Cantor Confusion
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From: Virgil on 26 Jan 2007 17:43 In article <1169804234.437199.89870(a)l53g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > S can be and is larger than L. S is not countable. > > > > > That again is an unjustified and provably wrong assertion. > > > The sequences belong to the countable union of the finite trees. > > > > Take any sequence s in S. We know that each initial segment > > of a s belongs to a finite tree t(s). However there is no single > > finite tree t_D such that every initial segment of s belongs to t_D. > > s does not correspond to a single finite tree, > > s corresponds to a sequence of finite trees. > > We are not interested in a finite single tree. Every initial segment s > of a path has lengths n, where n is in N. If the union of all initial > segments of a path is infinite, then you should try to explain how an > infinite number could creep in and can exist unrecognized as a finite > natural number among the natural numbers. In the same way that the union of the nested sequence of finite sets, the naturals, can be a set, N, so the union of a nested sequence of finite paths can be a path. > > > > > > *Everything* in this union is countable. Then WM's definition of "countable" differ from everyrone else's. If, for example, WM claims that the set of all endless binary sequences is countable, he should be able prove it by to providing a surjection from N to the set of all such strings. I, for one, have not seen him do this! And unless he has, his claim that such a set is countable is no more than a conjecture. > > That is a different matter. Each of our sequences has a length n. Their > union is finite (i.e, not acually infinite) unless there is an infinte > number in N, which is impossible. What is WM's definition of "finite? There are two possible ones in general usage: (1) a set, S, is finite if ( and only if) there is an initial sequence of naturals bounded above by a natural number n such that the set. S, can be bijected with this sequence. (2) A set, S, is finite if there does not exist any injection from S to any proper subset of S. Does WM have some other definition to offer which is not equivalent to either of these? If so we deserve to be let in on his little secret. According to either of these definitions, the union of sequences of length n for all n in N is a not-finite set. So what defnition of "finite" is WM using? Or is he just winging it?
From: Virgil on 26 Jan 2007 17:47 In article <1169806028.241439.210810(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > On 25 Jan., 15:57, Franziska Neugebauer > > > <Franziska-Neugeba...(a)neugeb.dnsalias.net> wrote: > > > WM ] > > >> | The union of two natural numbers is defined to be the larger one. > > >> | This is a set theoretic union. > > >> `---- > > >> > > >> So the union of two natural numbers is _not_ defined to be the larger > > >> one but the "larger" of two distint naturals (sets) a and b is > > >> defined to be a iff > > >> > > >> b c a > > >> > > >> or b iff > > >> > > >> b c a. > > >> > > >> So the order _is_ relevant to successfully prove you wrong. > > > > > > The union of two different natural numbers is defined to be the larger > > > one. > > > > Repeating a falsified claim. "Union" in contemporary set theory is > > generally defined (not restricted to natural numbers). This definition > > does by no means use a notion of "larger". > > > > Using e. g. von Neumann ordinals you can define the less than relation > > by the subset-property. Not the other way round! > > Purset nonsense. WM doesn't even spell his falsehoods correctly. > Using the most basic property of numbers, namely their > unary representability, the relation < is defined by c. Everything else > is simulating this fact. Unless "c" means something other than subset, or proper subset, WM is simultaneously denying and agreeing with FN. Wm really is mixed up!
From: Virgil on 26 Jan 2007 17:56 In article <1169806783.611087.268930(a)a75g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 26 Jan., 02:41, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1169740568.082762.116...(a)v33g2000cwv.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > On 25 Jan., 15:57, Franziska > > Neugebauer > <Franziska-Neugeba...(a)neugeb.dnsalias.net> wrote: > WM ] > > > > | The union of two natural numbers is defined to be the larger one. > > > > | This is a set theoretic union. > > ... > > > The union of two different natural numbers is defined to be the larger > > > one. > > > > In the von Neumann model. But that is a model about ordinal numbers, that > > start at 0. There are a few subtle differences. I think you are a bit > > confused here. In the von Neumann model we have that an ordinal number > > 'k' is the set of all its predecessors ( {0, 1, 2, ..., k-1), if k is not > > a limit ordinal). And so the union of two ordinals is the larger one. > > In this case, the ordinal 'k', is also the ordinal number of the set of > > predecessors. When you shift to '1' base, the latter statement is no > > longer true. > > It has nothing to do with sophisticated models at all. They only have > to simulate the basic fact. A natural number in its most basic > representation, namely unary or unadic, simply is the superset of all > smaller numbers and a subset of all larger numbers. That is the origin > which has to be simulated by every correct theory of natural numbers. WM is confusing strings with sets. In his "unary" representation, each positive natural is represented by a string. One still needs a notion of successor in order to order these strings appropriately, and the Peano properties are still necessary to get the system up and running. > II is a subset of III. As sets, this is false, as both sets contain only a single element. > This kowledge as to be taken into account when > denoting these numbers by 2 and 3. It has been done by the notation 2 < > 3. > > Everything which Zermelo or v. Neuman or others had to say about this > topic is not so new as you want to make me believe. It merely repeats > the basics. Which WM apparently has not yet learned, so that repetition of them, at least until WM has learned them, is appropriate.
From: Virgil on 26 Jan 2007 18:06 In article <1169822674.776815.294960(a)q2g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > For natural numbers c is the same as <. For natural numbers defined as equivalence classes of bijectables, they are not at all the same. 2 c 3 is the same as 2 < 3. The class of all sets bijectable with {0,1} is not a subset of the class of all sets bijectable with {0,1,2}. > This is proven by II + I = III. Only in Ancient Rome does II + I = III. >Perhaps some present mathematicians do > not remember these things As we do not often use Roman numerals for anything but labeling monuments and sundials, we have little need to remember. > That is the origin
From: Virgil on 26 Jan 2007 18:09
In article <1169823054.596842.310630(a)j27g2000cwj.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 26 Jan., 11:11, Franziska Neugebauer > <Franziska-Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 25 Jan., 15:57, Franziska Neugebauer > > > <Franziska-Neugeba...(a)neugeb.dnsalias.net> wrote: > > > > >> > If T(oo) as defined by me is T as defined by me, concerning the set > > >> > of nodes and edges, > > > > >>There is no room in equality for "concerning" this or that. Either two > > >> symbols refer to the same entity or they do not. > > > > > That is not so obvious. > > That *is* obvious. > > Indeed? Every function denoted by f is the same one? Even every pair of > functions with f(x_1) = g(x_1) is identical? WM as usual, has hold of the wrong end of the stick. The name is not the thing named. WM could benefit by reading Korzybski. > > However, concerning trees, Virgil is obviously wrong. I think so too. WM claiming to think about anything is a oxymoron. |