Cantor Confusion
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From: mueckenh on 10 Nov 2006 05:43 MoeBlee schrieb: > > This number does not exist. The diagonal has not omega positions. > > So I guess the answer to my question is that you are not prepared to > show that my proof has anything more than first order logic applied to > Z set theory nor are you prepared to demonstrate a contradiction in Z > set theory. Your proof shows either a contradiction in Z or it shows that Z is false. Z contains a contradiction to your proof, if one can show in Z that a diagonal of a matrix has not more elements than every line. Z is false, if one cannot show that. So, why further bother about Z? Regards, WM
From: mueckenh on 10 Nov 2006 05:48 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. This is about the same as the vase which is empty at noon or Tristram Shandy who completes his diary. No reason to bother about this. Regards, WM
From: mueckenh on 10 Nov 2006 06:05 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > > > > Wrong.. You have not understood the meaning. We have: The set of all > > > > sets does not exist. But if all mathematical entities including all > > > > sets do exist in a Platonist universe, how can it be that the set of > > > > all sets does not exist? > > > > > > That is NOT Fraenkel, Bar-Hillel, and Levy's point at all! > > > > You admitted that you do not understand their philosophical remarks. So > > let it be. > > No, please do not misparaphrase me. I said that I admit that I don't > fully MASTER their philosophical remarks. But I said that I do know > basically what they're talking about and what they're saying. You may think so, but that is obviously false (see below). > > > And they don't advocate it at all. > > > > They mention this point of view as a possible one, and, as far as I > > remember, as the only possibility for a platonist. > > WHERE? Where in that book do they mention the point of view that a set > can grow or have different members at different times and that that is > the only possible view for a platonist? You've got it completely > backwards. It is very much a platonist view that sets are NOT as just > described. p. 118, considering that the set of all sets does not exist, they write: a reasonable way to make this conform to a Platonistic point of view is to look at the universe of all sets not as a fixed entity but as an entity capable of "growing". Why should they write "reasonable" if they found it unreasonable? I did not say strictly that this is the only possible Platonist point of view, I said "as far as I remember". But I will not reread this book only because you did not understand it. Regards, WM
From: mueckenh on 10 Nov 2006 06:10 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. > > This seems to be the place where you disagree with everyone else. > You make an assertion, but you do not give a proof of that assertion. > Everyone else says you are wrong in that assertion and use the current > example as a counter-example. Everyone may say that a circle with diameter larger that circumference is no contradiction in ZFC. Neverheless it is a contradiction. Regards, WM
From: mueckenh on 10 Nov 2006 06:23
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > About a couple of weeks ago you presented an argument about trees, and > > > as you presented your argument, as you described it, it was clearly > > > reasonable to regard you as intending that as a proof in set theory (it > > > would have been UN reasonable to think the contrary). But later you > > > switched, so that your argument was not to be taken as in set theory. > > > > Why don't you try to find out how it could be presented in set theory? > > Or why that cannot be done? > > 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. > then you switched away from that. It's > not MY job to vindicate your argument as a set theoretic proof. > > Yet, STILL, I am open to seeing your argument as set theoretic if you > would just cooperate. I can't even begin to evalute your argument in > terms of set theory if you won't give me a mathematical definition of > the relation you first mention in your argument. And from there, I > cannot even begin to evaluate your argument without confirming that > your use of mathematical terminology is based on the same definitions I > would have. Moreover, definitions in graph theory and tree theory > differ even among different authors. I am interested in knowing the > precise mathematics of your argument. And I humbly represent myself as > not an expert in graph theory or trees, which is all the MORE reason > that the definitions need to be clear to me. 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. Continuing in this manner in infinity, we see by the infinite recursion f(n+1) = 1 + f(n)/2 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. Cantor's diagonal argument does not apply in this case, because the tree contains all binary representations of real numbers within [0, 1], some of them even twice, like 1.000... and 0.111... . Therefore we have a contradiction: |IR| > |IN| || || |{paths}| =< |{edges}| > > I do not prohibit to use the axiom of infinity, but I show that the > > consequences of its use are absurd. > > Now you're switching AGAIN! Previously, you objected that I got my > conclusion only because I availed myself of the axiom of infinity. NOW > you're switching to saying that you don't begrudge use of the axiom of > infinity but that now you point to (what you claim to be) an absurdity > that results from using the axiom of infinity. > > But what absurdity? Not a contradiction IN set theory. That is unimportant. An absurdity remains so, whether in or outside of set theory. Regards, WM |