Cantor Confusion
in [Math]
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From: Virgil on 10 May 2007 15:46 In article <1178808468.554108.46820(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 03:59, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > There is a slight difference. Finite definitions do not imply finitely > > definable. Consider the computable numbers as they are presented by > > halting > > Turing machines. The definition is indeed finite. > > The definition of the set is finite, not of the numbers. > > > Nevertheless they are > > considered not to be finitely defined because it is impossible to find out > > whether a particular Turing machine indeed *does* halt. > > Therefore only few of the numbers are finitely definable. They all are, but one just cannot tell which ones are the ones that are. > > > > > > But that a set of entities is finitely definable does *not* > > > > mean that each element of the set is finitely definable. Consider > > > > Cauchy > > > > sequences. By (a finite) definition a Cauchy sequence denotes a real > > > > number. This is a *finite* definition of the set of real numbers. > > > > > > Yes, of the set. But it is only a definition of those real numbers for > > > which Cauchy sequences can be finitely defined. > > > > No. It is a definition of the set. It is not a definition of the real > > numbers at all. > > 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. MatheRealism is irrelevant to mathematics. It is a philosophical dead end of no interest. > Please read: > I gave a bijection between the set of nodes and the set of branching- > offs of paths bunches. You have not given a bijection. or surjection, from the set of nodes to the set of paths for a CIBT, and cannot. > > In each node the number of path bunches increases. I am not at all certain what "path bunches" are supposed to signify, But at each successive node, the number of paths through it must either decrease or stay the same. In finite trees, it is easy to see that this number decreases, but in CIBT's it stays the same, so the number of path bunches is irrelevant to the number of paths. > If we add one bunch Bunches are irrelevant to the number of paths.
From: Virgil on 10 May 2007 15:47 In article <1178808886.504395.245480(a)w5g2000hsg.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 04:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1178722003.108344.38...(a)y80g2000hsf.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > > On 9 Mai, 03:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > > Sometimes it is preferable to have a calm atmosphere. > > > > > > > > I have first to sign up, next I get probably an interface that can > > > > only be > > > > handled with a mouse. No, thanks, I have already too many problems > > > > with > > > > mouse use. > > > > > > I have got the same interface as here. > > > > I have not. I do not use a web-browser for discussions. But you > > apparently > > do not understand the principles of Usenet newsgroups. > > To a certain extent you are right. I am not an expert in IT. But, > nevertheless, it was claimed that I had influenced some remote > computer! (I am glad if my own does what I want it to do.) > > > > > 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 of some use to putzers.
From: Virgil on 10 May 2007 16:04 In article <1178815921.746264.51330(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 04:10, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1178722674.540579.289...(a)e51g2000hsg.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > > On 9 Mai, 04:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > > Even in the *infinite* tree a path cannot be distinguished from all > > > > > other paths. > > > > > > > > Why not? For each other path there is a finite node where it goes a > > > > different way. > > > > > > Obviously not. For each node there is a co-path p' which has been with > > > p all the way long. > > > > Where is the contradiction? > > If you say > 1) for every path p', there is a node where p' leaves p > and > 2) there is no node where every path p' has left p Quite so. Every path in a CIBT is essentially an endless binary string of left/right branchings, with every possible such string appearing in the CIBT. > then you must be able to name a node K(n) such that for nodes K(m < n) > at least two paths p' and p'' are required to accompany p. > Because, if for all nodes which you can enumerate, one path p' is > sufficient to accompany p, why should anybody share your belieflthat > for all paths you can enumerate, not one path p' is sufficient to > accompany p? Cutting off the same finite number of left/right branchings from the head of all path strings leaves a set bijectable with the original. So WM's weird imaginings and "path bunchngs" are irrelevant to counting paths. > > > > > > > That is simply opion. Indeed, only countably many can be described by > > > > a > > > > finite formula (as is clear from the work of Turing). In mathematics > > > > that > > > > does *not* mean that the other numbers do not exist. > > > > > > How do you address, represent or use it in any other form "in > > > mathematics"? > > > > By having a set of it. Why is representation needed? > > Because you cannot use a number without address (= representation). > You can only use the set. And if the set is enough to represent it thent the set represents it. > > > > > > > Cantor's diagonal proof fails, because the diagonal number is never > > > > > distinguished from all other real numbers (if uncountably many real > > > > > numbers exist). > > > > > > > > But that is not the