Cantor Confusion
in [Math]
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From: Dik T. Winter on 15 Apr 2007 20:52 In article <1176300643.280458.120840(a)e65g2000hsc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 11 Apr., 03:29, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > 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. > > You could define a triangle with four corners. It would be as > meaningful and as possible. Of course. Definitions are not wrong, in principle. But by the common definitions natural numbers are limited to be finite. Just a case of defining things. There is nothing in nature, or reality, that (at least for mathematics) *mandates* that they should be finite. It is their mathematical definition that makes them finite. > > 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? > > Physicists will never look for a red quark because of wavelength > problems. So what is the distinction between a red quark and a blue or green one? Moreover, quarks change colour all the time. Are the terms 'red', 'blue' and 'green' in this case not so much chosen because of intrinsic properties but more based on similarities? But you critique mathematics for this. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 15 Apr 2007 23:38 In article <JGKFCB.J6D(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article <1176300393.408149.165530(a)w1g2000hsg.googlegroups.com> > mueckenh(a)rz.fh-augsburg.de writes: > > On 11 Apr., 03:56, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > > In article <1175968689.240804.64...(a)e65g2000hsc.googlegroups.com> > > > mueck...(a)rz.fh-augsburg.de writes: > ... > > > > The mirror is not "just virtual" unless you claim that the infinite > > > > paths exist only virtually. > > > > > > That is not what I state. In your mirrored sequence you have a first > > > element on the right. Well if you wish to go with the right-left > > > writing people, do so. But that does not change the sequence at all, > > > Only the way you look at it. > > > > In the geometric series there is no last element. Nevertheless you > > think the series has a limit value. > > RIght, there are precise definitions for that. At least when the common ratio (of any term to the previous term) is strictly between -1 and 1. Otherwise the series diverges. But it is in either case irrelevant to any "limits" in set theory.
From: mueckenh on 16 Apr 2007 07:45 On 16 Apr., 02:19, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1176298903.609533.227...(a)d57g2000hsg.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 11 Apr., 03:17, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > 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. > > > > I think that numbers belong to mathematics. > > But the question "where do they exist" is not a mathematical question. If you consider the tree, it is as mathematical as if you consider a "list". > > > > 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. > > > > But not all real numbers coexist in the tree? Only a countable number > > of them is admitted simultaneously? > > Why? There cannot be more separated paths in the whole tree than are points of separation in the whole tree, (unless there is more than one separation per point of separation which, however, can be excluded by the construction of the tree). Regards, WM
From: mueckenh on 16 Apr 2007 07:53 On 16 Apr., 02:43, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1176300393.408149.165...(a)w1g2000hsg.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > If all paths exist simultaneously, then there must exist uncountably > > many in the infinite tree. > > But there are. Without the chance that uncountably many are separated in the whole tree? > > > > So every finite part of the path does *not* > > > mean all parts of the path. > > > > You did not talk about such strange parts before. And this explanation > > does not fit my original statement: > > "Every finite node of a path means the whole path because there are no > > infinite nodes". > > Yes, I never contradicted *that*. > > > Nevertheless, we can agree upon these things if we > > clarify their meaning. Does "Every finite part of the path" mean "the > > whole path"? Does "Every finite part of the path" include or cover > > also that special part you just invented? > > And now you switch again from "finite node" to "finite part of the path". > As "every finite part of the path" would mean to me (without further > definition) a set of parts of paths, I would state: no, because a path > is not a set of parts of paths but (by your own definition) a set of nodes. > And even as a set of parts of paths, "every finite part of the path" does > *not* contain the path itself as an element (if the path is not finite). > What you do mean with the word "cover" escapes me. Every node n is in bijection with the set of nodes from the first to n. The latter is a finite part of a path. If every finite node of a path means the whole path (you never contradicted *that*), then "every finite part of the path" means "the whole path". Regards, WM
From: mueckenh on 16 Apr 2007 08:25
On 16 Apr., 02:52, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1176300643.280458.120...(a)e65g2000hsc.googlegroups.com> mueck...(a)rz.fh-augsburg.de writes: > > > On 11 Apr., 03:29, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > ... > > > > 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. > > > > You could define a triangle with four corners. It would be as > > meaningful and as possible. > > Of course. Definitions are not wrong, in principle. But by the common > definitions natural numbers are limited to be finite. Just a case of > defining things. There is nothing in nature, or reality, that (at least > for mathematics) *mandates* that they should be finite. It is their > mathematical definition that makes them finite. No. It is that they are obtained from natural sets of distinguishable elements. But there are no infinite sets in nature or reality. > > > > 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? > > > > Physicists will never look for a red quark because of wavelength > > problems. > > So what is the distinction between a red quark and a blue or green one? > Moreover, quarks change colour all the time. Are the terms 'red', 'blue' > and 'green' in this case not so much chosen because of intrinsic properties > but more based on similarities? But you critique mathematics for this. Quarks are far too small to emit or absorb light of 700 nm wavelength. So there is no red quark. The colour labels of QCD are only attached because the system of three propertiesf its the observations very well. Regards, WM |