Cantor Confusion
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From: Randy Poe on 10 Nov 2006 07:39 mueckenh(a)rz.fh-augsburg.de wrote: > Daniel Grubb schrieb: > > > >> Since the length of the diagonal equals the length of the first column, > > >> you are also saying that there is some line whose length is greater than > > >> or equal to the length of the first column. Is that correct? > > > > >The diagonal of actually infinite length omega cannot exist without the > > >existence of a line of actually infinite length omega. > > > > Why not? Do you have a proof of this claim? > > The diagonal of a matrix is defined as consisting of elements of this > matrix. For a diagonal longer than every line (or every column) this > is impossible. Is that supposed to be an argument? Here's a matrix: [1 2] [3 4] Here are some things composed of elements of that matrix: [4 3 1 2] [1 1 4 3 2 4] So how again does "consisting of elements of this matrix" imply "can't be longer than any line"? Could your "argument" be missing a few steps? - Randy
From: William Hughes on 10 Nov 2006 07:42 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David Marcus schrieb: > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > David Marcus schrieb: > > > > > > > > > > > > > > > The set of natural numbers is an infinite set that contains only finite > > > > > > > > numbers. > > > > > > > > > > > > > > Please do not assert over and over again this unsubstantiated nonsense > > > > > > > (this word means exactly what you think) but give an example, please, > > > > > > > of a natural number which does not belong to a finite sequence. If you > > > > > > > cannot do so, then it is obviously unnecessary to consider N as an > > > > > > > infinite sequence, because all its members belong to finite sequences. > > > > > > > > > > > > I didn't say anything about sequences, finite or otherwise. So, your > > > > > > request is irrelevant to my statement. > > > > > > > > > > The sequence of natural numbers is not comprehensible in ZFC? Neither > > > > > is the sequence of partial sums of a converging series? Nor are the > > > > > finite sequences which are called (initial) segments of sequences which > > > > > are ordered sets. Also the expression "extended sequence" for an > > > > > uncountable ordered set is new to you? > > > > > > > > Non sequitor. > > > > > > ? > > > I did not yet conclude anything but asked some questions. > > > > > > > > Let's make it simple. I'll give a statement and you say whether you > > > > think it is provable in ZFC. Is > > > > > > > > The set of natural numbers is infinite > > > > > > > > provable in ZFC? Please answer "Yes" or "No". > > > > > > > > > > Don't you think that you should label all your posts as > > > > > > "NON-STANDARD MATHEMATICS"? > > > > > > > > > > Cantor invented omega and defined omega as a whole number. > > > > > Who changed this standard meaning? > > > > > Why do you think this meaning was changed? > > > > > When do you think the contrary meaning became standard? > > > > > What is the contrary meaning? > > > > > Do you agree that A n: n < omega is incorrect? > > > > > If not, why do you complain about non-standard meaning on Cantor's > > > > > definition of omega as a whole number? > > > > > > > > Since Cantor predates axiomatic set theory, if you write anything that > > > > uses Cantor's definitions without checking whether the definitions are > > > > still standard is "Non-Standard Mathematics". > > > > > > Therefore I put above list of questions in order to find out what your > > > understanding of the standard is. If you say: My position is standard, > > > that is fine for you, but it is not sufficient to show anything but > > > orthodoxy. > > > > > > > If you want to discuss > > > > history, that is fine, but you should label your posts as such. This is > > > > simple courtesy. If you use words without defining them, readers assume > > > > you are using them in their current meanings. If you are using > > > > historical meanings, then either say so or use a different word. > > > > > > > > As for the current definition of omega, Kunen's book is a good > > > > reference. According to Kunen, omega is not a natural number. > > > > > > That is out of any question. > > > > > > > I'm > > > > guessing that by "whole number" you mean natural number, but I really > > > > don't know, since you seem to have your own language for everything and > > > > you never give definitions for any of the words that you use. > > > > > > "Whole number" is Cantor's name for his creation. > > > > > > My question is : Do you maintain omega > n for all n e N? I know that > > > modern set theory says so. If something can be larger than a number, > > > then it must be a number. > > > > Omega is not an element of N. However you can compare omega with > > any element of n. > > Therefore we can compare the diagonal with every line. We find that the > diagonal is longer than every line. Yes. However. as the length of the diagonal is the supremum of the lengths of the lines, this is not a contradiction. - William Hughes
From: David Marcus on 10 Nov 2006 08:17 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > That is completely beside the point. You represented that you were > > giving us a set theoretic proof, > > Never! I gave a mathematical proof. That has nothing to with set thery. If you think that your "mathematical proof" is valid and you prove something that is absurd, this only shows that your mathematical reasoning leads to absurdities. Everyone who uses mathematical reasoning consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, you will stop proving absurdities. > We need no experts in trees. This tree does nothing else but represent > the real numbers of an interval. Here it is in a form which can be > understod by ever mathamatician. > > Consider a binary tree which has (no finite paths but only) infinite > paths representing the real numbers between 0 and 1 as binary strings. > The edges (like a, b, and c below) connect the nodes, i.e., the binary > digits 0 or 1. > > > > 0. > > /a \ > > 0 1 > > /b \c / \ > > 0 1 0 1 > > .......................... > > > > The set of edges is countable, because we can enumerate them. Now we > set up a relation between paths and edges. Relate edge a to all paths > which begin with 0.0. Relate edge b to all paths which begin with 0.00 > and relate edge c to all paths which begin with 0.01. Half of edge a is > inherited by all paths which begin with 0.00, the other half of edge a > is inherited by all paths which begin with 0.01. Half of the paths include edge a. One quarter of the paths include edge b. Suppose we have a finite tree of height n. Then there are 2^n paths. Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the paths. We can define a real-value function on paths by g(e1,e2,...,en) = sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... > Continuing in this > manner in infinity, we see by the infinite recursion > > f(n+1) = 1 + f(n)/2 What is f(n)? > with f(1) = 1 that for n --> oo > > 1 + 1/2 + 1/ 4 + ... = 2 > > edges are related to every single infinite path which are not related > to any other path. (By the way, the recursion would yield the limit > value 2 for any starting value f(1).) The load of 2 edges is only > related to infinite paths because any finite segment of a path with n > edges will carry a load of > > (1 - 1/2^n)/(1 - 1/2) < 2 > > edges. The set of paths is uncountable, but as we have seen, it > contains less elements than the set of edges. Except your reasoning appears murky at best and probably invalid in standard mathematics. > > But what absurdity? Not a contradiction IN set theory. > > That is unimportant. An absurdity remains so, whether in or outside of > set theory. Indeed. That is why no one wishes to use your reasoning: it leads to absurdities. If we stick to standard mathematical reasoning, we don't get your absurdities. -- David Marcus
From: Bob Kolker on 10 Nov 2006 08:51 Dik T. Winter wrote: > > What surprises me is that opposition against it in this way comes from > especially the German posters in this group. They all want to stress > the difference between actual infinity and potential infinity, that > are indeed not mathematical terms, and most other posted do not know > the difference. Is it still the old religious issue? I think so, > based on the postings of at least one of the German posters. If Cantor's last name had been Shickelgrueber or von Richthoffen, none of this would be written. Bob Kolker
From: Franziska Neugebauer on 10 Nov 2006 09:32
David Marcus wrote: > [...] you are working with an infinite triangle [...] ,----[ http://en.wikipedia.org/wiki/Triangle ] | A triangle is one of the basic shapes of geometry: a polygon with | three vertices [...] `---- Are there really three vertices in WM's "triangle"? F. N. -- xyz |