Cantor Confusion
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From: Andy Smith on 28 Jan 2007 05:05 G. Frege <nomail(a)invalid.?.invalid> writes >On 27 Jan 2007 21:10:06 -0800, davidmarcus(a)alum.mit.edu wrote: > >>>> >>>> I think that Cantor's diagonalisation argument has a terminal flaw. >>>> >>> You have earned crank status now. Congratulations! >>> >> It would seem so. Andy keeps posting the same argument without taking >> into account the comments made on his previous posts. >> >Right. I've also noticed that. Btw, a rather interesting experience: >to meet a crank in statu nascendi! > Fair enough. But most cranks are probably convinced that they are right .... -- Andy Smith
From: G. Frege on 28 Jan 2007 05:16 On Sun, 28 Jan 2007 10:05:25 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: > > [...] most cranks are probably convinced that they are right > That's for sure! :-) http://en.wikipedia.org/wiki/Crank_(person) F. -- E-mail: info<at>simple-line<dot>de
From: mueckenh on 28 Jan 2007 07:19 On 27 Jan., 20:27, Virgil <vir...(a)comcast.net> wrote: > In article <1169915856.541369.8...(a)p10g2000cwp.googlegroups.com>, > > > The name is not the thing named. > > > If there really is a *thing*, this may be true. > Even if there isn't, it is true. Because a name which does exist cannot > be a thing which does not exist. But then the name is all that the name describes. > Depends on how it is stated. If one's induction is of the form: > There is a set S such that > (1) The first natural is a member of S, and > (2) The successor of every member of S is a member of S > Then one's conclusion should be that N is a subset of S. 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)? > > >I have a different understanding of what concrete means. Do you mean the > > > concrete in your head? > >You seem to know only one aspect of many things which have more. >> There are three meanings in English: noun, verb and adjective. > There are also prepositions and adverbs, etc. Adverb? Yes. But what, concretely, means the preposition "concrete"? > A set is finite if there does not exist any injection from it into > any of its proper subsets. Now apply this to paths. C> Not as a member, but the union itself may be a non-natural number as > bnoted above. And how about the paths of a tree? > Fact is that p, like N as the first limit ordinal, can be a set not > satisfying any definition of finiteness. The length of p is the union of the lengths of all finite paths. > What about the fact that it isn't a fact? Do you complain? You are familiar with facts that aren't. > Equivalently, a path in a tree is a maximal sequence of linked nodes in > that tree with each successive node being the child of its immediate > predecessor node. And this definition, which is equivalent to mine, does not tolerate different complete sets of paths in one and the same tree. > It is the maximality in its given tree that is essential for a path, as > anything less is not a path in that tree. Correct. Think about it. > Any set which can be listed in its entirety is, by definition cuntable > and any set which is not so listable is, by definition, uncountable. > Anything else is irrelevant. Where can I see the set N be listed in its entirety? > Cantor showed with two entirely different proofs that the reals cannot > be listed. So Andy is talking nonsense. Everybody not completely mindbended knows that the naturals cannot be listed too. > Already False! There is no (finite) binary index for any of those > uncountably many reals whose binary expansions require infinitely many > 1's. For example 1/3. There is a finite binary index to any listed sequence of the form 0.010101 [...] 01 which ever will be written. Regards, WM
From: mueckenh on 28 Jan 2007 07:20 On 27 Jan., 22:52, Franziska Neugebauer <Franziska- Neugeba...(a)neugeb.dnsalias.net> wrote: > mueck...(a)rz.fh-augsburg.de wrote: > > I am interested in the fact that every set of natural numbers has a > > finite maximum.Then you should perhaps not talk to contemporary set theorists who are > accustomed to the fixed idea being far from being a > fact that there *are* sets of natural numbers which do > not have maxima at all. > > F. N. > -- > xyz
From: mueckenh on 28 Jan 2007 07:21
William Hughes schrieb: > You claim that if something is true for every set L_n, > then it is true for N. I do not talk about N. This symbol has become the aim of heaviest abuse. > > We know something about the maximum that is true > for every set L_n. > > So you do want to prove something about the maximum > of the set N. I want to see whether the union of all finite numbers can be an infinite number. This question was raised in the framework of the infinite tree. Set theorists asserted that a union of finite paths cannot be / contain any infinite path. Regards, WM |