Cantor Confusion
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From: David Marcus on 19 Oct 2006 00:31 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. I've never seen "potential infinity" or "actual infinity" in any textbook I've used. -- David Marcus
From: David Marcus on 19 Oct 2006 00:40 Dik T. Winter wrote: > In article <1161183237.727249.154740(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > > In article <1161079802.120515.175530(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > ... > > > > 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. > > > > > > There are quite a few obvious reasons. > > > (1) 1) is not because of continuity > > The reason that continuity plays no role is because the function of the > number of balls in the vase when written as a function of t is discontinuous > at infinitely many positions. > > > Why then? > > Depending on the person who writes about the problem. One possibility is > defining a set of functions dependent on time t that tells whether at time > t (not t now goes from -1 to 0) ball n is in the vase or not. Addition of > those functions gives the number of balls present in the vase at time t. > But all depends on the exact mathematical formulation of the problem. There > is none (in my opinion, but I have already had discussions with others > about this point, so I will not repeat them). > > > > (3) no continuity reasoning can lead to the result that the balls > > > remaining at noon is the set N. > > > > But more than 1. > > No continuity reasoning can lead to anything. The functions are inherently > not continuous. When we look at the functions when we let t go from -1 > to 0 we see: > Balls added since time t = -1: > entier(- log_2(- t)) * 10 > Balls removed since time t = -1: > entier(- log_2(- t)) > Balls remaining since time t = -1: > entier(- log_2(- t)) * 9 > None of these functions is continuous, and in fact they have discontinuties > infinitely often within any time frame that contains t = 0. Moreover, none > is defined at t = 0. They all have a singularity at t = 0 (and are not > extensible in a sensible way beyond t = 0, unless you allow for imaginary > balls, but I have no idea what the entier function does with imaginary > values). > > You can *model* the problem in different ways in mathematics, and one of > those models leads to 0 balls in the vase at time 0. And there are > presumably other models that give different results. So, unless the > problem is stated in a mathematically proper way, actually nothing can > be said about it. Indeed. Before anyone gives their solution to the problem, they should first restate the problem in Mathematics. We try to teach students to do this in high school when they work "word problems". It is something that should always be done as the first step of any mathematical analysis. -- David Marcus
From: David Marcus on 19 Oct 2006 00:48 MoeBlee wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > but very rarely the important distinction between potential > > and actual infinity, > > Of course no such distinction is mentioned, since they are not > predicates used in set theory. If you want them, then it's up to you to > propose a theory that has them as primitive or defined. Indeed. A basic principle of Mathematics is the necessity to define your terms. There are standard definitions for many words, but many people posting to this thread don't seem to know them and/or think that many words that don't have standard definitions will be understood by everyone (or perhaps don't even know what a mathematical definition is). A careful writer always defines any terms that may be unclear. -- David Marcus
From: David Marcus on 19 Oct 2006 00:50 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > First you say the notion of 'rational relation' (whatever that means) > > "cannot be expressed by mathematical notion". Then you challenge me to > > say what part of your proof is in conflict with set theory. What is the > > notion of 'rational relation' that "cannot be expressed by mathematical > > notion"? Are defining a certain relation in set theory or are you > > definining a relation you claim not to exist in set theory? > > Meanwhile there are many who understand the binary tree. Perhaps you > will follow the discussion, then you may understand it too. But, no one seems to understand your binary tree. Please state your claim and its proof using standard terminology and words that you clearly define. For example, use the terminology that Halmos uses in "Naive Set Theory". If you don't like that book, then pick a book you like and tell us what it is. -- David Marcus
From: David Marcus on 19 Oct 2006 00:57
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 (as do all theories of quantum mechanics), agrees with all experiments, and doesn't produce negative probabilities. So, it seems to be a better theory than the one you constructed. -- David Marcus |