Cantor Confusion
in [Math]
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From: WM on 17 May 2007 06:39 On 17 Mai, 03:03, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1179346754.744571.316...(a)k79g2000hse.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 16 Mai, 04:17, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1178808886.504395.245...(a)w5g2000hsg.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > > > On 10 Mai, 04:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > ... > > > > > > I will answer there. > > > > > > > > > > I will not see it, so you clearly cut short the discussion. > > > > > > > > Please try, we could reduce the noise significantly and could kill any > > > > polemics and insults. If you are not able to see it, I will repeat it > > > > here. But I wonder why Google introduced this technique if it remains > > > > inaccessible to most users and even to experts. > > > > > > It is not inaccessible to me. > > > > So you saw my answer? > > No. I say it is not inaccessible to me. Not that I do access it. > > > This set F of functions can be bijected with the > > set R of reals. The reals form an intercession with the rationals Q. > > The rationals form a bijection with the naturals N. > > Sorry, makes no sense without context. The context was your due posting. Either you recall it or you access my group or yu vcannot get an answer on your question which uttered an unjustified doubt. > > > > But I do not want to sign up with google > > > just for the privilege of accessing it. And I do not want to use a web > > > browser to access discussions. Why google introduced it beats me. I > > > prefer my keyboard based access to newsgroups, no mouse for me. > > > > If you are interested in the discussion which we had about Cantor's > > mistake or not mistake concerning his second diagonal proof, you may > > enter > >http://groups.google.com/group/sci.math.research/browse_frm/thread/5d... > > If I look correctly, again, it is not the case that Cantor made an error. > It is my opinion that what I wrote about it was right, the interpretation > being that Cantor gave an additional prove that there are sets with > cardinality larger than the naturals. But in essence this is quite > irrelevant. Cantor may have erred at times, that is *not* related to the > current ways of set theory. It is. Set theory is simply biased. Consider the list 0.666... 0.3666... 0.33666... 0.333666... .... If the diagonal number is defined by "replace 6 by 3", then we have two answers none of which can be preferred by logic, but the second of which is suppressed by convention. 1) Every entry of the list differs at some place from the diagonal number. 2) Every initial segment of the diagonal number is represented by an entry of the list. As no number has a last digit, we can only look for initial sequences from positions 1 to n. By that method we find that (2) is correct for all. Why do you think the first answer be more important? Regards, WM
From: William Hughes on 17 May 2007 07:21 On May 16, 3:55 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > On 14 Mai, 22:09, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On May 14, 3:17 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > Which of the following statements is not true > > > Every element of R is different from p. > > > Every node of p is in some element of R > > Back to the facts: Find a path p and a set of paths R such that p is not in R, but every node in p is in some element of R. > > 1) No element r =/= p of R is required to cover any node of p. (I can > show of every such element r =/= p which you may name that it is not > required.) So What? That every element of R was necessary was not required and I never claimed it was true. > 2) No element of R is sufficient to cove all nodes of p. So What? That no one element of R is sufficient was not required and indeed I have noted it is impossible. > Hence your set R is not suitable to demonstrate your claims. Does not follow. Which of these statements is not true. Every element of R is different from p. Every node of p is in some element of R - William Hughes
From: Virgil on 17 May 2007 12:40 In article <1179397419.898330.160410(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Mai, 01:42, Virgil <vir...(a)comcast.net> wrote: > > In article <1179345320.393851.200...(a)u30g2000hsc.googlegroups.com>, > > > > Let R be the set of all finite initial subsets of N. > > No member of R covers N yet R itself does. > > The first assertion is right, the second is wrong. So that WM claims that there is a member of N that is not a member of any finite initial subset of N? How quaint!
From: Virgil on 17 May 2007 12:55 In article <1179398355.204677.123430(a)o5g2000hsb.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > > If I look correctly, again, it is not the case that Cantor made an error. > > It is my opinion that what I wrote about it was right, the interpretation > > being that Cantor gave an additional prove that there are sets with > > cardinality larger than the naturals. But in essence this is quite > > irrelevant. Cantor may have erred at times, that is *not* related to the > > current ways of set theory. > > It is. Set theory is simply biased. Biased only in favor of strict logic in place of unorganized assuptions and claims. > Consider the list > > 0.666... > 0.3666... > 0.33666... > 0.333666... > ... > > If the diagonal number is defined by "replace 6 by 3", then we have > two answers none of which can be preferred by logic, but the second of > which is suppressed by convention. > > 1) Every entry of the list differs at some place from the diagonal > number. > 2) Every initial segment of the diagonal number is represented by an > entry of the list. Both can be quite true without conflict, because there are other true statements to consider: 3) every member of the list contains a '6' at some digit position. 4) the diagonal does not contain a '6' at any digit position. So that while, the sequence listed may converge to the diagonal described, no member of the sequence EQUALS the diagonal. Which is all that Cantor needed. > > As no number has a last digit, we can only look for initial sequences > from positions 1 to n. By that method we find that (2) is correct for > all. > > Why do you think the first answer be more important? Because those difference establish that the diagonal is not listed, which is precisely what was to be proved! Nothing was said about whether the list values converged to the diagonal, only whether one of them is exactly equal to the diagonal. WM, as usual, not only has the wrong end of the stick, he has the wrong stick.
