An uncountable countable set
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From: Franziska Neugebauer on 12 Jul 2006 16:50 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> > >> >> > To prove the existence and uniqueness of (|Q_+, <)? >> >> >> >> I want to know what the application of all of your conditional >> >> transpositions to the ordered set *means*. I ask for a definition >> >> of the *result*. And it would be nice if you show if it exists and >> >> is unique. This is not (yet) proven "by the axioms". >> > >> > Many things cannot be proven by ZFC+FOPL. >> >> But you still have not proven your claim, that it is - at least - >> meaningful to write about application of *all* conditional >> transpositions to a given sequence. Until then it is not meaningful. > > The transpositions are a set of order omega. The existence of this set > is guaranteed by the axiom of infinity. Noone doubts the existence of the set of conditional transpositions. > That the transpositions are conditional does not disturb the proof. I proof cannot be "disturbed". A proof is not describing a process in time. What you did not proof is the result. There is no limit of the *application* of your sequence of transpositions *on* a given series. > Cantor's [...] Forbidden word encountered. F. N. -- xyz
From: Virgil on 12 Jul 2006 16:56 In article <1152735445.836339.213400(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > >> > To prove the existence and uniqueness of (|Q +, <)? > > >> > > >> I want to know what the application of all of your conditional > > >> transpositions to the ordered set *means*. I ask for a definition of > > >> the *result*. And it would be nice if you show if it exists and is > > >> unique. This is not (yet) proven "by the axioms". > > > > > > Many things cannot be proven by ZFC+FOPL. > > > > But you still have not proven your claim, that it is - at least - > > meaningful to write about application of *all* conditional > > transpositions to a given sequence. Until then it is not meaningful. > > The transpositions are a set of order omega. The existence of this set > is guaranteed by the axiom of infinity. Which axiom and set "mueckenh" has repeatedly rejected. >That the transpositions are > conditional does not disturb the proof. It does mean that they must be performed in order though. And that does distrub the proof. > Cantor's diagonal argument is > also conditional. But there are no conditions on any digit which depend on any other digit having been calculated previously, i.e., no sequential conditions. Thus one can have a general rule that does eachdigit without reference to any others. so is valid for all in one rule and one step. > That the conditions have to be executed one after the > other does not disturb the proof. It means that the"mueckenh" process, as described, can never end. > Cantor's diagonal argument requires > counting which is a sequential act too. Not at all. Getting the list in the first place may require counting, though that is dubious, but once it is presented, no further counting is required. > > So I am on the safe side with respect to each and every objection which > could be raised by jealous people. No one is jealous of your idiocy, merely appalled by it.
From: Virgil on 12 Jul 2006 16:58 In article <1152735810.530270.91150(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The axiom says "there is an infinite set". It does not say that > 0.111... does belong to that set, in particular because all numbers > which in fact do belong to the set are different from 0.111... . But 0.111... in effect IS that set to which it does not belong.
From: Virgil on 12 Jul 2006 17:09 In article <1152736018.461492.108050(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > The binary tree: All paths, not yet disperged, start through the root > edge a. There is no need for a root edge at all. the tree can quite comforably be rooted in a rot node from which two edges branch out to two cild nodes, and so on. (Let us denote them as a single path as long as they are > together.) As paths are sets of edges, this does not work. Map this root edge a on this single path. In the next level > the path splits in two paths. Map half of edge a on each of them. The > right one passing through edge b, gets b mapped on it and it inherits > half of a. After splitting again, each of the paths gets the next edge, > say c, half of b and quarter of a. > > | a > o > / \ b > o o > / \ / \c This tree contains 4 paths indicated by sequences of two branchings, left or right right, so that {LL,LR, RL, RR} represents the set of all paths for the tree as shown. For larger binary trees one gets more and longer strings of left to right branchings. For infinite binary trees each such list of branchings is an infinite sequence. And there are uncountably many possible such infinite sequences.
From: Dik T. Winter on 12 Jul 2006 19:18
In article <J2AqAx.G18(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <1152708585.848976.307310(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > > I did not say that Cantor's strings were binary numbers. I said that > > > > they might be *interpreted* as binary (dyadische, dual) > > > > representations, safely. Cantor and Zermelo at least did so. .... > You stated "binary representations". Now I am a bit at a loss, what you > mean with that. Representations of *what*? I just reread what you wrote: > Cantor does not at all use real numbers but merely infinite sequences > with only two different symbols w and m (which might be interpreted as > binary representations but were not). But that there is no satisfactory > limit consideration becomes clear from the following: We know that > 0.999... = 1.000... This leads to the result that a change of 1 in the > limit where the digit number goes to oo does not have the effect which > would be required in order to distinguish the diagonal number from the > list numbers. Still at a loss with what you mean with "binary representations" in this context. But in Cantor's article limits are not needed, and your comparing 0.999... with 1.000... is a red herring in the context of Cantor's article. In that context the sequences: E1 = (m, w, w, w, w, ...) and E2 = (w, m, m, m, m, ...) are different. However, some say that Cantor has also written an article about the uncountability of real numbers using decimal notation, but I have not yet found it. I am starting to doubt that. What I find is the following: 1. Cantor's first diagonal argument: proves (amongst others) that the rational numbers are countable. 2. Cantor's second diagonal argument: proves that there are sets with cardinality greater than the cardinality of the natural numbers (the proof we are discussing here). 3. His proof that the reals are not countable (published in 1874). Of course, Zermelo did show that (2) can be transformed to a proof that the reals are not countable, but I do not think that Cantor did do that transformation. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |