Cantor Confusion
in [Math]
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From: William Hughes on 10 Apr 2007 14:52 On Apr 10, 2:32 pm, mueck...(a)rz.fh-augsburg.de wrote: > Do all paths exist separated from one another in the infinite binary > tree or not? Yes, but no level separates an uncountable number of paths. i. No level of the infinite binary tree separates an uncountable number of paths. ii. The union of all countable levels is the union of countable levels. And we get the usual logical jump iii. The union of all countable levels does not separate an uncountable number of paths Which bit of "i and ii do not imply iii" do you fail to understand? - William Hughes
From: Virgil on 10 Apr 2007 15:42 In article <1176230179.923106.250510(a)q75g2000hsh.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 7 Apr., 20:54, Virgil <vir...(a)comcast.net> wrote: > > In article <1175967792.541552.222...(a)q75g2000hsh.googlegroups.com>, > > > > > > Not by definition, but by nature. > > > > > > Natural numbers have nothing to do with nature. From the definition it > > > > immediately follows that all natural numbers are finite. > > > > > Without nature (or reality, as we say today) there would not be any > > > natural number. > > > > Without nature, there would not be any people to argue about natural > > numbers, but without people's imaginations, there would be no place for > > those naturals to exist.- Zitierten Text ausblenden - > > The solar system would have eight planets with and without any > people's imaginations. Perhaps nobody would call the numer "e�ght". > But it would be the very number which is also represented by XXXXXXXX > or by oooooooo . Without anyone to decide which objects are planets and which are not, there can be no such thing as counting them. And even with people, the "number" of planets seems a bit variable. So WM's example argues against him.
From: Virgil on 10 Apr 2007 16:02 In article <1176230527.931928.12420(a)y5g2000hsa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 7 Apr., 21:21, Virgil <vir...(a)comcast.net> wrote: > > In article <1175968689.240804.64...(a)e65g2000hsc.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 4 Apr., 16:00, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > In article <1175681058.802334.28...(a)e65g2000hsc.googlegroups.com> > > > > mueck...(a)rz.fh-augsburg.de writes: > > > > > > > On 2 Apr., 14:43, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > ... > > > > > > > And if the series is absolutely converging, then you can > > > > > > > exchange > > > > > > > all > > > > > > > terms you like. The result is independent of the order. > > > > > > > > Not arbitrarily. The result must also be a sequence. If I start > > > > > > with > > > > > > the > > > > > > sequence (1, 1/2, 1/4, 1/8, ...) I have a convergent series. If I > > > > > > apply in sequence the transpositions (1 2), (2 3), (3 4), ... I do > > > > > > not > > > > > > know what I get because it is not defined in mathematics. On the > > > > > > other > > > > > > hand, the only reasonable definition for a result I can come up > > > > > > with > > > > > > is > > > > > > the sequence (1/2, 1/4, 1/8, ...), because there is *no* place for > > > > > > the > > > > > > element "1". So the sum is changed. > > > > > > > You can mirror the series without change of value. You can decide to > > > > > write from right to left. > > > > > > Yes. In that case the mirroring is just virtual and the first number > > > > is > > > > on the right. But that means that the '1' applies to the *first* node > > > > in the path. I do not think you want that... > > > > > The mirror is not "just virtual" unless you claim that the infinite > > > paths exist only virtually. > > > > All mathematics is equally virtual. Finite trees and their finite paths > > are as virtual as infinite trees and their infinite paths, as they exist > > only in imaginations. > > Wrong. The finite tree > 0. > /\ > 0 1 > > does exist here, on your screen. That is only a picture of a tree, and is no more an actual tree than a picture of a triangle is an actual triangle. 'Actual' trees, like 'actual' triangles are only actual in one's imagination. > > > > You take them all at once, by taking an "arbitrary" entry. As the > > > > entry is > > > > arbitrary the proof goes for each and every entry, so in effect you > > > > have > > > > proven it for all entries. > > > > > Wrong. It has been proven always only for a finite part of the list. > > > > If one proves something true for ALL naturall numbers, WM has just > > claimed that there are natural numbers to which that proof does not > > apply! > > > > Which members of the N does such a proof not cover, WM? > > Cantor's proof does not cover the number behind the last one proved. Since the proof covers all members of N simultaneously, there is no "last one" > Same as: > The binary tree does not contain all paths, because otherwise it must > contain a level with uncountably many nodes. Not in my world: (1) A tree without all paths is not actually a tree at all. (2) each partition of WM's set of countably many node levels into an ordered pair of two sets determines a path in which the first set of the pair contains those nodes levels from which the path branches left and the other contains the set of node levels from which the path branches right. This works equally well for finite or infinite trees. For a finite tree with n levels, excluding leaves, this produces 2^n paths. For an infinite tree with aleph_0 levels, there being no leaves, this produces 2^aleph_0 paths. That's how it works in mathematics, including ZF and NBG and most other systems. If that is not how it works in WM's system, then WM's system is no part of any standard mathematics.
From: Dik T. Winter on 10 Apr 2007 21:17 In article <1175967715.832820.167690(a)q75g2000hsh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 4 Apr., 15:47, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > > Such laws show that there is no separated path without a separation > > > point. > > > > Indeed. But that does *not* prove your statement. As I have said time > > and time again, for every two paths there is a single node where they > > separate. On the other hand, there is not a single level where all the > > paths are separated. > > Do you claim that all paths exist as separated entities? If so: how or > where do they exist? Philosophy? I would expect mathematics in this newsgroup. > Can you explain what you mean by existence of > real numbers at all? Dedekind-cuts, Cauchy-sequences, whatever you want. > How are they related to separated paths? Each path is (in principle) a Cauchy-sequence. > Are the > numbers represented by paths which are completely within the > infinitely many levels of the tree or not? This sentence is quite difficult to pars. Yes, each real number is represented by a path which is completely within the infinitely many levels of the tree. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 10 Apr 2007 21:29
In article <1175967792.541552.222680(a)q75g2000hsh.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 4 Apr., 15:48, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > > > If you unite natural numbers, you cannot get an infinite number but > > > > > only an infinite set of natural numbers. Why is this different for > > > > > paths? > > > > > > > > Careless again. You get an infinite number, but not an infinite > > > > natural number. The reason is simply that natural numbers are > > > > finite by definition. > > > > > > Not by definition, but by nature. > > > > Natural numbers have nothing to do with nature. From the definition it > > immediately follows that all natural numbers are finite. > > Without nature (or reality, as we say today) there would not be any > natural number. And, so what? If I now define that the natural numbers are the ordinal numbers, there is nothing wrong with that. It is just contrary to common nomenclature. So, no, finiteness of the natural numbers is *not* because of nature, but it is because of definition. I may put in your mind another system where '0' is also called a natural number. Mathematicians use terms as they see fit. It is the same with physicians. How red is a red quark? And in what way is it red? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |