Cantor Confusion
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From: mueckenh on 25 Jan 2007 04:53 Virgil schrieb: > In article <1169639256.391493.236600(a)s48g2000cws.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > On 23 Jan., 13:35, Franziska Neugebauer > > <Franziska-Neugeba...(a)neugeb.dnsalias.net> wrote: > > > > > ** The std-union of the set of sets of paths of every finite binary > > > std-tree is not identical to the set of paths of the infinite > > > binary std-tree G. ** > > > > Summary > > > > 1) Every complete infinite binary tree T (containing all nodes and > > edges) contains all paths. > > 2) The union tree T(oo) of all finite trees is well defined (as I have > > shown elsewhere) > > What WM has constructed is not a union of trees. > Call it as you like. Concerning nodes and edges it is the complete tree. > > > > and yields the complete infinite binary tree > > containing all nodes and edges: T = T(oo). > > 3) The union of all finite trees includes the union of all nodes and, > > with it, the union of all such subsets which are paths (because every > > path is a well defined subset of the set of nodes if the structure of > > the tree is well defined). > > 4) The set of paths in T(oo) is a subset of the countable set of finite > > sets of all paths in the finite trees. > > Except that > (1) it contains paths not in any finite tree, > such as paths which evantually alternate directions of branching The union of finite trees cannot contain an infinite path. > (2) It is not countable, as Cantor proved. Compare (5). > > 5) A countable union of countable sets is a countable set (according to > > ZF with AC). > > The set of nodes of the total binary tree is countable and the set of > edges is countable but the set of paths is provably not. All the paths of T(oo) are paths from a finite tree and, hence, from a finite set of paths. > > > > > Going on, we can say: > > > > 6) T(oo) = T contains only finite paths. > Then it does not contain all paths, not even the infinite paths of his > own weeping willow trees. Yes, the union given above it the union over cut trees. > > > 7) T(oo) = T contains all paths including all infinite paths. > > ==> There are no infinite paths. (There are no irrational numbers.) > > That is an irrational conclusion based on an irrational argument. Contrary, the irratinal is expelled from mahematics by this rational argument. > > > > Nothing further remains to say. Regards, WM
From: G. Frege on 25 Jan 2007 04:54 On 25 Jan 2007 01:47:48 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > I am interested whether there are irrational numbers. > Well, fact is, no one has ever seen one of them so far. (On the other hand, the same is true for natural numbers, integers, and rational numbers, too. :-) F. -- E-mail: info<at>simple-line<dot>de
From: mueckenh on 25 Jan 2007 05:23 On 25 Jan., 04:05, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1169637406.690911.177...(a)j27g2000cwj.googlegroups.com> mueck....(a)rz.fh-augsburg.de writes: > On 24 Jan., 04:36, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > About Wolfgang Mückenheim. > > > > > Actually I am getting the impression that what he presents here is not > > > what he presents in the mathematics courses at his University. (A > > > Fachhochschule is called a University in the Netherlands.) > > > > The official name is "University of aApplied Sciences" > > Here something like that is called a "Technical University". But, what is > in a name. Yes, what is a name? (But in Germany there is a difference. )The University of Applied Sciences, Augsburg has 7 Faculties. We educate (produce) Bachelors and Masters in several technical disciplines including Electrical eng. and computer studies, but also business, international management etc. Besides these classical studies there are some new developments. The Faculty of Design and the Faculty of Computer Studies offer a course in Multimedia Studies. There is a course in Environmental Technology and Mechanical Engineering. The Faculty of Architecture and Construction Studies offer a course in Construction and Building Site Management. There is also a range of additional courses including Environmental Engineering Emission Control and Construction Management. A Master of Engineering course is also offered in collaboration with the University of Ulster. > > > > I received > > > his book and I found no errors in the first four chapters I did read (of > > > the ten in all). I am only a bit unlucky about his distinction between > > > (indeed) actual and potential infinity, but that can be clarified later. > > > > See chapter 8. In short: Potential infinite never ends. So Cantor's > > diagonal is never completed. > > I am not yet there. I just read chapter 5. Chaper 8 is a bit tedius, you will see, but I tried to get it as complete as I could. But as a compensation, chapters 9 and 10 are exciting (as the lengh of the present thread also poves). > > > > (*): The reason appears to be that irrationals can only be given by a rule > > > about how to compute it. But I think that: > > > 0.142857142857... > > > is also nothing more than a rule how to compute it. > > > > I agree. This sequence does exist as little as does 3.1415... But > > contrary to the latter, the first number might have an existing > > representation in base 142857 (and others, where trichotomy applies) as > > 0.111.... Nevertheless even that might reasonably be doubted. (See > > chapter 10 and the present discussion.) > > Indeed. You will see more remarks on this in my review. > > > Anyhow, Dik, thanks for the fair report. If you notice misprints or > > errors in the first 8 chapters, please notify me. (Errors which you > > might encounter in the last two chapters are no errors.) > > You think so ;-). But indeed, I try to be fair. Regards, WM
From: Tez on 25 Jan 2007 06:13 Andy Smith wrote: > In message <1169644806.887614.311180(a)13g2000cwe.googlegroups.com>, Tez > <terence.hoosen(a)gmail.com> writes > > [snip my slightly sarcastic example] > > > This is probably aimed at me, fair enough. > > Consider a bit sequence b_n where b_0 is the least significant bit and > b_n = 1 A n e N > > and the number defined by m = b_0 +2*b_1 + ... 2^nb_n ... > > Is m a natural number? (I would say not) Well, loosely I'd say m is not a natural number. More precisely, I'd say the your sum, doesn't converge. The difference is, if I simply say that m isn't a natural number, you might be inclined to infer that the sum is something well-defined, just that it equals some different kind of number. Well, not really. the "m = " part on the left hand side of your expression is misleading if the sum diverges (doesn't converge), which is the case here. Usually, the way these infinite sums (infinite series, technically) are thought about is to consider the infinite sequence whose nth term is the sum of the first n terms of the series, ie, consider c_n = sigma (2^n b_n) for k = 0 to n and determine whether c_n converges. > -- > Andy Smith -Tez
From: Dik T. Winter on 25 Jan 2007 06:26
In article <1169713460.685982.294480(a)k78g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 24 Jan., 15:51, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1169642941.554024.145...(a)k78g2000cwa.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > On 23 Jan., 19:27, Dave Seaman <dsea...(a)no.such.host> wrote: > > > > On Tue, 23 Jan 2007 17:15:52 GMT, Andy Smith wrote: > > > > > > > By definition, a set is "finite" if it has the size of some natural > > > > number. If a set isn't finite, then it's called "infinite". > > > > > > If a colour is not red, then it is called green by set theorists. > > > > Wrong. not-red. > > "Not red" is blue or green in the RGB axiom system. But you exclude > blue (potential infinity), hence you say green (green stands for > "hope", hoping on actually finishing infinity) when you can't see red. Wrong. Not-red is any colour that is not red. It can be blue, green, orange, yellow, purple, or whatever. As long as it is not a colour whose name is red. But apparently you do not know what the prefix "in" in its many forms means. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |