Cantor Confusion
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From: Virgil on 1 Feb 2007 14:30 In article <1170331739.136815.10990(a)v33g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 20:40, Virgil <vir...(a)comcast.net> wrote: > > In article <1170157336.490678.144...(a)v45g2000cwv.googlegroups.com>, > > > > > > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 28 Jan., 20:44, Virgil <vir...(a)comcast.net> wrote:> > In fact? If > > > there is a path of lengths n then there is a path of > > > > > length n+1. And there is a path of length 1. What is the length of > > > > > the union of all these paths (which contains only finite paths)? > > > > > > If that union is to be regarded as a path at all > > > > > Yes, please regard it as such! > > > > > > then it cannot be a > > > > finite path as it would have to be be longer than every finite path > > > > (for every path of length n, the union wold have to be of length at > > > > least n+1 ). So its length is not finite. > > > > > On the other hand, it must not be longer than every natural number > > > because it is simply the union of all natural numers. > > > > So that WM claims something that must be longer than any natural but can > > not be longer than every natural? Good luck finding it! > > It has been known for centuries. Something which must be strictly larger than itself? Not hardly! At leas not in any actual mathematical world. WM apparently lives in a looking glass world where the red queen reigns.
From: Virgil on 1 Feb 2007 14:34 In article <1170332245.102511.34970(a)v33g2000cwv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 30 Jan., 21:50, Virgil <vir...(a)comcast.net> wrote: > > In article <1170164725.794981.290...(a)a34g2000cwb.googlegroups.com>, > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > > > Continued > > > > > > 2. BECAUSE it is _undefined_ until the "sum of all natural numbers" is > > > > defined. > > > > > This sum is defined by > > > I > > > II > > > III > > > ... > > > > WRONG! As usual. > > > > A listing of all natural numbers is not the same as the summing of all > > natural numbers, and WM has, at most, produced a listing. > > The sum is a listing: II + III = IIIII = 1,2,3,4,5. > > The sum of all natural numbers can be obtained (if all natural nunbers > exist) by the following "diaogonalization" > > 1 > 23 > 654 > ... > It is countable. WM is using some definition of "sum" incompatible with any standard meaning of the word. For a "sum" of infinitely many numbers to exist in mathematics, the sequence of finite "partial sums" must converge, which no sequence of positive naturals can do.
From: Virgil on 1 Feb 2007 14:36 In article <1170332611.025673.158500(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 1 Feb., 13:12, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > On 30 Jan., 17:18, Franziska Neugebauer <Franziska- > > > Neugeba...(a)neugeb.dnsalias.net> wrote: > > > > >> >> 2. BECAUSE it is _undefined_ until the "sum of all natural > > >> >> numbers" is defined. > > > > >> > It is defined if the set of all natural numbers s defined, > > > > > Here is the definition of the set N > > > > > 1 > > > 2 > > > 3 > > > ... > > > > > Here is the definition of the sum of all elements of N > > > > > 1 > > > 23 > > > 456 > > > ... > > > > Abstract art? > > > > F. N. > > -- > > xyz- Zitierten Text ausblenden - > > > > - Zitierten Text anzeigen - > > For people with less capabilities of abstract thinking: Imagine a rope > and then thread the numbers: > > 1 > 32 > 456 > ... > > Better now? Still not a sum.
From: Virgil on 1 Feb 2007 14:38 In article <1170337216.102374.272080(a)k78g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 1 Feb., 13:38, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > On Feb 1, 7:02 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > No, the potentially infinite set of natural numbers exists. > > Yes, but it does not exist actually. It exists as actually as the number 1 exists. Nothing of mathematics "actually" exists, it is all in the mind, and in the minds of mathematicians, N can be as actual as 1.
From: Virgil on 1 Feb 2007 14:47
In article <1170337797.588981.135470(a)q2g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 1 Feb., 14:09, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1170328870.612961.224...(a)v33g2000cwv.googlegroups.com> > > > Because there is nothing else in the union but natural numbers. > > > You cannot get an apple if you collect nuts. > > > > The union of all finite natural numbers is not a natural number. Why > > do you think it is? > > The union of paths is a path. The projection of a finite number on a > pathlength remains a finite number. The projection of all finite > numbers on a pathlength remains a finite number. Are these perpendicular projections? Why are you projecting numbers on paths? Each finite path is a set of a finite number of nodes and/or edges. Unions of them are also sets of nodes or edges, but need not be finite. > > > > > Nevertheless, if you assume that the length of the union of all paths > > > (not the cardinal number !) is infinite, then you advocate infinite > > > number sizes are required for an infinite union of numbers. No more so that the union of all finite naturals being an infinite ordinal but not a natural. > It is only unfamiliar to think about that. > > > But indeed, the infinite union of > > all natural numbers (considered as sets) is not a natural number. > > Maybe. But the natural numbers consideed as paths remain natural > numbers in any case. The union of all of them doesn't > > > On the other hand many people assert that there is an infinite union > > > of finite pathlengths without containing or establishing an infinite > > > pathlength. > > > > Eh? Who? > > Everybody who says that there is "an infinite set of finite > pathlengths". On the contrary, those are exactly the people who insist on having an infinitely long path. It is WM who maintains otherwise. > > Regards, WM |