Cantor Confusion
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From: mueckenh on 13 Nov 2006 16:53 Virgil schrieb: > In article <1163428970.972473.215570(a)i42g2000cwa.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > Every column has a fixed number omega of terms. You can add 1 further > > > > term and you will get omega + 1 terms in every column. Every line has > > > > less than omega terms. If you add 1 term to every line, the number of > > > > terms in every line remains less than omega. What about the diagonal? > > > > It has to satisfy both conditions, which is impossible. > > > > > > No. Adding one element to each line does not change > > > the supremum of the lengths of the lines, so it does > > > not change the length of the diagonal. > > > > > > The number of terms in each line, n, is less than omega. > > > The number of columns is the supremum of the number of terms > > > in each line. The number of columns is omega. > > > The number of lines is omega. > > > The number of columns is equal to the number of lines. > > > > > > > > > Now add one term to each line. > > > > > > The number of terms in each line, n+1, is less than omega. > > > The number of columns is the supremum of the number of terms > > > in each line. The number of columns is omega. > > > The number of lines is omega. > > > The number of columns is equal to the number of lines. > > > > PS: You forgot to consider the case that one term is added to each > > column. The ordinal of the columns is then larger than omega. > > But the cardinal is not. Which is what you get. If bijecting ordered sets, I am interested in the ordinal number. Without order no cardinal. Did you forget why Cantor strived to order any set? Regards, WM
From: Lester Zick on 13 Nov 2006 17:23 On 12 Nov 2006 16:28:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 10 Nov 2006 15:18:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> Your way or the highway huh, Moe(x). >> >If he has an argument that he thinks can be put in set theory, then I'm >> >interested in his argument; If he doesn't think his argument can be put >> >in set theory, then I'm not interested. He can post about his argument >> >all he wants, but I'm not obligated to study his argument. >> >> No one suggests you are. The problem I see is that one might cast an >> argument in such terms as are acceptable to you and still not satisfy >> exactly the same criteria on the part of others. I mean unless you are >> the generally acknowledged expert in the field. Otherwise it would >> look to me like you're just trying to take control of the discussion >> in terms you find acceptable whether or not others do. > >Nothing of the kind. HE suggested to ME that I can see if his argument >can be put into set theory. If I am take MY time and effort to do that, >then I have every prerogative to set my own terms for doing it. Oookay. Then the question still occurs whether your comprehension of set theory is sufficient unto the task. If not the issue is moot. >> Let me see if I can simplify how the issue is or ought to be argued. >> You posit certain properties of an infinite however you define it. So >> the question then becomes whether your claim is or can be true. Now >> one way to show it's actually true would be to produce some entity >> with the properties you posit of an infinite. Otherwise you'd have to >> find some other way to get at the truth of what you claim unless you >> just intend to claim it's true because you or others say so. >> >> Now as I understand WM's argument he suggests you can never actually >> produce any physical infinite because the physical universe is finite. > >WM and were talking about his tree argument. The finititude of the >physical universe is a separate subject. OK. Withdrawn. >> However he then apparently concludes from this that there can be no >> infinites at all because there can be no physical infinites if the >> universe is finite. >> >> Now personally I find most of the arguments disingenuous on both >> sides.And I see no special merit to your definition for the properties >> of infinites you recommend to the exclusion of others. But they are >> specific properties you can't demonstrate through exemplification so >> if you wish to show that the characteristics you assign to infinites >> can be true you have to approach the proof some other way than >> empirical exemplification. > >MoeBlee ~v~~
From: MoeBlee on 13 Nov 2006 17:32 Lester Zick wrote: > On 12 Nov 2006 16:28:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> On 10 Nov 2006 15:18:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >> > >> >Lester Zick wrote: > >> >> Your way or the highway huh, Moe(x). > > >> >If he has an argument that he thinks can be put in set theory, then I'm > >> >interested in his argument; If he doesn't think his argument can be put > >> >in set theory, then I'm not interested. He can post about his argument > >> >all he wants, but I'm not obligated to study his argument. > >> > >> No one suggests you are. The problem I see is that one might cast an > >> argument in such terms as are acceptable to you and still not satisfy > >> exactly the same criteria on the part of others. I mean unless you are > >> the generally acknowledged expert in the field. Otherwise it would > >> look to me like you're just trying to take control of the discussion > >> in terms you find acceptable whether or not others do. > > > >Nothing of the kind. HE suggested to ME that I can see if his argument > >can be put into set theory. If I am take MY time and effort to do that, > >then I have every prerogative to set my own terms for doing it. > > Oookay. Then the question still occurs whether your comprehension of > set theory is sufficient unto the task. If not the issue is moot. It's very possible that someone else would succeed where I would fail. But when I am invited to see whether I can put an argument into set theory, then I can only bring the knowledge that I do have. If I fail to put the argument into set theory, then I would not claim that as a demonstration that the argument cannot be put into set theory, but if I do succeed in putting the argument into set theory, then the argument can be put into set theory, so a confirmation may come from me that the argument can be put into set theory, even though I would not claim that a failure by me would be refutation of a claim that the argument can be put into set theory. MoeBlee
