Cantor Confusion
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From: Virgil on 8 Dec 2006 16:52 In article <4579B4C7.8040501(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:14 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> Nevertheless, all these arguments of mine are most likely flawless. > > > > Wow! That has got to be your most absurd statement yet! > > Weak statements invite for refutation. > > Good luck Anyone who claims to be as flawless as EB claims in an area in which he is an untutored amateur, already has made one error.
From: mueckenh on 8 Dec 2006 16:54 Virgil schrieb: [concerning Cantor's first proof of uncountability of the real numbers] > > In the reals, any subset which has a real upper bound has a real least > upper bound and, similarly , any subset which has a real lower bound has > a real greatest lower bound. > > The only subsets of the reals for which there is a similar property are > real intervals. > > Thus it is only for the set of all reals, or for real intervals, that > the proof appplies. Yes. And it does not apply if only one single element of the investigated real interval is missing. As the uncontability property of this interval cannot depend on this single element, the whole proof fails. > > On the other hand, the proof can show the uncountability of a countable > > set. If, for instance, the alternating harmonic sequence > > (-1)^n/ n --> 0 > > is taken as sequence (1), yielding the intervals (-1 , 1/2), (-1/3 , > > 1/4), ... we find that > > its limit 0 does not belong to the sequence, although the set of > > numbers involved is obviously > > denumerable. > > That doesn't work it unless you show that NO sequence can be made > including all of the values of S = {0} union { (-1)^n/n: n in N} And Cantor's proof doesn't work unless one can show that there is a missing number in every sequence. But just this cannot be done for a real interval with only one real number missing. Regards, WM
From: Virgil on 8 Dec 2006 16:55 In article <4579B60F.1090206(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 8:03 AM, David Marcus wrote: > > Quite right: neither "potential infinity" nor "actual infinity" occur in > > modern mathematics. Time to leave the Antiquarian Bookshop and join the > > 21st century. > > The basic errors mainly date back to 1872 and German megalomania. German megalomania seems well represented here in late 2006, and will, no doubt, continue unabated in 2007.
From: mueckenh on 8 Dec 2006 16:57 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Dik T. Winter schrieb: > >> >> > Everybody knows what the number of ther EC states is. > >> [...] > >> > The number of EC states is "the number of EC states". > >> > >> This is hardly a definition. > >> > >> > It is simply a notion which can be equal to a natural number. > >> > >> Which may _evaluate_ to a number. > > > > No. It evaluates to a number as little as 6 evaluates to a number. It > > *is* a number, though not a fixed number. > > Mathematically one modells such "not-fixed numbers" as functions. > Conclusively this function has value 6 at 1968. > > > That is a matter of definition of the word "number". > > Provide one. Don't forget to provide a definition of "not-fixed" number > and "not-fixed" set. And please show that one gains advantage over the > function concept. > A function is a set of ordered pairs and as such it is not variable. The expression "variable" is merely a relict from ancient times when people knew that the objects of mathematics do not exist in some nirvana but have to be present in a mind where not everything can be present simultaneously. Regards, WM
From: Virgil on 8 Dec 2006 16:58
In article <4579B665.9010009(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 7:54 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> Notice: Cantor's untennable definition > > > > Which definition are you referring to? > > See an other reply of mine today. None of those relies describe any definition by Cantor that is mathematically or logically untenable. So one must conclude that EB is faking it. |