Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: mueckenh on 13 Nov 2006 10:58 Dik T. Winter schrieb: > In article <1163353613.745777.63980(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > > The first falseness is his assumption that every set can be > > > > > well-ordered. > > > > > > > > This assumption is as wrong as the official assumption made by Zermelo. > > > > > > Assumption? > > > > AC is taken to be an axiom. That is nothing but an arbitrary > > assumption. > > Or it is taken not to be an axiom. So you eather reason with AC or > without AC. > > > Well-ordering is so easy connected with AC that we can > > state: Well-ordering has been assumed. > > Not in set theory. AC is an axiom that may or may not hold, depending > on the branch you are following. Zermelo considered it as "has to be taken". > > > > Do you know what that term means in mathematics? It means from a > > > chosen set of axioms using ordinary logic. > > > > The chosing of the set is an arbitrary act. It is not reduced to first > > principles in modern math. Cantor had first principles! > > They were in fact axioms, although he did not state it as such. And the > first principles he did chose where of course arbitrary. Oh, he would rotate in his grave if he heard you. Of course he only assumed those first principles which were true in nature or reality. > > > > > Cantor considered well-ordering as a first principle, > > > > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > > > > Zermelo was right? > > > > > > Cantor did state it without suggesting either that it was a first > > > principle or something else. He just assumed it. And he was wrong > > > with that assumption. > > > > He was as wrong as Zermelo, not a bit more. > > Oh, well, do set theory without AC. No problem. There is a number of > people doing it. Zermelo did not belong to this group. Regards, WM
From: mueckenh on 13 Nov 2006 11:00 Dik T. Winter schrieb: > In article <1163426814.926776.54020(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > ... > > May be so with some sets. A set of natural numbers includes its > > cardinal number. > > I did not know that. Care to prove that the set {1, 3, 5, 7} includes > the cardinal 4? Thank you for your attention. Correction: An initial segment of natural numbers includes its cardinal number. Regards, WM
From: Franziska Neugebauer on 13 Nov 2006 11:43 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Franziska Neugebauer schrieb: >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> >> [...] >> >> >> Are there really three vertices in WM's "triangle"? >> >> > If finished infinities [...] >> >> Verbiage. >> > Yes. But, sorry to see, it is the fundament of modern mathematics. >> "Finished infinities" is your wording. > Precisely describing the fundament of modern mathematics. Cbjre Bs Oryvrs. F. N. -- xyz
From: Lester Zick on 13 Nov 2006 13:18 On Sat, 11 Nov 2006 13:50:38 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >stephen(a)nomail.com wrote: >> 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". > >You could be right. Although, it seems unfair of the cranks to dictate >what words mathematicians can appropriate. It is hard to make up good >names. We have enough names like "second category" as it is. Once more we have this sloppy word usage on the part of those who self righteously proclaim their mathematical rectitude. What exactly does "crank" mean besides "crank(x)=disagree(u)"? Modern mathematikers routinely appropriate words and make up private definitions for them as if they were the sole arbiters of truth in mathematical terms. Of course it's hard to make up good names especially when mathematikers insist on private definitions cast in parochial terms of modern math. ~v~~
From: Lester Zick on 13 Nov 2006 13:36
On Sat, 11 Nov 2006 15:29:18 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> 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. >> >> 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). "Provability" of what pray tell? If you're not concerned with proving the truth of what you say in mathematics exactly when are you not discussing philosophy every time you say anything in mathematics? >> 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. 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. > He seems to be saying that >the notion of a completed infinity leads to either absurdities or >contradictions. 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. Niether seems very convincing yet adherents of each pretend they are. >> 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. ~v~~ |