Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: David Marcus on 19 Oct 2006 00:59 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > You have an interpretation of a thought experiment that differs from > > > > the interpretation of other people. That doesn't make set theory > > > > inconsistent. It just makes set theory not suitable for your intuitions > > > > regarding the thought experiment. > > > > > > The inconsistency is that > > > 1) For the balls inserted until noon, you can find the result: It is > > > the set N. > > > 2) For the balls removed until noon, you can find the result: It is the > > > set N. > > > 3) For the balls remaining at noon, the same arguments of continuity > > > which lead to (1) and (2) cannot apply. > > > > > > This is the contradiction. No matter what the result (3) may be. > > > > In other words, just as I said, you have an interpretation of a thought > > experiment that differs from the interpretation of other people, > > And according to your intuition, the axioms of ZFC and the rules of > logics are more likely to believe and do not contradict the fact that > accumulating numbers will not yield an empty set? Here is the problem: For n = 1,2,..., define A_n = 12 - 1 / 2^(floor((n-1)/10)), R_n = 12 - 1 / 2^(n-1). For n = 1,2,..., define a function B_n by B_n(t) = 1 if A_n < t < R_n, 0 if t < A_n or t > R_n, undefined if t = A_n or t = R_n. Let V(t) = sum{n=1}^infty B_n(t). What is V(12)? Are you saying that you can prove that V(12) = 0? If not, then please state precisely your claim. -- David Marcus
From: David Marcus on 19 Oct 2006 01:25 David Marcus wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Han de Bruijn wrote: > > > > David Marcus wrote: > > > > > cbrown(a)cbrownsystems.com wrote: > > > > >>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 David Marcus is an adherent of some rather outdated Quantum Mechanical 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. > > > > Hidden variables do exist in exactly the same manner as the well-order > > of the reals. > > Waves have no fixed positions or velocities. > > > > I developed a theory describing hidden variables obeying the Bell > > inequalities (A Review of Extended Probabilities, Phys. Rep. 133 (1986) > > 337). Alas, it turned out that negative probabilities were required. > > Bohmian Mechanics is a deterministic theory that avoids the measurement > problem, satisfies Bell's Inequality Sorry. I should have said "violates Bell's Inequality". > (as do all theories of quantum mechanics), > agrees with all experiments, and doesn't produce negative > probabilities. And, Bohmian Mechanics is non-relativistic. > So, it seems to be a better theory than the one you > constructed. -- David Marcus
From: cbrown on 19 Oct 2006 01:37 David Marcus wrote: > 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. Until an experiment which distingushes between two similar but distinct theories is proposed and perfomed, there's no scientific way of knowing whther adherents to either theory are confused or correct. > > > > 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. And I'm surprised that I wasn't exposed to it at MIT. > However, many > physicists seem to prefer the illogical "Copenhagen" explanations of > Bohr. > I would claim that the Copenhagen interpretation is not illogical /on its own terms/ - it's simply inconsistent with other interpretations (including "common sense"). > > 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. > Non-locality isn't a particularly vexing issue in a non-relativistic setting, because the question of what we mean by "event A is simultaneous with event B" is perfectly clear. In a relativistic setting "event A is simultaneous with event B" is much more complicated; so when we say "non-locality implies that some event A simultaneously affects event B", this is much more complicated. I might be wrong, but isn't that the point of the EPR experiment? Not that I claim to fully understand (after my 30 minutes of study) Bohm's theory, but I can imagine that the idea that changes in the pilot wave's state propagate "at an infinite speed" (i.e., is non-local) is going to be a real issue here when we consider relativistic frames of reference. > 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) > This would then be the "GRW" that is described at the link you supplied. > As for non-locality, Bell showed that non-locality is required by > experiment, so we are stuck with it. > Sure; just as we are stuck with relativity. > > Besides, it can't be true. What would Deepak Chopra write about if we > > removed indeterminancy?! > > Hmm. > > -- > David Marcus
From: Han de Bruijn on 19 Oct 2006 03:47 Dik T. Winter wrote: > I think that, compared to Cantor, in modern set theory potential and actual > infinity are split up again. The contents of the set N form only a potential > infinity, on the other hand, the *size* is an actual infinity. And you call _that_ "thinking" ?! Han de Bruijn
From: Han de Bruijn on 19 Oct 2006 03:52
David Marcus wrote: > Han de Bruijn wrote: > >>David Marcus wrote: >> >>>Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> >>>>David Marcus schreef: >>>> >>>>>stephen(a)nomail.com wrote: >>>>> >>>>>>What does it mean for a thing in the real world to be true? >>>>>>How do you know if a thing in the real world is true? >>>>>> >>>>>>Consider the twin slit experiment. Is the fact that none of >>>>>>the following accurately describe the situation an inconsistency? >>>>>> a) the photon goes through one slit >>>>>> b) the photon goes through both slits >>>>>> c) the photon goes through neither slit >>>>> >>>>>In Bohmian Mechanics (and similar theories), the photon goes through >>>>>only one slit. Physicists could learn something about logical thinking >>>> >>>>>from mathematicians. >>>> >>>>Sure, theories. Can't you talk about something else but "theories"? >>> >>>In the real world, the photon undoubtedly goes through one slit. >> >>Not in my real world of physics. >> >>>However, discussing physics with you is undoubtedly even more useless >>>than discussing mathematics. >> >>BTW. Are you a physicist? > > Why do you ask? Maybe because it's handsome to have an impression of the LoOrHigh level at which I can debate with you about physics. Han de Bruijn |