Cantor Confusion
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From: David Marcus on 16 Nov 2006 02:44 Dik T. Winter wrote: > In article <1163469888.057589.117040(a)k70g2000cwa.googlegroups.com> "William Hughes" <wpihughes(a)hotmail.com> writes: > > So lets count the set of all natural numbers {1,2,3,...} > > There are no natural number left. So we stop > > using natural numbers and use ordinals > > (and to nobody's surprise a few things change). > > This is wrong. There is no ordinal needed to count the elements of the > set of all natural numbers. You can count until you weigh an ounce (;-)) > but you will never finish. Neither the elements you wish to count will > be exhausted nor the numbers with which you count. I think this is > potential infinity. On the other hand, when you ask "how many" elements > there are in N, you need an infinity (and this is, I think, actual > infinity). But all this hinges quite closely on the semantics of the > word "count". If seen as a process, you do not need an infinity; when > seen as the result of a process, you do need an infinity. In many > languages (German and Dutch amongst others) there are different words > for the two meanings, but the meanings are conflated in English. Are you discussing languages or math? What would a mathematical meaning for "count the set of all natural numbers" be? -- David Marcus
From: David Marcus on 16 Nov 2006 02:56 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > > > What is wrong with my reasoning? > > > > > > > > > > Nothing. Wrong is only the assumption that "goes on forever" could be > > > > > considered a finished infinity, i.e., could be denoted by a fixed > > > > > cardinal number being larger than any natural number. > > > > > > > > I'm not sure what you mean. In particular, I'm not sure what > > > > "considered", "finished" (or "finished infinity"), "denoted", and > > > > "fixed" mean here. > > > > > > I see. But recently you used the word "completed infinity". > > > > I don't think I ever said that. Do you have a quote? > > Here it is: In the below post, I was just trying to paraphrase what you are saying. I didn't say I would say that or that I understood what you were trying to say. In fact, I don't know what you you mean by the phrase. Did you really misunderstand what I wrote? > ================ > 2306 Von: David Marcus - Profil anzeigen > Datum: Sa 11 Nov. 2006 21:29 > E-Mail: David Marcus <DavidMar...(a)alumdotmit.edu> > Gruppen: sci.math > Noch nicht bewertetBewertung: > Optionen anzeigen > Antworten | Antwort an Autor | Weiterleiten | Drucken | Einzelne > Nachricht | Original anzeigen | Missbrauch melden | Nachrichten dieses > Autors suchen > > > > > - Zitierten Text ausblenden - > - Zitierten Text anzeigen - > > Lester Zick wrote: > > On 10 Nov 2006 15:18:07 -0800, "MoeBlee" <jazzm...(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. > > > > 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. > > > > What does "is or can be true" mean? In mathematics, we are normally > only > concerned with provability (unless discussing philosphy). > > > > 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. > > 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. > > That doesn't seem to be what WM is saying. He seems to be saying that > the notion of a completed infinity > ... > ======================== > > > > > Be sure that my finished infinity means the same. > > > > > > > > If you look at the diagonal, you always see that it is only there where > > > > > lines are. Therefore you will always see that it is of finite length. > > > > > "It goes on forever" does not mean actually infinite length. The > > > > > lengths of the lines also increase from line to line forever. > > > > > Nevertheless, an infinite length will never be reached. > > > > > > > > To me, "infinite length" just means "goes on forever". To you, the two > > > > phrases have different meanings, it seems. So, what do you mean by the > > > > phrase "infinite length"? > > > > > > "Goes on forever" is a property of the set of natural numbers, as well > > > of the elements n as of initial the segments 1,2,3,...,n. > > > > Yes. > > > > > It does not > > > make you believe that there are infinite natural numbers n. > > > > If "infinite natural number" means a natural number which is larger than > > every natural number, then I don't believe there are infinite natural > > numbers. > > But there is a complete initial segment N of N which is larger than any > other segment of N? Not sure what you are asking. What do you mean by "complete initial segment N" and "other segment"? > > > Why does it > > > make you believe that there is an infinite initial segment 1,2,3... ? > > > > Sorry. I don't understand. "1,2,3,..." is the set of natural numbers. > > You just wrote a few lines above that the natural numbers "go on > > forever", and I agreed to it. You seem to be asking me why I believe the > > set of natural numbers goes on forever. But, we just agreed that was > > true. So, what are you asking me? > > If it only goes on forever without being completet anywhere, then there > is no chance to find out whether all natural numbers are sufficient to > enumerate all real numbers or not. I don't know what you mean by "completed anywhere". -- David Marcus
From: imaginatorium on 16 Nov 2006 03:41 Dik T. Winter wrote: > In article <1163602667.089204.113210(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > In > > > principle no axiom is necessary. But you need a few to have some start > > > to work with. > > > > That's the question. By means of axioms you can produce conditional > > truth at most. I am interested in absolute truth. Axioms will not help > > us to find it. I don't think we need any axioms. > > If you want to find absolute truth you should not look at mathematics. Really? There are two groups of order 4; could any truth be more absolute than that? Brian Chandler http://imaginatorium.org
From: Virgil on 16 Nov 2006 04:06 In article <455C15B9.90504(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/16/2006 2:17 AM, Dik T. Winter wrote: > What are the things that represent numbers? > > My humble trial to answer this question is: EB humble? Not bloody likely.
From: Virgil on 16 Nov 2006 04:10
In article <MPG.1fc5d0ae1682037a98990a(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Virgil wrote: > > In article <1163426009.651510.237050(a)h48g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > It is difficult to answer this question, because the expression "set" > > > is occupied in modern mathematics by collections of elements which are > > > actually there (you don't know what that means, imagine just a set as > > > you know it). Such infinite sets do not exist. > > > > While infinite collections in any physical sense are not possible, why > > are imaginary infinities, such as sets of numbers must be, unimaginable? > > For that matter, we can always switch from Platonism to formalism and > declare the question of whether sets really exist to be a philosophical > question. I suspect that WM will be as bad at philosophy as he is at mathematics, but at least it will no longer be a mathematical problem. |