An uncountable countable set
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From: Tony Orlow on 24 Aug 2006 08:47 Dik T. Winter wrote: > In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > The first part of his first proof shows that a complete ordered field > > > has cardinality larger than the natural numbers. In his proof he did > > > not rely an any properties of the reals other than that they form a > > > complete ordered field (he uses reals to exemplify). > > > > Specifically, he demonstrates that there will always exist an > > unaccessible real in any finite interval, given that we are only allowed > > a finite (natural, that is) number of iterations. > > Sorry, you have got that wrong. We are allowed an infinite number of > iterations indexed with finite numbers (whatever that may mean). Thagt means that at no time have we ever completed an infinite number of iterations. That's why we cannot arrive at that intermediate value. > > > Therefore, indeed, the > > number of reals in a finite interval is greater than any finite natural. > > That proof is essentially valid, but it's not a proof that the reals > > cannot be linearly ordered in a discrete manner. > > This is approximately right. The number is larger than the number of > finite naturals. Whether the reals can be linearly ordered in a > discrete manner (eh? quite a few buzzwords here without definition) > depends on the axiom of choice, and a few more things. Using the axiom of choice, it is provable that every set can be well ordered, which entails a linear order, but this proof gives no clue as to how this can be accomplished. But, well-ordering aside, the H-riffic numbers give a sequential list of all reals though exponentiation. The other approach is Ross, which relies on the notion of infinitesimally different reals, sequential in their natural order. > > > > (The second > > > part shows that the cardinality of the set of subsets of a set is > > > strictly larger than that of the original set. > > Actually this is the second part of again another article. I will look > it up when I am close to the book again. > > > > This proof comes > > > close to the proof provided by Hessenberg, but is, in my opinion, > > > a bit less strict.) > > > > It is related to the powerset through |P(S)|=2^|S|, but he did it in > > decimal, no? > > No. Nowhere in his papers did Cantor use decimals. Perhaps it is just the explanations of his second proof of uncountability which I have seen. He did use SOME digital representation of his list of reals. Was it binary? I have not read his original papers like you have. > > > In any case, the powerset relation boils down to the > > symbolc equation N=S^L, where S is the number of logical states allowed > > (there can be more than two in various systems) and where L ultimately > > corresponds to the size of the root set. So, the two are related. > > You make here as much sense as in much earlier articles. What part of does not make sense? I suppose it's hard to put that concept into a sentence or two and be readily understood. > > > > What he states in the quote provided by Mueckenheim is that that proof > > > can be easily used to show that the cardinality of any set of reals in > > > an interval is larger than the cardinality of the natural numbers. > > > > I don't disagree that the first is infinite while the second is > > unbounded bt finite, and therefore smaller. > > Pray, for once, provide a definition. A set is either finite or infinite. > And if a set is finite, by the definitions there is a largest element. > So, how do you define "unbounded but finite"? Not with the Dedekind definition. A truly infinite set must have elements infinitely many position beyond other elements, the way I see it. > > > > His second proof is about sets of sequences, and his diagonal proof > > > shows that the cardinality of the set of infinite sequences with > > > two possible elements is strictly larger than the cardinality of > > > the natural numbers. Zermelo has indicated how that can be converted > > > to a proof that the reals have cardinality larger than that of the > > > natural numbers. But Cantor had *no* intention to prove that at that > > > point at all, and did nowhere write that it could be applied for that > > > purpose. > > > > Yes, the second is really a proof about power set and/or symbolic > > representations of quantites. > > Not at all. There is no question about "representation of quantities". In the second proof of uncountability of the reals, there are no digital strings? What list is he deriving the antidiagonal from, a shopping list? > > > > Cantor's one and only purpose was a proof of the theorem: > > > There are sets with cardinality larger than that of the natural numbers. > > > > Considering that the set of finite naturals is ultimately finite, as > > Wolfgang has been trying to express, that's not a surprising result. > > What's surprising is Dr. M's reluctance to admit the actually infinite > > in the real interval. > > What does not surprise me at all is that apparently you do not understand > the discussions at all. If you say so. It seem that you're the one saying you don't understand what I'm saying, but perhaps that's a matter of perspective. > > > > Has the book by Zermelo on Cantor's work ever been translated into English? > > > If not, the non-German speakers are certainly missing something. As with > > > the untranslated book by O. Perron on continued fractions. (BTW, with > > > Zermelo's book also the non-French speaking are also missing something; > > > parts of it are in French.) > > > > I dunno. Did I miss anything? > > Quite a lot. When was the last time you did read a book about set theory? Quite a while back. I wonder how many books on set theory Cantor read? :)
