Cantor Confusion
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From: David Marcus on 10 Nov 2006 17:33 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > > Exactly this is done by Cantor's definition given above: "It is allowed > > > to understand the new number omega as limit to which the (natural) > > > numbers n grow". This is a definition, at least for those > > > mathematicians who know what "limit, number, grow" means. > > > > I seriously doubt that Cantor considered it to be a definition. If he > > did, he wouldn't have said, "It is allowed to understand". Regardless, > > it is not a definition by modern standards. > > Who judges what modern standards are, in your opinion? A silly question. You haven't even bothered to learn the current meaning of the word "definition". If you would actually read a textbook or take a math course at a university, you might learn something. > > Nor, is it the defintion of > > omega in modern mathematics. And, since the set of mathematicians who > > know the mathematical meaning of the words "limit", "number", "grow" is > > probably empty, your final sentence seems vacuous. > > You carried out a survey? Or is that your guess? I know some > mathematicians who know what a mathematician should understand by > "limit", "number", "grow". Please post their definitions of the words. > > Your changing "finished entity" to "finished infinity" is a bit of a > > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > > comment, not discussing mathematics. > > They were discussing mathematics. Philosophy belongs to mathematics. That's nonsense. > > So, it remains true that "finished > > infinity" does not have a mathematical meaning. If you wish to use the > > term in a mathematical discussion, it is incumbent on you to define it. > > This term has been used in a mathematical discussion by Hrbacek and > Jech. I don't see why I shouldn't do the same. Several reasons: They were discussing philosophy of mathematics, not mathematics itself. You don't understand what they were saying. And, you haven't given a definition of the term. You can use any term you wish, but only if you define it. That is one of the rules of the game. If you don't want to play, then don't, but you can't unilaterally change the rules. > > No, that is still wrong. That's not what the word "definition" means in > > Mathematics. > > Which mathematics do you allude to? That which is taught in school, explained in textbooks, published in journals, and discussed by some in this newsgroup. You appear to be unfamiliar with all these sources. It must have taken concerted effort to have avoided all those opportunities to learn some mathematics. -- David Marcus
From: David Marcus on 10 Nov 2006 17:39 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > 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. I was asking you whether *you* agree with them, not what mathematicians say. I'll try again: 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. > > 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. That's a "square", but what is a "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. I have no idea what you mean by "number cannot be known" or "not fixed". These are not standard terms. I believe I already told you that if you use nonstandard terms, you must define them. Let's try some easier questions: Is the number of lines finite? Is the number of lines not finite? > > According to you, is there a last line? > > Of course, but we cannot know it. It is not fixed but depends on > several circumstances. I have no idea what "we cannot know it", "not fixed", or "depends on several circumstances" mean. As I've already told you, if you use nonstandard terms, you must define them. So, please define them. -- David Marcus
From: David Marcus on 10 Nov 2006 17:42 mueckenh(a)rz.fh-augsburg.de wrote: > 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". You need not be dismayed, since so far we haven't found any evidence that you think at all. -- David Marcus
From: David Marcus on 10 Nov 2006 17:47 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > 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? It is just a name. Why are you called "WM"? > > > "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 da=DF omega die erste ganze > > > Zahl sein soll, welche auf alle Zahlen nu folgt, d. h. gr=F6=DFer 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? You are confusing "Set Theory" as a branch of mathematics with whether a certain axiomatic theory can be used as a basis for contemporary mathematics. Note that Dik said that topology is "built on set theory". That is the same thing that Hrbacek and Jech are saying. You seem to have quite a bit of trouble with English. -- David Marcus
From: David Marcus on 10 Nov 2006 17:48
Lester Zick wrote: > It's amazing that 21st century mathematikers cannot analyze the truth > of contentions without playing cards:the race card, the religion card, > the homeland card, etc. My last name is Zick. I'm American. Period. > When it comes to mathematics I'm universalist. A pity you don't know any mathematics, though. > Patriotism and religion > among others are the last resort of scoundrels. In mathematics these > kinds of issues are historical anachronisms unless you can't establish > truth in universal terms which it would seem mathematikers can't. -- David Marcus |