proof. The proof is that the diagonal number is > > > > distinguished from the real numbers in the list (which are countably > > > > many). > > > > > > That's your (and others') error. > > > > Did you ever correctly read the proof? > > More than once.And I wonder, why the case 1 = 0.999... must be > excluded, if only finite positions played a role. Why should numbers containing 0's or 9's be relevant to a diagonal rule which prohibits 0's and 1's. The diagonal is prohibited by that rule from ever equaling any number with dual representation. > > > > > The proof in the tree is, that for > > > every path p there is another path *existing in the tree* which is not > > > different from p. > > > > Nonsense, if two paths are different there is a node where they differ. > Do you believe that there is a real number between 0 and 1, which is > not represented in the tree? No. And some are represented twice. > > > > > And if you apply Cantor's proof in the tree, by > > > forming a "diagonal" by switching a bit for every path, then it is > > > undisputed that the constructed diagonal number is represented by a > > > path in the tree. > > > > I need a list of paths before I can even try to start this. The set of > > paths is not a list. So, first, provide me with a list of paths and > > I will come up with a path not in the list. > > > > > Why should this be different in Cantor's list? > > > > Because it is a list. A tree is not a list. > > It is possible to change a digit of every number, because the nodes of > the tree are a countable set. You can even change as many nodes as you > like (also in a decimal tree) and construct a number from the changed > tree. That number which you construct has already been in the tree. Present is with a list of all paths in a CIBT or cease claiming you can do so. It is easy to present a bijection between the set of paths in a CIBT and the set of infinite binary strings (set of all mappings from N to {L,R}) which Cantor has proved uncountable.
From: WM on 10 May 2007 16:06 On 10 Mai, 13:52, William Hughes <wpihug...(a)hotmail.com> wrote: > On May 10, 6:32 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > <snip> > > > Give me two levels for > > which two different paths are required. > > There do not exist two levels for which two different paths > are required. And this is true for all levels which belong to the tree and for all nodes any path consist of. > > Therefore > > Any finite set of paths can be replaced by a single path. > > Look! Over There! A pink elephant! > > Any infinite set of paths can be replaced by a single path. If you say 1) for every path p', there is a node where p' leaves p and 2) there is no node where every path p' has left p then you must be able to name a node K(n) such that for nodes K(m < n) at least two paths p' and p'' are required to accompany p. Because, if for all nodes which you can enumerate, one path p' is sufficient to accompany p, why should anybody share your belief lthat for all paths you can enumerate, not one single path p' is sufficient to accompany p? Note: Finity or infinity does not play a role. The only relevant property is: "for all nodes which you can enumerate". Your swollen speech of "hark, the angels singing: an INFINITE SET !!! is present" would better be expressed by "abracadabra". Regards, WM
From: Virgil on 10 May 2007 16:10
In article <1178817355.571399.114020(a)y80g2000hsf.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 10 Mai, 04:20, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1178723452.432944.32...(a)e51g2000hsg.googlegroups.com> WM > > <mueck...(a)rz.fh-augsburg.de> writes: > > > On 9 Mai, 15:50, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > But that is not the definition, their formal definition is: > > > > "A binary relation F is called a function (or mapping, > > > > correspondence), if > > > > aFb_1 and aFb_2 imply b_1 = b_2 for any a, b_1, and b_2." > > > > > > And how do you think the a's and b's are selected? And from what sets > > > do you think are they selected? > > > > The first thing you should do is find how they define "binary relation". > > But if you skip introductory material you can be lead to errors. > > Be sure, I studied every page very carefully, so carefully that I > typed every letter (some time ago). Some results are available here. > > 2. RELATIONS > > Mathematicians often study relations between mathematical objects. > Relations between objects of two sorts occur most frequently; we call > them binary relations. > ... > A binary relation is, therefore, determined by giving all ordered > pairs of objects in that relation; it dos not matter by what property > the set of these ordered pairs is described. > > !!!!!!!!!!!! But it obviously does matter *that* it is > described. !!!!!!!!!!!!!!! That depends on what one wants to do with the set. In dealing with sets of ordered pairs in general, one need not describe it as anything but a set of ordered pairs. > > > > > > It is ridiculous to > > > follow this discussion. I can assure you, if one of my students would > > > not know that a function is a formula (or rule or whatever) together > > > with a domain where it is defined and a range, then he or she would > > > not pass the exame. And this is the same in the better math courses in > > > Germany. > > > > If that is true, it tells us a lot about mathematics education in Germany. > > Functions in set theory are *not* the same as functions in analysis. > > I talked about mathematics, not about set theory. Actually, much of what WM claims as mathematics, is not, but is merely his personal pseudomathematical pseudophilosophy. |