From: Dik T. Winter on 24 Aug 2006 10:21 In article <J4HBpI.Cq1(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Dik T. Winter wrote: > ... > > > The first part of his first proof shows that a complete ordered field > > > has cardinality larger than the natural numbers. In his proof he did > > > not rely an any properties of the reals other than that they form a > > > complete ordered field (he uses reals to exemplify). .... > > > (The second > > > part shows that the cardinality of the set of subsets of a set is > > > strictly larger than that of the original set. > > Actually this is the second part of again another article. I will look > it up when I am close to the book again. This is the second part of the second article, not of the first... -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 24 Aug 2006 10:26 In article <44ed99b7(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: .... > > > > Well, you and he are not. The logic is flawed. > > > > > > For 1, 2, and 3, order and value are identical, even according to your > > > logic, I suppose. Where does the first deviation happen? Which one does > > > deviate first, i.e., which one does first be larger than the other? And > > > why? > > > > It is the case when the set size is not a natural number. > > And, with the addition of which natural number to the set does the set > achieve this unnatural size? There is no such specific natural number. It is when we have them all, but as there is no largest number, this can not be achieved by taking them one by one. > > > > And his proof is not a proof. > > > > > > O course not, because every proof which has an unpleasant result is not > > > a proof. > > > > Then show that the set of all natural numbers does not have cardinal number > > aleph-0 and ordinal number w. A proof please. > > I have already shown how the set of bit positions in the binary naturals > has no cardinal or ordinal that can be assigned to it. Yes, but your showing was not a proof. > > > > > For my part, I agree that the set of finite > > > > > naturals is finite, though unbounded, > > > > > > > > In that case you are not using standard mathematical terminology. I > > > > have no idea what a finite but unbounded set is. > > > > > > That's why you cannot understand mathematics. You fall back behind > > > Cantor. He knew it. > > > > Oh, perhaps *you* do not understand mathematics? Earlier I have already > > written that a few things that Cantor has written do not conform with > > current thinking. But see <http://mathworld.wolfram.com/FiniteSet.html>: > > A set X whose elements can be numbered from 1 to n, for some positive > > integer n. > > Yes, given the current understanding, any unbounded set is infinite. Sorry, that is not current understanding, that is using the definitions and terminology. Understanding may change, but within the definitions and the terminology any unbounded set will remain infinite. > > A set is infinite if it is not finite. A set is Dedekind infinite if > > it can be mapped to a proper subset of itself, and it is Dedekind > > finite if it is not Dedekind infinite. When we assume the axiom of > > choice, the two notions are identical. Without that axiom there > > can be infinite sets that are Dedekind finite. Do you want to know > > more about set theory? > > > > Now, using, this terminology (pretty standard), what is a "finite but > > unbounded" set? > > Using that system, the phrase is senseless, but that system is not the > real universe. It's a concoction. Well, that is just a statement that (in my opinion) is senseless. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 24 Aug 2006 11:10 In article <44ed9fcb(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > Dik T. Winter wrote: > > In article <ecig8i$ta$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > > Dik T. Winter wrote: > > ... > > > > The first part of his first proof shows that a complete ordered field > > > > has cardinality larger than the natural numbers. In his proof he did > > > > not rely an any properties of the reals other than that they form a > > > > complete ordered field (he uses reals to exemplify). > > > > > > Specifically, he demonstrates that there will always exist an > > > unaccessible real in any finite interval, given that we are only allowed > > > a finite (natural, that is) number of iterations. > > > > Sorry, you have got that wrong. We are allowed an infinite number of > > iterations indexed with finite numbers (whatever that may mean). > > Thagt means that at no time have we ever completed an infinite number of > iterations. That's why we cannot arrive at that intermediate value. Indeed. Not step by step. And so? I have said this before. I am quite sure others have said that before. You can not get the set of naturals by adding the one by one. It is the axiom of infinity that asserts that that set does exist. > > > Therefore, indeed, the > > > number of reals in a finite interval is greater than any finite natural. > > > That proof is essentially valid, but it's not a proof that the reals > > > cannot be linearly ordered in a discrete manner. > > > > This is approximately right. The number is larger than the number of > > finite naturals. Whether the reals can be linearly ordered in a > > discrete manner (eh? quite a few buzzwords here without definition) > > depends on the axiom of choice, and a few more things. > > Using the axiom of choice, it is provable that every set can be well > ordered, which entails a linear order, but this proof gives no clue as > to how this can be accomplished. But, well-ordering aside, the H-riffic > numbers give a sequential list of all reals though exponentiation. The > other approach is