From: Lester Zick on 13 Nov 2006 17:57 On 13 Nov 2006 13:48:28 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > >Lester Zick schrieb: > >> >> Now as I understand WM's argument he suggests you can never actually >> >> produce any physical infinite because the physical universe is finite. > >Even if it is not finite, the domain accessible to us will remain >finite forever. Which was exactly the point of my "The Transfinite Zen Abacus" thread. However I don't necessarily agree that infinities cannot be inferred from collateral considerations. To the best of my knowledge the calculus does not rely on infinities for the idea of infinitesimals. >> >> However he then apparently concludes from this that there can be no >> >> infinites at all because there can be no physical infinites if the >> >> universe is finite. > >Everything of our thinking including mathematics relies on physical >fundaments (writing, speaking, thinking). Therefore physical >restrictions imply mathematical restrictions. This is incorrect to the extent writing, speaking, thinking, etc. do not rely on physical fundaments which in turn relies on what the term "physical" means. When you categorically restrict mathematics to the "physical" the actual significance of the term becomes critical. I don't consider we need abaci, zen or otherwise, to do mathematics. >> >That doesn't seem to be what WM is saying. > >I do not use the physical argument against set theory. It only makes me >ultimately sure that I am right. >> >> I don't consider myself to be an expert on what WM is saying. The >> above is just what I've gleaned from past comments. If it's incorrect >> I apologize. I've just tried to make the best argument for whatever he >> actually may be saying. > >Well done. Thanks. >> > He seems to be saying that >> >the notion of a completed infinity leads to either absurdities or >> >contradictions. > >"Completed infinity" is an absurdity. >> >> Well as far as I can tell it does. Most notably the containment of >> sets and subsets. I don't remember as the calculus ever requires >> infinites. >> >> > Perhaps he thinks the way to avoid these absurdities is >> >to only consider things that can be physically produced. >> >> Well that would certainly be one way. Another would be to stop using >> finite infinites. Infinites only make sense in relation to one another >> and not in relation to finites. >> >> >> Now personally I find most of the arguments disingenuous on both >> >> sides. >> > >> >what is a standard mathematical argument that you find disingenuous? >> >> Insistence on the rectitude of arguments which can't be demonstrated >> true. WM's arguments seem to rely on a finite universe. Opposition to >> his arguments seem to rely on problematic assumptons of set theory. >> Neither seems very convincing yet adherents of each pretend they are. > >Try to construct as many numbers as you can using only 100 bits. Then >increase to 10^10 bits, then increase to 10^100 bits. More is not >available. I find this very convincing. I don't. It's a problematic argument at best. Based once again on a hypothetical finitude of the "physical" universe whatever that means. ~v~~
From: Virgil on 13 Nov 2006 19:07
In article <1163453699.401521.95280(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Daniel Grubb schrieb: > > > >> make it into this matrix: > > >> > > >> 1 0 0 0 0 0 .... > > >> 2 3 0 0 0 0 ... > > >> 4 5 6 0 0 0 ... > > >> .... > > >> .... > > >> > > >> In which case, it is a square matrix and every line > > >> has infinite length. So the diagonal, which has infinite > > >> length, has the same length as every row and every column. > > > > >There is no line without trailing zeros. The diagonal has no zeros. > > > > I don't see the relevance of this. The question wasn't about whether > > there are zeros, the question was about the length of the diagonal versus > > the > > number of rows or columns. > > The diagonal consists of numbers =/= 0. Therefore it must touch every > line at such a position =/= 0. I do not see that it need "touch" any line at all, by any common meaning of "touch". > > Please drop the zeros. > Please increase every column by 1 digit. You will obtain omega +1 > digits and the ordinal of the length of the matrix will then be omega > + 1. But its cardinal will still be aleph_0. And it is only the cardinality that is relevant. > Please increase every line by 1 digit. You will obtain less than omega > digits and the ordinal of the width of the matrix will then be omega. But its cardinal will still be aleph_0. And it is only the cardinality that is relevant. > > But if you do not drop the zeros, then every line will contain zeros > before the last (added)element, such that the diagonal will not be > possible without zeros. On the contrary, the diagonal need not partake of any of those trailing zeros. If it is to contain zeroes, it will do so for the finite lines also. > > That's a pretty bold claim. Why do you assert this? While *you* > > might not have any ideas what an 'actual infinity' is, I think > > that I do. > > What is your idea about the actual infinite? That one version is like the set of all possible natural numbers. > > > > >Would you believe that the diagonal of our infinite triangle can be > > >longer than the first column? > > > > Until I saw that there was a bijection from the entries of > > the first column to the entries of the diagonal, I would entertain > > the idea. After I saw that bijection, I would know they are the > > same cardinality. > > The bijection is d_nn <--> d_n1. > The diagonal d_11, d_22, ...,d_nn, ... with omega digits is bijected to > the first column > d_11, d_21, ...,d_n1, ... with omega digits. The number of digits is more alpha than omega. > > Please try the same now with each line and each column and the diagonal > extended by 1 digit. > The diagonal > d_11, d_22, ...,d_nn, ..., d_ww > is bijected to the first column which represents the length of the > matrix > d_11, d_21, ...,d_n1, ..., d_w1 > The diagonal, however, cannot be bijected to the first or any other > line because there is no line with omega + 1 elements. The number of digits is more alpha than omega. And Card(omega) = Card(omega + 1). |