Cantor Confusion
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From: Virgil on 18 Dec 2006 02:11 In article <1166420023.961801.207150(a)73g2000cwn.googlegroups.com>, "Newberry" <newberry(a)ureach.com> wrote: > Sorry that I joined a bit late. Are you saying that (in an infinite > binary tree) the set of paths is uncountable but the set of edges is > countable? That is certainly the case within set theories like ZFC or NBG. But for those whose versions of set theories prohibit anything but finite sets, there can be no infinite binary trees anyway.
From: Virgil on 18 Dec 2006 02:17 In article <1166423927.544160.38800(a)80g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > You misinterpret L_D. L_D is that line which contains all numbers > contained in the diagonal. What makes you assume that there is any such line? In ZFC and NBG, every line is finite but the diagonal, being in a sense the union of infinitely many finite lines of ever greater sizes, is not finite. So now WM is requiring existence of a finite set, L_D, having infinitely many members?
From: Virgil on 18 Dec 2006 02:35 In article <1166424789.737230.23790(a)f1g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > You have claimed that you can use the binary tree and your "rational > > relation" to prove this. This proof should be independent of any > > /other/ proof that the reals are countable. > > And so it is. Not by mathematical standards. Every attempt WM has made at such a proof has presumed statements contradicting those of both ZFC and NBG. > > > > > Your edges are not > > > > being broken into 2 "shares" each; they are being divided into an > > > > infinite number of shares, and that is very different from 2 shares! > > > > > > For this sake we have mathematics, I mean really correct mathematics, > > > not based on dubious infinite sets, which states that 1 + 1/2 + 1/4 + > > > ... = 2. > > > > > > > "Really correct mathematics" doesn't simply /state/ that "1 + 1/2 + 1/4 > > + ... = 2". Mathematics /defines/ what the series of symbols "1 + 1/2 + > > 1/4 + ... = 2" means, and then /proves/ that it is true from these > > definitions. > > > That is the nonsense en vogue today. Really correct mathematics > computes that the sum is 2. Except that mathematicians say that it is not correct mathematics, much less "really correct" mathematics, and only such non-mathematicians as WM claim that it is correct mathematics. I prefer the judgements of real mathematicians on whet "really correct mathematics" is. > > > Similarly, I ask that you don't simply /state/ "there is a rational > > relation which is a surjection of edges onto paths". I ask that you > > /define/ what that statement means; and then /prove/ that that > > statement is true from those definitions. > > I define and proved. Neither defined not proves in any mathematically way. > That I did not use the language of set theory is > because this theory has no value. That you did not use either the rigor of definition or the rigor of logic required by mathematics makes your language mathematically invalid. > > > > > So the above "drawing" is /not/ sufficient to say "(C) proves a > > surjection"; we must /also/ know the /sizes/ of the shares, as well as > > "what they are mapped to", before we can say "we have proven a > > surjection". > > O course. In my appliation of his technique the shares are equally > divided. Then you are dividing each edge into infinitely many shares. > Shares of different sizes are a complication which I did not > introduce in order to keep things as simple as possible (and because > they are not required for my purposes). Countably many shares of one edge each of equal size are impossible. > We do not need this assumption. We need not talk about R at all as long > as arguing in the tree. We argue only > 1 + 1/2 + 1/4 + ... = 2. > Nothing else! As that proves nothing at all, besides itself. WM's arguments fail again. > > It is you who tries to doubt this simple and clear formula on the > grounds of unjustified assumptiobn about real numbers. It is not the correctness so much as the relevance of the formula which we question. It does not provide any method for mapping the set of edges (or the set of nodes) onto the set of paths of an infinite binary tree. WM keep claiming the ability to do it and failing to deliver on that claim.
From: Han.deBruijn on 18 Dec 2006 03:20 William Hughes schreef: > Han de Bruijn wrote: > > William Hughes wrote: > > > > > Han de Bruijn wrote: > > > > > >>William Hughes wrote: > > >> > > >>>Han de Bruijn wrote: > > >>> > > >>>>William Hughes wrote: > > >>>> > > >>>>>Han de Bruijn wrote: > > >>>>> > > >>>>>>Let's repeat the question. Does there exist more than _one_ concept of > > >>>>>>infinity? Isn't unbounded the same as infinite = not finite = unlimited > > >>>>>>= without a limit? Please clarify to us what your "honest" thoughts are. > > >>>>> > > >>>>>You are *way* in deficit on clear answers. Try answering > > >>>>>the following question with yes or no. > > >>>>> > > >>>>> Is there a largest natural number? > > >>>> > > >>>>No. > > >>> > > >>>I there an unbounded set of natural numbers? > > >> > > >>Suppose you mean "Is". What does it mean that a set is unbounded? > > > > > > An unbounded set of natural numbers is a set of natural > > > numbers that does not have a largest element. > > > Please answer yes or no. > > > > No. > > Does the "potentailly infinite set" of natural numbers > exist? A potentially infinite set is a function on sets > that takes on the values true and false. The > potentially infinite set of natural numbers > takes on the value true for a set containing only > natural numbers, and false for any other set. > (i.e. is there a way of recognizing a set of > natural numbers?) I don't understand this question. Han de Bruijn
From: Han de Bruijn on 18 Dec 2006 03:41
William Hughes wrote: > Does the "potentailly infinite set" of natural numbers > exist? A potentially infinite set is a function on sets > that takes on the values true and false. The > potentially infinite set of natural numbers > takes on the value true for a set containing only > natural numbers, and false for any other set. > (i.e. is there a way of recognizing a set of > natural numbers?) I don't understand the question (but maybe already posted this). Han de Bruijn |