Cantor Confusion
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From: Dik T. Winter on 17 Oct 2006 20:55 In article <1161080567.156919.211680(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > And there is no complete list of computable numbers. But they are > countable. Hence the diagonal argument does not prove anything. You are, again, missing the essential information. There *is* such a list, but it is not computable. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 17 Oct 2006 21:12 In article <eh2fe1$j3e$1(a)mailhub227.itcs.purdue.edu> Dave Seaman <dseaman(a)no.such.host> writes: > On Tue, 17 Oct 2006 02:48:55 GMT, Dik T. Winter wrote: .... > > One of the most serious errors can he found in the statement that > > "to count sets of first cardinality you need ordinals of the second > > class" > > Are you sure you are quoting him correctly? Cantor did say (in fact, > this is a section heading in boldface): It was in an article explaining transfinite "counting". And I am quite sure I did quote him reasonably correct. But the book is on my desk at work, so I will try to find it tomorrow. > > where (I may have first and second wrong), first cardinality means (in > > current terminology) aleph-0 and second class ordinals means (in current > > terminology) omega and larger (until omega^omega or somesuch). The > > error is of course that there is a set of cardinality aleph-0 that can > > be counted using finite ordinals only (the natural numbers). > > No, the second number class according to Cantor is the set of all > ordinals having cardinality aleph_0. So the smallest is omega, but it > goes well beyond omega^omega, all the way up to (but not including) > omega_1. And he is quite explicit about stating that omega is the > smallest member of the second number class. Yes, my upper-bound was wrong. However, I have commented earlier in this newsgroup about the error mentioned above. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 17 Oct 2006 21:31 In article <1161080477.835597.271060(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > stephen(a)nomail.com wrote: > > > b) the photon goes through both slits .... > Without that > unrelativistic assumption case (b) is correct. Traces of Banach-Tarski? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 17 Oct 2006 22:01 cbrown(a)cbrownsystems.com wrote: > I took HdB's statement as "it is not possible to have a theory that is > empirically supported and states 'A and not A, simultaneously' ". > > The fact that there are empirically supported theories which state "it > is not possible for A and ~A to be true simultaneously" doesn't negate > the fact that there are equally empirically supported theories that > state the opposite. Perhaps. Although, I suppose it depends on whether you think the empirical evidence really supports the theory or the people claiming it does are just confused. > > Try this: > > > > http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/bmstartE.htm > > > > I haven't been exposed to Bohm before (FWIW, I studied undergrad > physics in the 70's); Amazingly enough, Bohm developed his theory before then. However, many physicists seem to prefer the illogical "Copenhagen" explanations of Bohr. > it's certainly interesting. See also: > > http://plato.stanford.edu/entries/qm-bohm/ > > for some more interesting philosophical implications at the next level > of description. Thanks. I think I read the first version, but it seems he's revised it. Goldstein has lots of very interesting papers on his website. There are also quite a few interesting books by various people on the topic. > The main limitation I can see in his theory (from my exhaustive 30 > minute study :-) ) is that it seems to rely on the assumption of > non-locality, in a theory that isn't relativistic. > That seems a /lot/ easier to swallow than it would be in a relativistic > theory. I'm not quite sure I follow what you mean. It is very true that standard Bohmian mechanics is not relativistic (Lorentz invariant). From what the experts say, it seems that there isn't an obvious way to make it Lorentz invariant. However, some recent papers of Goldstein and/or his collaborators (I forget exactly which papers) seem to show progress in developing a Lorentz invariant version of the Ghirardi?Rimini?Weber theory (which is similar in many ways to Bohmian mechanics) As for non-locality, Bell showed that non-locality is required by experiment, so we are stuck with it. > Besides, it can't be true. What would Deepak Chopra write about if we > removed indeterminancy?! Hmm. -- David Marcus
From: David Marcus on 17 Oct 2006 22:04
Han de Bruijn wrote: > David Marcus wrote: > > cbrown(a)cbrownsystems.com wrote: > >>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> > >>>I talked about the real world, physics as an empirical science, > >>>not about artifical theoretical constructs. In the real world, > >>>Schrodinger's cat is dead :-( > >> > >>I thought you kept up with physics? > >> > >>http://physicsweb.org/articles/news/4/7/2 > >> > >>The device is conducting electricity in a clockwise fashion; and the > >>device is not conducting electricity in a clockwise fashion. > > > > That interpretation of the experiment is probably dependent on theory. > > Try this: > > > > http://www.mathematik.uni-muenchen.de/~bohmmech/BohmHome/bmstartE.htm > > David Marcus is an adherent of some rather outdated Quantum Mechanical > theories, as have been proposed in the middle of the past century, by > David Bohm. Especially Bohm's theory of "hidden variables", which have > never been found. (And IMO will never be found) Since the "hidden variables" are the positions of the particles, I think we find them all the time. Kind of bizarre to call a particle's position a "hidden variable". Nice to see you are just as illogical in your beliefs about physics as about mathematics. -- David Marcus |