Cantor Confusion
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From: David Marcus on 11 Nov 2006 15:29 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. > > 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 leads to either absurdities or contradictions. Perhaps he thinks the way to avoid these absurdities is to only consider things that can be physically produced. > Now personally I find most of the arguments disingenuous on both > sides. what is a standard mathematical argument that you find disingenuous? > 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. -- David Marcus
From: stephen on 11 Nov 2006 15:39 stephen(a)nomail.com wrote: > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: >> In article <4rjeevFrhvopU2(a)mid.individual.net> Bob Kolker writes: >> > Dik T. Winter wrote: >> > > What surprises me is that opposition against it in this way comes from >> > > especially the German posters in this group. They all want to stress >> > > the difference between actual infinity and potential infinity, that >> > > are indeed not mathematical terms, and most other posted do not know >> > > the difference. Is it still the old religious issue? I think so, >> > > based on the postings of at least one of the German posters. >> > >> > If Cantor's last name had been Shickelgrueber or von Richthoffen, none >> > of this would be written. >> Perhaps. Though I do not think so. But what I am wondering about is the >> preoccupation with "actual infinity" and "potential infinity" with some >> of the posters. I think you are objecting to my "German posters". >> I can shrink this to "German physicists". In this newsgroup there are >> actually three posters that refer to these terms. You may try to find out >> who those three are. > I think a lot of this "opposition" would go away if the word > "transfinite" instead of "infinite" had been used to describe > a set that can be put into a one-to-one correspondence with > a proper subset of itself. The word "infinite" sends people > down strange philosophical paths, as does the word "infinity" > despite the fact that it is not really even used in set theory. > Noone would argue about "transfinity". > Stephen By the way, what is the German word for "transfinite"? I had thought that native German would have less trouble with "infinite" given that "Unendlich" obviously means "unending" whereas "infinite" is derived from Latin and that gives it mysterious properties from the get go. :) I always thought it was rather sensible of the Germans to construct new German words from old German words, instead of the English tradition of plundering other languages. :) Stephen
From: David Marcus on 11 Nov 2006 15:41 Lester Zick wrote: > On Fri, 10 Nov 2006 17:33:16 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >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. > > Curious coming from one whose definitions aren't demonstrably true. Care to define "demonstrably true"? -- David Marcus
From: David Marcus on 11 Nov 2006 15:44 Franziska Neugebauer wrote: > David Marcus wrote: > > > Franziska Neugebauer wrote: > [...] > >> So you are writing about abstract entities like functions and domains > >> whereas WM does not. In WM's view a /notation/ like > >> > >> 1 > >> 1 2 > >> 1 2 3 > >> ... > >> > >> *is* the object under consideration whereas for you it is merely an > >> illustration or reference the abstract entity. > > > > Are you sure? If you ask him if the the above object has less than > > five lines, I think he will say no. > > LOL. I am pretty sure that he will not agree that the above object "can > have" aleph_0 lines. I think you are right. So, for WM, it has more than five, but less than aleph_0. Very curious. I think WM would not agree that the object "can have" any number of lines. -- David Marcus
From: David Marcus on 11 Nov 2006 15:53
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 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. > 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? > Are you going to prove they're true? I don't understand the question. > Clearly like most trained in the modern mathematical arts > you don't take words seriously enough to form critical thoughts. 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. Mathematics is a language. People who learn the language communicate using that language. People who are not fluent in the language may misunderstand what is being said, since many of the same words are used as in English. However, the meaning of many words is different in the two languages. -- David Marcus |