Ross, which relies on the notion of infinitesimally > different reals, sequential in their natural order. But the last is irrelevant in the context of the paragraph above. There is no proof whether the reals can be linearly ordered or not. Unless you assume the axiom of choice. And under other discussions you can prove that the reals can not be linearly ordered (I think). So *of course* Cantors proof is not a proof that the reals cannot be linearly ordered. > > > > This proof comes > > > > close to the proof provided by Hessenberg, but is, in my opinion, > > > > a bit less strict.) > > > > > > It is related to the powerset through |P(S)|=2^|S|, but he did it in > > > decimal, no? > > > > No. Nowhere in his papers did Cantor use decimals. > > Perhaps it is just the explanations of his second proof of > uncountability which I have seen. He did use SOME digital representation > of his list of reals. Was it binary? None at all. The second proof (the diagonal proof) is *not* about reals. Zermelo has indicated how it *can* be transformed to a proof for the reals using binary notation and noted the probles in that case with dual representations and how to solve it. Later somebody has transformed it to a proof using decimals, but I do not know who has done that. > > > In any case, the powerset relation boils down to the > > > symbolc equation N=S^L, where S is the number of logical states allowed > > > (there can be more than two in various systems) and where L ultimately > > > corresponds to the size of the root set. So, the two are related. > > > > You make here as much sense as in much earlier articles. > > What part of does not make sense? I suppose it's hard to put that > concept into a sentence or two and be readily understood. It works for finite sequences, but your conclusions do *not* work in the infinite, because transfinite induction fails. > > > I don't disagree that the first is infinite while the second is > > > unbounded bt finite, and therefore smaller. > > > > Pray, for once, provide a definition. A set is either finite or infinite. > > And if a set is finite, by the definitions there is a largest element. > > So, how do you define "unbounded but finite"? > > Not with the Dedekind definition. A truly infinite set must have > elements infinitely many position beyond other elements, the way I see it. Yes, but in that case you are not using standard mathematical terminology. And I have yet to see a *definition* of you about *infinite*. > > > Yes, the second is really a proof about power set and/or symbolic > > > representations of quantites. > > > > Not at all. There is no question about "representation of quantities". > > In the second proof of uncountability of the reals, there are no digital > strings? What list is he deriving the antidiagonal from, a shopping list? Pray read. His second proof is *not* about the uncountability of the reals. There are no reals involved at all. It is about (infinite) sequences of symbols (he uses 'w' and 'm'). > > > > Cantor's one and only purpose was a proof of the theorem: > > > > There are sets with cardinality larger than that of the > > > > natural numbers. > > > > > > Considering that the set of finite naturals is ultimately finite, as > > > Wolfgang has been trying to express, that's not a surprising result. > > > What's surprising is Dr. M's reluctance to admit the actually infinite > > > in the real interval. > > > > What does not surprise me at all is that apparently you do not understand > > the discussions at all. > > If you say so. It seem that you're the one saying you don't understand > what I'm saying, but perhaps that's a matter of perspective. You are using words without definition, using words from standard terminology but with (apparently) not the standard definition. That is what makes you on occasion not understandable. > > > > Has the book by Zermelo on Cantor's work ever been translated into > > > > English? If not, the non-German speakers are ce
From: MoeBlee on 24 Aug 2006 12:47
Tony Orlow wrote: > MoeBlee wrote: > > Please say what sentence and its negation you believe are both theorems > > of set theory. > > I'm not sure how this would be proven in set theory (I don't think it > is), So you don't know that set theory is inconsistent (by 'inconsistent' I mean the usual definition). > but it appears to be a belief, anyway, that all sets can be > classified through cardinality. With suitable axioms, we can define a cardinality operation 'card' so that we get a theorem: card(x)=card(y) <-> x equinumerous with y. > However, the set of bit positions > required to list the naturals in binary defies classification in this > system. Whatever you mean by that, based on remark that you know of no proof of a contradiction in set theory, it is a problem of your own misconception and not a contradiction in set theory. Fortunately, we don't obligate ourselves to the burdens of your own misconceptions. > > No you haven't. You've posted disconnected pieces of undefined > > terminology. > > No, I've shown how IFR works with the notion of Big'un. Like I've said all along, you have not stated a logistic system, primitives, or a set of axioms. >> See Enderton's mathematical > > logic textbook. The full formalization is culminated in the exercise in > > which you are asked to make sets of functions from the variables into > > the universe the values of a certain recursively defined function. > Okay. That's not a very fundamental definition, as far as I can tell. You can't tell anything about it, since you don't know anything about it. The definition is a rigorous realization, for classcial first order logic (and can be generalized for higher order logic), of the fundamental principle of Tarki's theory of truth. MoeBlee |