From: WM on 17 May 2007 14:50
On 17 Mai, 02:49, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1179346617.111313.267...(a)o5g2000hsb.googlegroups.com> WM <mueck...(a)rz.fh-augsburg.de> writes: > > On 16 Mai, 04:13, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > Each of the numbers is finitely definable, because each is a halting Turing > > > machine, and Turing machines are finitely definable. It is the set that is > > > not finitely definable. > > > > If each number is finitely defined, then there are only finitely many > > numbers, because there are only finitely many Turing machines. Of > > course, this finite set then is finitely defined too. > > Back again to how you started. There are only finitely many natural numbers... > That is not mathematics. That is the obvious basis which can be proven by mathematics. > > > > > A well defined Cauchy sequence, for pi, for instance, makes pi well > > > > defined - not as a number in the sense of MatheRealism, but as a > > > > number in the sense of mathematics. > > > > > > Now you come up again with a new term. 'Well defined'. What is > > > 'well defined'? > > > > A number is well defined if it is defined by a finite number of words > > such that, in principle (i.e., given an infinite amount ressources) > > the Cauchy epsilon can be made arbitrary small. > > So pi is well defined as the limit of the circumference of the inscribed > n-gon? The sequence of those circumferences *is* a Cauchy sequence... According to current mathematics, pi is well defined. Even according to MatheRealism pi is well defined (as an idea). > > > > > Please read: > > > > I gave a bijection between the set of nodes and the set of branching- > > > > offs of paths bunches. > > > > > > Yes. I do not understand. 0.1010101010... is a path bunch. So there is > > > a branching-off of this path bunch. And you claim a bijection with the > > > nodes. > > > > I showed that there are countable sets which cannot be bijected with > > the set of natural numbers, for instance the set of finitely definable > > numbers. > > But that set *can* be bijected with the set of natural numbers. Wrong. > It is > indeed easy to show that there is an injection from that set to the set > of natural numbers. Consider all finite sentences over some alphabet > (let's say the 26 latin letters plus a space). Each such sentence can > be considered as a base-27 number, so we have an injection. Claim of injection is correct. Claim of bijection is wrong. (pi for instance is defined by many different definitions). An injection is also possible for the set of all paths into the set of all nodes. (There are two nodes per path.) > > > > > > What node is bijected with the branch-off of 0.101010101010...? For an injection you can choose whatever node you want. Obviously for every number represented in the tree there is a node. And there is no diagonal construction possible in the tree. > > > > I claim that there are no more branching-offs than odes and that there > > are no more real numbers represented in the tree than are branching- > > offs. There is no path ever finished, but it is only branching off > > from other paths in infinity. But the number of paths separated from > > other paths cannot surpass the number of branching-offs. > > The number of paths is the same from the root node, because every path > starts at the root. Or are you suggesting that there are paths *not* > starting at the root? Every path starts at the root node. But in order to count the paths, they must be distinguishable, i.e. separated. > > > > Makes no sense. As all bunches start at the root, there cannot be more > > > bunches that come out of a node than go in. By your own definitions a > > > bunch that comes out of a node, also comes in it (when it arrives there > > > from the root). > > > > One bunch goes in, because the two which come out have not yet been > > separated when the paths which they consist of, went in. > > So it is your opinion that bunches that start of the root nevertheless do > not come in at some node but only come out? Every bunch starts at the root node. But in order to count the bunches, they must be distinguishable, i.e. thy must be separated bunches. The number of separated bunches is doubled at every level. > > Another question about chapter 10. Do you understand what a normal number > is? I think not. Off-hand I do not know whether there are normal numbers > that are normal with respect to all bases (although it is expected that pi > is one). Such numbers are called absolutely normal. But to know that is neither required for the readers of my book in order to understand MatheRealism nor would it be useful to expand the number of pages and the price of the book by a large factor. > But if a number is normal with respect to some base that does > *not* mean that the digits are unpredictable. Nor does unpredictability > of digits mean that a number is normal. > -- There are different notions (for instance weakly normal numbers and absolutely normal numbers). Of course normal numbers can be constructed, one of the simplest cases is the rational number 0.12012012... with respect to base 3, but as there must be included also normally distributed frequencies of 10^100-tuples and larger tuples most normal numbers cannot be constructed. Regards, WM |