then it has a limit. Thus every countable decimal expansion represents a real number. In particular, the anti-diagonal of any given countable sequence of denumerable decimal expansions is a countable decimal expansion and thus represents a real number. And we prove it does not represent any real number in the given countable sequence of decimal expansions. Argue that you don't like the axioms of set theory, if you like. Argue that you don't ascribe to the principles of reasoning codified by first order logic, if you like. But it is not in any way rationally arguable that the uncountability of the reals is not a theorem of the stated axioms. And more simply, there is no theory that you have proposed in which there exists a function from the counting numbers onto the carrier set of a complete ordered field. > Cantor argues that you must not have the limit. We must not have the limit of what? And where does Cantor argue this? Morevover, no matter what Cantor did or did not write, we've moved on to axiomatic theories that are not at all beholden to Cantor. > But with the same idea > you can e.g. construct a kind of an [anti-]diagonal (natural) number of any > list of natural numbers. So what? If it's a denumerable list of natural numbers, then there is no anti-diagonal that is a finite sequence or a denumerable that is all 0's after some position, hence no such anti-diagonal represents a natural number, since natural numbers are represented only by finite sequences or by sequences that are all 0's after some position. > In > consequence the set of the natural numbers is uncountable. That is incorrect. See above paragraph. > You can proof a lot of strange things with the idea of Cantor. Many of the ideas originate with Cantor, but the axiomatizations are not beholden to Cantor. > But > these things are not very useful. They axiomatize ordinary calculus used for science and engineering. Moreover, mathematical logic and set theory were and are very useful in the development of the digital computer, as you are using such a digital computer to declare that these are ideas are not very useful. MoeBlee
From: Tonico on 3 Oct 2006 13:59
> ............................................................................. > Your consideration is really strange to me. > I will try to explain my claim again: There is only a diagonal number > which is proveable not in the list if there is a real number which is > build up out of all the real numbers in the list. But for an infinite > list you can't end the diagonal number. You may have a sequence which > converge. But you never have a limit. ************************************** ??? What do you mean by this?? "Never have a limit"?? The diagonal number built by Cantor's method (by the way, one out of MANY...many possible ones that can be built out of one given "list") is perfectly well and completely well constructed: just ask me what is its n-th decimal entry and I will tell you...) ****************************************************************************************** > Cantor argues that you must not have the limit. But with the same idea > you can e.g. construct a kind of an diagonal (natural) number of any > list of natural numbers (this idea is from Russell Easterly). ***************************************************************** Really?? I'd love to see what natural number can you construct from the "list" of natural numbers 1,2,3,4,.... which is NOT included in this list... I think that Russell Easterly must be pretty confused, or else he didn't express that idea and you misunderstood it....or else I am going to learn something new and VERY interesting. Regards Tonio ************************************************************************** In > consequence the set of the natural numbers is uncountable... > You can proof a lot of strange things with the idea of Cantor. But > these things are not very useful. > > L. Renfro |