Cantor Confusion
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From: mueckenh on 14 Oct 2006 10:08 David Marcus schrieb: > Han de Bruijn wrote: > > David Marcus wrote: > > > Is your claim only that set theory is not useful or is contrary to > > > common sense? Or, are you claiming something more, e.g., that set theory > > > is mathematically inconsistent? > > > > I said that set theory is not *very* useful. I have developed (limited) > > set theoretic applications myself, so I don't say it is useless. > > > Yes, a great deal of set theory is contrary to common sense. Especially > > the infinitary part of it (: say cardinals, ordinals, aleph_0). > > > > I'm not interested in the question whether set theory is mathematically > > inconsistent. What bothers me is whether it is _physically_ inconsistent > > and I think - worse: I know - that it is. > > What does "physically inconsistent" mean? Wouldn't your comments be > better posted to sci.physics? Most people in sci.math are (or at least > think they are) discussing mathematics. Even worse, most of them truly believe their ideas on mathematics and the functions (of mathematics as well as of their brains) were independent of physics Regards, WM
From: William Hughes on 14 Oct 2006 10:09 mueckenh(a)rz.fh-augsburg.de wrote: > David R Tribble schrieb: > > > mueckenh wrote: > > >> Yes, but the assertion of Fraenkel and Levy was: "but if he lived > > >> forever then no part of his biography would remain unwritten". That is > > >> wrong, because the major part remains unwritten. > > > > > > > David R Tribble wrote: > > >> What part? > > > > > > > mueckenh wrote: > > >> That part accumulated to year t, i.e., 364*t. > > > > > > > David R Tribble schrieb: > > >> It's stated that he lives forever, so what value of t you are using? > > > > > > > mueckenh wrote: > > > You can use any positive value of t and prove that the unwritten part > > > n(t) for t > t_0 is larger than the unwritten part for t_0. You can > > > even use the formal convergence criterion for the convergent function > > > 1/n(t). There is no room for he assumption that the written part could > > > ever surpass the unwritten part. > > > > If I use any positive value for t, then there is still the positive > > value t+1 (and 2t, t^2, and all the rest), none of which satisfies the > > "lives forever" part. So I can't use any positive value of t. > > > Obtain the limit of the balls in the vase for t --> noon like the limit > of n from Lim {n --> oo} (1/n) = 0. > > > > mueckenh wrote: > > >> If you think Lim {t-->oo} 364*t = 0, we need not continue to discuss. > > > > > > > David R Tribble schrieb: > > >> I don't think anyone has said that. I merely asked which pages (days) > > >> in the "major part" of the book don't get written. Do you have a > > >> certain t in mind? > > > > > > > mueckenh wrote: > > > I merely answer that it is completely irrelevant to speak of certain t. > > > > Then why did you say "use any positive value of t"? > > > > > The paradox is raised only by the asumption that the set of all t did > > > exist. > > > > What paradox? > > The result Lim{n-->oo} 9n = 0 where mathematics leads to Lim{n-->oo} > 1/9n = 0. No, in mathematics Lim{n->oo} 9n = oo. Since the number of balls in the vase at noon has nothing to do with Lim{n->oo} 9n, a claim that the number of balls in the vase at noon is 0 is not a claim that Lim{n->oo} 9n = 0. - William Hughes > > Regards, WM
From: mueckenh on 14 Oct 2006 10:11 Virgil schrieb: > I do not object to the constraints of the mathematics of physics when > doing physics, but why should I be so constrained when not doing physics? Because whatever you are doing, you are doing something, and "doing" means utilizing and applying physics. Regards, WM
From: mueckenh on 14 Oct 2006 10:13 David Marcus schrieb: > Han de Bruijn wrote: > > Existence in physics is given by nature itself. And I am a physicist by > > education. The mantra is: A little bit of Physics would be NO idleness > > in Mathematics. (A bit cryptic - so it seems - but it will do.) > > So, "completed infinity" does exist (in standard mathematics, the topic > of discussion). If you think that what mueckenh wrote is correct (in > standard mathematics), please give a (mathematical) reason. Potential infinity cannot be surpassed by another infinity. Infinite sets with different cardinals aleph_0 and 2^aleph_0 are either actually existing or non-existing. Regards, WM
From: mueckenh on 14 Oct 2006 10:16
Dik T. Winter schrieb: > In article <ddeb9$452e55fe$82a1e228$16456(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > Dik T. Winter wrote: > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > It is not > > > > contradictory to say that in a finite set of numbers there need not be > > > > a largest. > > > > > > It contradicts the definition of "finite set". But I know that you are > > > not interested in definitions. > > > > Set Theory is simply not very useful. Yes. The finite parts are useful, the transfinite part is useless. > > Oh. So you think that Banach spaces are not very useful? You think that > a book like "The Algebraic Eigenvalue Problem" is not very useful? You > may note that both are heavily based on set theory. Only by accident, mainly because of Bourbaki. In, say, 2020, there will be Algebra but without any transfinite set theory. Regards, WM |