Cantor Confusion
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From: mueckenh on 10 Nov 2006 12:53 David Marcus schrieb: > > Here we can set up a bijection, expressing the lines by the natural > > indexes of the elements: > > > > 1 1 > > 2 1,2 > > 3 1,2,3 > > ... ... > > n 1,2,3,...,n > > ... ... > > omega 1,2,3,... > > > > which shows that omega or aleph_0 does not exist. > > Do you agree with the following statements? > > For each natural number, we have a line. > Each line is assigned a natural number. > > The length of the first column is omega. > The number of lines is omega. In set theory, of course. It is shown by the scheme above after eliminating the last line. Regards, WM > What is your definition of a "square matrix"? Same length and width. Or: the diagonal contains an element of each line and of each column. > So, you are saying that a column can't be longer than every line. Does > your argument also apply to the following (finite) list? > 1 > 1 > 1 > Here we have an array that has three lines and one column. The length of > the column is three, which is greater than the length of every line (the > latter being one). This is not a square matrix. > Consider the following list. > 1 > 1 > 1 > ... > There is a line for each natural number. How many lines do you say there > are? *I say*: There are many lines, the number cannot be known because it is not fixed. > According to you, is there a last line? Of course, but we cannot know it. It is not fixed but depends on several circumstances. Regards, WM
From: mueckenh on 10 Nov 2006 12:59 Dik T. Winter schrieb: > No. When we start with finite triangles, also each column has a finite > length. So when we have "completed" the triangle, the height of the > triangle it the supremum of the set of the number of lines, and is > also the supremum of the lengths of the finite columns. The width > is the supremum of the lengths of the lines. The hight (column) has actually many digits (maximum). All numbers are there, yielding a set of cardinality omega., The width (lines) has not actually many digits n(supremum). All numbers are there, but no one has infinitely many digits. The diagonal must have both, but cannot. Regards, WM
From: mueckenh on 10 Nov 2006 13:04 David Marcus schrieb: > WM's logic (if we can call it that) Please do not! You have no balls at noon. You have Tristram Shandy complete his diary. You have a diagonal longer than every line. You can make two balls of one. You have a countable model of an uncountable theory. I would be dismayed if you found any parallel between my thinking and your "logic". Regards, WM
From: mueckenh on 10 Nov 2006 13:13 Dik T. Winter schrieb: > > Comparison of numbers by size is one of the first tasks of mathematics. > > You think so. As I said above, comparing weights and lengths does not > involve mathematics. > > > > But at least weight is not a mathematical entity. Length is, > > > but not in the way you see it. > > > > The numbers counting the units of weight or length are mathematical > > entities. > > They are. But in mathematics comparing numbers comes in quite late in > the process of definition. In mathematics counting and comparing numbers comes before most other things. Not because I think so, but because mathematics developed this way. > > > > becauses you > > do not know that limits are inside set theory. > > Where are they? That there are "limit ordinals" does not mean there are > "limits" in set theory. Why are they called so? > > > "Es ist sogar erlaubt, sich die neugeschaffene Zahl omega als > > ****Grenze**** zu denken, welcher die Zahlen nu zustreben, wenn > > darunter nichts anderes verstanden wird, als daß omega die erste ganze > > Zahl sein soll, welche auf alle Zahlen nu folgt, d. h. größer zu > > nennen ist als jede der Zahlen nu" [Cantor, Collected works, p. 195]. > > Since Cantor quite a bit has changed. Limits are *not* part of set > theory, they belong to topology and other things build on set theory. Experience has shown that practically all notions used in contemporary mathematics can be defined, and their mathematical properties derived, in this axiomatic system. (Hrbacek and Jech, p. 3) But this does not include limits? Regards, WM
From: William Hughes on 10 Nov 2006 13:14
mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > No. When we start with finite triangles, also each column has a finite > > length. So when we have "completed" the triangle, the height of the > > triangle it the supremum of the set of the number of lines, and is > > also the supremum of the lengths of the finite columns. The width > > is the supremum of the lengths of the lines. > > The hight (column) has actually many digits (maximum). All numbers are > there, yielding a set of cardinality omega. All numbers are there, however every number that is there has finitely many digits. Omega is not a natural number, so omega is not there. The fact that omega is not there does not mean that there are less than omega numbers. , The width (lines) has not > actually many digits n(supremum). All numbers are there, but no one has > infinitely many digits. Correct. No number has infintely many digits. No number is omega. However, the fact that omega is not there does not mean that there are less than omega numbers. >The diagonal must have both, but cannot. No, the diagonal must have all natural numbers. Both the height and the width contain all natural numbers so this is not a problem. The diagonal cannot contain omega. However, the diagonal does not have to contain omega. The fact that the diagonal does not contain omega does not mean that the diagonal contains less than omega numbers. - William Hughes |