Cantor Confusion
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From: mueckenh on 20 Nov 2006 16:32 Randy Poe schrieb: > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > principle. > > In principle, if a is the sqrt(2) to 10^100 digits, then > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > What "in principle" prevents me from calculating 2/a, > or adding it to 1, or taking 0.5*(a + 2/a)? Lack of bits to represent 2/a if a already requires all bits available. If you don't believe me, simply try it. By about 330 calculations you should be able to produce the first 10^100 digits. If you have a slow computer you will need one second per calculation. So let it run for 5 minutes and you will know what prevents you from calculating 2/a. Regards, WM
From: mueckenh on 20 Nov 2006 16:38 Franziska Neugebauer schrieb: > > Experience has shown that practically all notions used in contemporary > > mathematics can be defined, and their mathematical properties derived, > > in this axiomatic system. In this sense, the axiomatic set theory > > serves as a satisfactory foundation for the other branches of > > mathematics. [Karel Hrbacek and Thomas Jech: "Introduction to Set > > Theory" Marcel Dekker Inc., New York, 1984, 2nd edition, p. 3] > > Again: Meet _your_ obligation and define "column[s]". You have > introduced this notion hence it is up to you to define it. No. The notion "column" is well known to anyone having studied the first semesters math. Further I defined it by means of the EIT as the union of all initial subsets of N. > > >> It is curious that you obviously don't like to give precise > >> definitions of certain notions even when you have explicitly been > >> asked for. > > > > I defined the EIT. A column is a vertical row. > > What is a row in the language of (which?) set theory? I use the language of mathematics. > Regards, WM
From: Virgil on 20 Nov 2006 16:41 In article <1164037587.537732.324270(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sara M schrieb: > > > > But the set of all lists is countable (as is any quantized or > > > discontinuous set), so is the set of all list entries. > > > > That's absurd. First of all, I'm not really clear what "discontinous" > > means when applied to a set, > > Why then do you call it absurd? > > Here are some explanations: > A list is a (injective) sequence. A list is a function from N to some set in which the listed objects are to be found. While one often wishes to have such functions injective, there is no inherent reason why an object may not be listed more than once is a list, and there are lists allowing this. > Discontinuous means: The difference between two elements (i.e. natural > numbers enumerating a list) has a finite minimum value. That is not a standard mathematical meaning for "continuous". Continuity/discontinuity is more standardly only defined for functions between topological spaces. There are only two commonly occurring topologies for the set of naturals, the discrete topology and the co-finite topology, and under neither of them does WM's definition make any sense. There is a property of orderings similar to what WM wants but the name of that property is "discreteness" not "discontinuity". An ordered set is discretely ordered when there are at most finitely many members of the set between any two given members of the set. A topological space is discrete when every subset is open. > Every list occupies a non vanishing part of the space of he universe. > The volume of the (accessible) universe is finite. Those are restrictions in physics, not in math. A mathematical list can be like a point in an uncountable universe of points.
From: Virgil on 20 Nov 2006 16:50 In article <1164038784.730374.278930(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > You are trying to argue > > > > The diagonal cannot be longer than every > > initial segment with a largest element because > > there is no set of lines without a largest line. > > No. The diagonal cannot be longer than every line because it consists > of line-ends only. The diagonal is clearly and unabiguously longer than each line for which there is a next line. Does WM declare that within ZF there is a line for which there is no next line? > The diagonal cannot be longer than every line > because it consists of line-ends only. Non-sequitur. there is nothing in "it consists of line-ends only" which requires "the diagonal cannot be longer than every line". > > If the diagonal has omega elements, then a line must have omega > elements. if no line has omega elements, then the diagonal cannot have > omega elements. Not in ZF. WM keeps assuming things which are false in ZF, and not provable without being assumed. So WM is assuming a set of axioms different from ZF, and what WM claims for his system is irrelevant for those using ZF. > > This is not circular but a bijection, the same bijection as between > first column and diagonal. The bijection between lines and initial > segments of the first column is brought about by the diagonal. The > reason for this bijection is the fact that natural numbers count > themselves. > > 1 > 2 > 3 > ... > <--> 1,2,3...n > > This bijection fails in case of the full column with omega elements but > no line with omega elements. > > That is so obvious that I cannot understand how one can miss it. Just lucky, I guess.
From: mueckenh on 20 Nov 2006 16:51
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> > I cannot understand your explanation given there. If you say "The > >> > cardinality of omega is |omega| not omega", so you must have had in > >> > mind |omega| =/= omega, > >> > >> No Way! If you want to misapprehend me do so, but don't confuse your > >> misapprehensions with theorems of set theory. > > > > You wrote: "The cardinality of omega is |omega| not omega." > > And what I meant was: "The cardinality of X is |X| not X". I have > already clarified that my wording was misleading. You tried to say "misleading", but your wording was wrong. The cardinality of omega is omega as well as |omega|. > > > Shall this sentence of yours express a difference between |omega| and > > omega or not? > > What exactly is so hard to understand? The cardinality of a set X is > (written) |X| and -- in the case of omega -- equals under the common > definition to omega. Therefore, the sentence "The cardinality of omega is |omega| not omega" is false? Or is it not false? > > > (Now I recognize why it is so difficult to convince the proponents sof > > set theory.) > >> > >> > Would it be meaningful to write "Y is X but not X"? > >> > >> This is not a "faithful" representation of my sentence you have > >> objected to because there is no character string which can be > >> substituted for X to get my original sentence back. > > > > If you meant |omega| = omega, then we have X = |omega| = omega. > > Let Y = "The cardinality of omega" and > > case 1: let X = "|omega|" "Y is X but not X" transforms into > "The cardinality of omega is |omega| but not |omega|", > > case 2: let X = "omega" "Y is X but not X" transforms into > "The cardinality of omega is omega but not omega". > > I did neither write nor mean what case 1 nor case 2 state. So you have *not* yet learned that omega = |omega|? Because considering this eqation, you effectively wrote "The cardinality of omega is |omega| but not |omega|". (Now I recognize why it is so difficult to convince the proponents sof set theory.) Regards, WM |