Cantor Confusion
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From: William Hughes on 16 Nov 2006 15:08 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No. The length of the diagonal is the supremum of the lengths > > of the lines (this is easy to show). So if there is no > > longest line the diagonal must be longer than any line. > > The length of the diagonal cannot surpass the length of any line. (This > is easy to show for every matrix.) As written this is piffle. If the lines get longer, it is certainly possible for the diagonal to surpass the length of a given line. 1 1,2 1,2,3 1,2,3,4 The diagoal has length 4. Not all lines have length 4. Perhaps you want to say that "The length of the diagonal cannot surpass the length of *every* line." This is false. The length of the diagonal can surpass the length of every line, if and only if the matrix does not have a longest line. [Or pehaps you want to return to the usual definition of a matrix where all lines have the same length. In this case we have to add 0's as place holders Our matrix is now 1,0,0,0,... 1,2,0,0,0,... 1,2,3,0,0,0,.. ..... Every line ends in an endless string of 0,s. What is the length of each line? omega. What is the length of each column, omega. What is the length of the diagonal, omega.] - William Hughes
From: Lester Zick on 16 Nov 2006 15:33 On Thu, 16 Nov 2006 02:28:14 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sat, 11 Nov 2006 15:53:40 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >Lester Zick wrote: >> >> On Fri, 10 Nov 2006 18:19:05 -0500, David Marcus >> >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >> >> >I think he has a bigger problem. He doesn't seem to agree that there are >> >> >infinite sets. It is very strange. >> >> >> >> You mean if the editorial "we" agree that there are infinite sets >> >> there are infinite sets? >> > >> >I don't know what you mean. What does "there are" mean in this context? >> >> You used the phrase and you're asking me? > >I know what I meant, but my meaning doesn't seem to make sense if I use >it in your sentence. Your meaning whatever it might be then just doesn't make sense. >> I can't really say what you >> meant when you used it. I mean capable of demonstration of truth. > >Sorry. I don't know what "demonstration of truth" means. Can you give an >example of demonstrating the truth of something? I can give an example of "non demonstration of truth" in everything you say. >> All >> I've seen people do in connection with infinites is assume certain >> properties and characteristics for infinites which they can't then >> demonstrate are actually true. Showing an entity which had such >> properties and characteristics would certainly be one way to prove >> their truth and show that "there are" infinites. The only alternative >> would seem to be some other form of demonstration. In neither case >> would the mere hypothetical assumption of truth demonstrate anything. > >Not sure what you mean. Apparently you never are. > In mathematics, all we do is pick some axioms, >then see what theorems we can prove. So let's just start out with square circles and two sided triangles. > Of course, we think the resulting >systems are useful and/or interesting, How clever of you. I should think square circles and two sided triangles would be very interesting. > but that doesn't "prove their >truth". And nothing you can say would. > The usefulness to other fields is demonstrated via the >scientific method, not by mathematical proof. You mean the empirical method not the scientific method. So mathematical axioms are to be empirically demonstrated now? >> >> You have a very curious sense of words in >> >> others but not in yourself. You claim to be able to prove things >> >> without being able to prove they're true. >> > >> >I'm using "prove" in its mathematical sense. I don't know what you mean >> >by "prove they're true". I suspect the meaning of the word "prove" is >> >different in the two senses. >> >> No doubt that's true. The problem I have is that every time you use a >> word, we have no idea whether what you're saying is supposed to be >> true or philosophical. So what if you "prove" something mathematically? >> Is the thing proven necessarily and universally true in mechanically >> exhaustive terms? If not it's a systematic philosophical exposition at >> best. That's why mathematikers coin definitions in ambiguous parochial >> terms that would embarrass a sixth grade school teacher. > >Don't know what you mean. Of course you don't. Get back to me when you learn to think for yourself. > Are you saying you don't know what the word >"proof" means in mathematics? I'm saying you can't prove the truth of whatever you say in or about mathematics. >> >> And what if one doesn't agree that there are infinite sets? >> > >> >If you mean you want to use different axioms for your mathematics, then >> >you are welcome to. It that's not what you mean, then I don't know what >> >you mean. What does "there are" mean in your sentence? >> >> I don't want to use any axioms for mathematics. That's the point. >> That's what got math into the pickle it's in with all its parochial >> axiomatic assumptions of truth and private ambiguous definitions. > >How can you do mathematics without axioms? Beats the hell outta me. I gave you the citation. Learn to read. > A major purpose of axioms is >to avoid ambiguity. The main purpose of axioms is to provide assumptions of truth without proof. >> >> Are you going to prove they're true? >> > >> >I don't understand the question. >> >> Are you going to illustrate the existence of infinites by production >> of one or more; or are you going to demonstrate the truth of their >> existence by some alternative means? You posit certain properties and >> characteristics of things you call "infinites" but don't show they can >> actually be realized in combination with one another. > >Sorry. Don't know what you mean. In particular, I don't know what you >mean by "illustrate the existence", "demonstrate the truth of their >existence", "actually be realized". Can you give an example? I should give you examples of the examples of infinites I asked you for? Boy do you have brass balls, slick. >All we do in mathematics is prove theorems from axioms. That's it. All you do in modern math is prove theorems from assumptions of truth. Not exactly overtaxing intellectually but there it is. >> >I take words seriously enough to be sure that I and the person I am >> >conversing with are using the words with the same meaning before I jump >> >to any conclusions. >> >> You jump to every conclusion like a twelve year old boy jacking off. >> How is it you verify you are using words with the same meaning if you >> can't demonstrate the truth of what you're saying in mechanically >> exhaustive terms capable of comprehension by others in identical >> terms? That's what mathematical formalisms are for. But just saying >> they're mathematical formalisms doesn't necessarily make them true and >> doesn't make them mechanically reducible in exhaustive terms. > >I can verify I'm using words with the same meanings as other people by >asking them what definitions they are using, then seeing if they are the >same as mine. Similarly, I can read a book, and see what definitions the >boo
From: david petry on 16 Nov 2006 16:09 Dik T. Winter wrote: > If you want to find absolute truth you should not look at mathematics. Perhaps we should replace "absolute truth" with "culturally neutral truth", or in other words, truth without any cultural, religious, or philosophical bias. We can reason about this concept by asking the question: what will the mathematics of advanced alien civilizations (i.e. from other planets) look like? Thinking about this question leads most of us to believe that there is a core of mathematics which every such civilization will accept.
From: Virgil on 16 Nov 2006 16:14 In article <1163670263.126778.26080(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > If there is no infinite number then there are not infinitely many > numbers. And, by definition, there is no infinite number. It is very > simple and independent of any religion (as far as I am concerned.) That seems to BE WM's religion. Until he presents us with a more complete and well thought out axiom system for his beliefs, all he has is "And, by definition, there is no infinite number." Which does not prevent trans-finites, hyper-finites, or all sorts of things without a definition of what an infinite number would be like if it had not been declared a deadly sin to talk about it.
From: Virgil on 16 Nov 2006 16:21
In article <1163704913.828089.12430(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > No. The length of the diagonal is the supremum of the lengths > > of the lines (this is easy to show). So if there is no > > longest line the diagonal must be longer than any line. > > The length of the diagonal cannot surpass the length of any line. (This > is easy to show for every matrix.) it has never been shown because it is false for a list in which the nth member is of length n. In such a list, for every n, the diagonal has "length" >= n, and in fact, has "length">= n+1 (as here is always a n+1'st line) so: For every n in N, length(diagonal) > n. |