Cantor Confusion
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From: Virgil on 7 Dec 2006 18:04 In article <1165531790.881743.134350(a)n67g2000cwd.googlegroups.com>, cbrown(a)cbrownsystems.com wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > cbrown(a)cbrownsystems.com schrieb: > > > > > > > So now you return to claiming that your mapping T /is/ a surjection > > > from edges onto paths. Perhaps there is some confusion regarding "what > > > is the domain/range of T?". > > > > > > To clarify, let e be any edge; say the first branch to the left in your > > > original diagram. According to your above statement, e is in the domain > > > of T. Which path is T(e)? > > > > You know that the two edges mapped on a path consist of shares of many > > edges. Why do you put your question? Why should I name a special edge? > > Because your argument is of the form: "There exists a bijection f > between the naturals and the edges. There exists a bijection h between > the paths and the reals. There exists a surjection T between edges and > paths. Therefore, the composition h o T o f is a surjection of the > naturals onto the reals; contradiction." > > I assume you know what "a surjection T from edges onto paths" is, yes? > It doesn't map 1/2 of an edge to 1/4 of a path! It is a function that > maps each edge /in the original diagram/ to a path; so that every path > is in the image of the function. > > If you claim to have constructed T, then you claim that you have > constructed a function T such that for any edge e, say the first edge > to the left, T(e) is a path. > > So how do you claim to have constructed T? How are we to determine > T(e), in principle at least? > > If you do /not/ claim to have constructed T as a function from edges > /in the original diagram/ to paths, then your argument poses no > contradiction: there is no composition h o T o f, because the domain of > T is not "edges /in the original diagram/", which is the range of f. > > > I have proved that two edges are collected in form of shares. > > > > No, you have proven that the function g: (paths x edges x N) -> R is > defined such that lim n->oo sum (over all edges) g(p, e, n) = 2 for > every path p. > > The range of the function g is not "the set edges /in the original > diagram/", nor is it "the set of full edges"; it is R. > > The real number "2" is not some specific edge /in the original > diagram/; nor is it "2 specific edges /in the original diagram/". It is > simply the real number, 2. > > You have yet to define what you mean by saying that, therefore, we can > use g to biject "shares of edges /in the original diagram/" to "shares > of full edges" to produce "full edges", which can then be > bijected/injected to "edges /in the original diagram/". > > Until you have you defined these ideas and constructed these mappings, > you have not proved that therefore there exists a surjection T (edges > /in the original diagram/ -> paths); you at best have a function > mapping something called "full edges" to paths. > > You need to actually /demonstrate/ that g can be used to /construct/ T > as required; instead of merely asserting it by waving your hands and > claiming that it is obvious. > > > > > I have proved it by rational relation and by a random mapping. > > > > > > You haven't proved it until you provide a surjective function T : > > > (edges -> paths). (And I am only interested in your "rational relation" > > > argument). > > > > It has been proved. See above. I do not see the necessity to do more > > than to show that there are enough shares. > > Since you refuse to describe what you mean by "shares" in a > mathematical sense, I can't assign any particular meaning to the > assertion "there are enough shares". How do you define this phrase so > that we can construct a surjection T (edges /in the original diagram/ > -> paths)? > > > > > > > > > > > > > > > I add an appendix to one of my papers, where this is underlined I > > > > > > (here > > > > > > the arguing is based on nodes instead of edges, but that doesn't > > > > > > matter > > > > > 4> much): > > > > > > > > > > > > > > > > Your appendix fails to address the key question: what is the domain > > > > > of > > > > > the function T? If e is an edge, what is the set of "shares of the > > > > > divided edge e"? > > > > > > > > What is your problem? The complete set of shares of one edge is the > > > > full edge. > > > > > > So you have a bijection S : (edges -> complete sets of shares of an > > > edge). But given an edge e, what are the /elements/ of the set of > > > shares, S(e)? How many elements does S(e) have? 1? 32? A countable > > > number? An uncountable number? > > > > Everything in this tree is countable. > > Could you instead answer my questions regarding S(e)? What is the set > of shares associated with edge e, where e is an edge in the original > diagram? You claim it is a set; so one assumes it has members. Is > "e/32" a member of this set of shares? Is "1/sqrt(2)" a member of this > set of shares? Is "0" a member of S(e)? > > Cheers - Chas WM will respond, if at all, with some way of mapping an infinite sequences of branches to a path. But as a path is already essentially an infinite seqeunce of branches (if we ignore the nodes) that does not produce anything supporting his wild claims.
From: David Marcus on 7 Dec 2006 18:45 Bob Kolker wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > You cannot imagine the integer [pi*10^10^100]. > > That is not an integer, dummkopf. It is an irrational real number. Perhaps he meant the brackets to indicate the greatest integer less than or equal to the bracketed quantity. Although, I don't find any difficulty in imagining that integer. -- David Marcus
From: David Marcus on 7 Dec 2006 18:54 Eckard Blumschein wrote: > On 12/7/2006 1:54 AM, Dik T. Winter wrote: > > > > Could you read a pdf version? > > > Yes. > > I will add it to M283 as soon as possible. I'm sure Dik can't wait. > > > It started with the question how to deal with the nil in case of > > > splitting IR into IR+ and IR-. I got as many different and definitve > > > answers as there are possibilities. I expected that there is only one > > > correct answer, and I found a reasoning that compellingly yields just > > > one answer in case of rationals and a different one in case of reals. > > > > Oh, well. In Bourbaki's mathematics R+ and R- both contain 0. So you > > are a follower of Bourbaki after all? But of course the 0's are the > > same. If they were different you would have quite a few problems with > > limits and continuity. > > According to my reasoning, any really real number is not unique but must > rather be void because even the tiniest interval is thought to contain > indefinitely not just many rational numbers but indefinitely much of > real numbers. That's gibberish. Trying speaking English. > Therefore, unreachable the very nil on top of the nested > intervals has not any significance at all. More gibberish. Or, maybe poetry. > It cannot even be > distinguished from numbers 0- and 0+ left and right from it, > respectively, because the diffence is zero. Are you saying that there are numbers 0- and 0+? Are these rational? Real? What are their definitions? > So I agree with the > Bourbakis perhaps for the first time: 0+ and 0- are indiscriminable in > IR. That's certainly not what Dik said Bourbaki said. > However among the rationals, the nil is the first negative number > according to my reasoning and my old encyclopedia. What is "nil"? Zero? Are you saying zero is negative? If so, define "negative". -- David Marcus
From: stephen on 7 Dec 2006 18:51 David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Bob Kolker wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > >> > You cannot imagine the integer [pi*10^10^100]. >> >> That is not an integer, dummkopf. It is an irrational real number. > Perhaps he meant the brackets to indicate the greatest integer less than > or equal to the bracketed quantity. Although, I don't find any > difficulty in imagining that integer. You only imagine that you can imagine that integer. :) Stephen
From: David Marcus on 7 Dec 2006 18:56
Franziska Neugebauer wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > >> > Everybody knows what the number of ther EC states is. > [...] > > The number of EC states is "the number of EC states". > > This is hardly a definition. WM has his own definition of the word "definition". -- David Marcus |