An uncountable countable set
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From: mueckenh on 12 Jul 2006 08:49 Dik T. Winter schrieb: > > The limit is the stable state where no further transpositions are > > applied, because there are no two elements remaining, which were not > > ordered by magnitude. > > That makes no sense as a definition, as you will never reach that stable > state. That is not my problem but the problem of non-existing infinite sets. > > > > There are no "steps" in a well-order of Q. There is no notion of at > > > which natural number something happens. A well-ordering of Q means a > > > precise set of rules that determine the natural number to which a > > > rational corresponds. It is your re-ordering that requires some notion > > > of limit. > > > > "Step" was meant here as counting from n to n+1. A precise set of rules > > is not yet established for the algebraic numbers, as far as I know. > > But it is for the rationals. And I think it can also be found for the > algebraic numbers, but it will take a bit more care. Is it for rationals *without repetition*? How does it read? > What do you mean with a sentence like "in which line of Cantor's list a certain > digit of the diagonal number will be placed"? How can you find out whether a certain real of Cantors list has the number 5782712398413208741? > > > > But what is known is that the n-th digital > > The mapping from the naturals to the list gives the n-th element. It is > in the *definition* of the list. Do you want to define a proof or to give a proof? > > > > A mapping that well-orders > > > the rationals is *not* a sequential process. The determination of the > > > n-th digit of the diagonal from a given list is *not* a sequential > > > process > > > > Wrong. You cannot know line number n without knowing line number n-1. > > The mapping gives that. > > > But even if you were right, your argument would be void. If infinity > > would exist, then an infinite set could be exhausted by a definition > > like that of Cantor or that of mine. > > Makes no sense. The natural numbers can not be exhausted by a sequential > process. Therefore you resist to count lines. But in case of proof (not definition of proof) you are forced to count all of them. > Nobody has a problem with that, as it is well known that > what is the case in the limit is not necessarily what is the case outside > the limit. Why don't you apply this knowledge in case of Canor's list? In the limit n --> oo it may well be that 4 cannot be distinguished from 5. > > > > > > > But that is not interesting. It is easy to see that the diagonal is a > > > > sequence of the same sort as are the list entries. Whether they are > > > > real numbers is uninteresting. > > > > > > But it is just that part that is interesting. Try the same with a > > > sequence of algebraic numbers. You need to prove that what you get is > > > also an algebraic number (and you cannot). So for algebraic numbers > > > the proof fails. For real numbers the proof goes through, because you > > > can prove that the resulting number is also a real number. > > > > I proved in my special list even that the diagonal number is a > > rational. > > I wonder whether it was a proof or just some handwaving. 0.0 0.1 0.11 0.111 .... replace 0 by 1. > > Why do they if you do not see why limits are needed in sequences? These > > are sequences and nothing else. Would they be different without your > > "special definition"? > > I am lost. Mathematically a sequence of symbols like 0.999... makes no > sense (as a number) unless there is a definition. 10 * 0.999... = 9.999... 9.999... - 9 = 0.999... That does not need special definition, after * and - are defined and can be applied digit by digit. 1/n becomes arbitrarily small for large enough n. Without further definition but that of /. > The sequence makes > perfect sense as a sequence. But as a sequence I can only state that > 1.000... is not equal to 0.999.... Only when we want to interprete them > as numbers we need a definition, and with the common definition those two > are the same *as numbers*. Do you really believe another definition would be possible in |R? > > > I did not say that Cantor's strings were binary numbers. I said that > > they might be *interpreted* as binary (dyadische, dual) > > representations, safely. Cantor and Zermelo at least did so. > > You can do so *if* you take dual representations in account. But at least > with Cantor I do not find any evidence that he *did* interprete them as > binary numbers. No, he did not. But when extending his proof to functions, he applied numbers 0 and 1, which obviously can also serve as m and w in the orignal version. Regards, WM
From: mueckenh on 12 Jul 2006 08:55 Franziska Neugebauer schrieb: > > To prove the existence and uniqueness of (|Q_+, <)? > > I want to know what the application of all of your conditional > transpositions to the ordered set *means*. I ask for a definition of > the *result*. And it would be nice if you show if it exists and is > unique. This is not (yet) proven "by the axioms". Many things cannot be proven by ZFC+FOPL. ZFC is not all, as even set-theorists increasingly recognize. (|Q_+, <) is the result, because if there was any q_i not yet ordered by magnitude then it would become ordered by magnitude by my definition (not by yours). Regards, WM
From: mueckenh on 12 Jul 2006 08:57 Virgil schrieb: > What "mueckenh" claims is no more possible that to have a natural number > which is simultaneously even and odd. Or to have an infinite set of finite numbers. > > Since if is clear that NO transposition can achieve this, "mueckenh"'s > alleged final result never results. Just that is the essence of my proof. It shows that infinity can never by reached and does not exist. Regards, WM
From: Franziska Neugebauer on 12 Jul 2006 09:11 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> > To prove the existence and uniqueness of (|Q_+, <)? >> >> I want to know what the application of all of your conditional >> transpositions to the ordered set *means*. I ask for a definition of >> the *result*. And it would be nice if you show if it exists and is >> unique. This is not (yet) proven "by the axioms". > > Many things cannot be proven by ZFC+FOPL. But you still have not proven your claim, that it is - at least - meaningful to write about application of *all* conditional transpositions to a given sequence. Until then it is not meaningful. > ZFC is not all, as even set-theorists increasingly recognize. You may chose any other axiomatic system. But please take care that it is not contradictory by design again. > (|Q_+, <) is the result, because if there was any q_i not yet ordered > by magnitude then it would become ordered by magnitude by my > definition (not by yours). You have still not shown, what the application of *all* of your conditional transpositions means. Writing about ordings of q_i requires a meaningful (q_i), which you still have not proven to exist. F. N. -- xyz
From: Dik T. Winter on 12 Jul 2006 10:44
In article <1152707645.284464.200140(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > you are arguing that 1/9 does not exist. > > 1/9 does exist. What does not exist is its representation 0.111... > which does not belong to the sequence 0.1, 0.11, 0.111, ... the 1's of > which are completely indexed by natural numbers. The representation 0.111... *does* exist. There is a specific definition for it (as I have already given), and an axiom through which you may prove that it does exist. But it is indeed not in the sequence 0.1, 0.11, etc. Again, I think you are arguing against the axiom of infinity. > > Sorry, this ia again not logical reasoning. Again, I ask what logical > > steps you take to get from > > For all p there is an n such that An[p] = K[p] > > to > > There is an n such that for all p An[p] = K[p], > > I would think you should be able to answer that simple question. > > I do not do this step! Already your antecedent "for all p there is an n > such that An = K[p]" *is wrong*. What is wrong about it? Take some particular p and take n any value larger than or equal to p. We see that An[p] = 1 = K[p]. So what is wrong? > I say: for those K[p] for which there > is an An, one and only one An is sufficient. That is not valid for all > p, because for all n we have > > 0.111... - 0.111...1 = 0.000...0111... > > such that there are more 1's in K than in any An. The last is true. But it is also true that for all p there is an An. Or please exhibit a p for which it is not wrong. > > > If you think the sentence "all positions of K = 0.111...are indexed by > > > list numbers" is not equivalent to the sentence "K > > > is in the list", then you seem to imply that more than one list number > > > is required to index the digits of K. > > > > Yes, each digit position of K can be indexed by a number in the list. And > > it is not equivalent to "K is in the list". The implication you mention > > I do not understand. > > All digits which are indexed by smaller list numbers can be indexed by > one larger list number. Therefore always only one is required. Care to explain the implication you mention above? > > > You do not need the first two list numbers 0.1 and 0.11, because the > > > third alone is sufficient: 0.111 does index the first three digits p = > > > 1 to 3 of 0.111... > > > > I am a bit at a loss here what you mean with "indexing". Again care to explain? How can a single number index three digits? > > > That can be carried on. p = 1 to 4 in 0.111... can > > > be indexed by 0.1111. The digits p = 1 to n can be indexed by list > > > number n = 0.111...1 with n 1's. If, as you seem to imagine, it may > > > happen, that two list numbers are required to index some p, then one of > > > the two is smaller than the other. So the other will suffice for the > > > task of the smaller. This proves: If a p is finite and, hence, can be > > > indexed by finite list number, then all digits < p can be indexed by > > > the same list number. Therefore, all digits which can be indexed can be > > > indeed by one finite list number alone. > > > > Pray prove the "therefore", because it does not follow. What is valid > > for the finite is not necessarily valid for the infinite. > > But it is not necessarily just the opposite (like, allegedly, in the > binary tree). No, indeed. So you have to prove it. > So let us find a resolution of this dilemma by mathematics. > > Define "*-" by > a_i *- b_i = a_i - b_i if a_i > b_i else a_i *- b_i = 0 > (subtraction down to zero but without negative numbers) > > Consider 0.111... and the list numbers as sequences or as vectors, such > that *- can be applied to each term separately. > > If you are right, you must maintain the result: > > 0.111... *- (0.1 + 0.11 + 0.111 + ...) = 0.000... I can make no sense of the second term. What do you *mean* with (0.1 + 0.11 + 0.111 + ...) ? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |