Cantor Confusion
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From: Virgil on 14 Oct 2006 17:52 In article <1160834930.893285.295850(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > 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 Why should mathematicians be tied down to the misunderstandings of physicists? Do physicists claim to have a unique channel to "truth"? Are they never wrong? Does "God" speak directly into their ears?
From: Virgil on 14 Oct 2006 17:57 In article <1160835084.478453.305880(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. That may be a physicists view of the world, but by that same measure, whenever one is doing physics, what he is really doing is math, so that mathematicians should rule.
From: Virgil on 14 Oct 2006 17:58 In article <1160835219.191147.313300(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Infinite > sets with different cardinals aleph_0 and 2^aleph_0 are either actually > existing or non-existing. So what else is new?
From: Virgil on 14 Oct 2006 18:02 In article <1160835375.441299.190640(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Dream on. Once Cantor opened that Pandora's box of the transfinite, nobody ever will have the power to close it again. We'll just have to learn to live with it.
From: David Marcus on 14 Oct 2006 18:04
mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David Marcus schrieb: > > > > > > > I am sure you are able to translate brief notions like "to enter, to > > > > > > > escape" etc. by yourself into terms of increasing or decreasing values > > > > > > > of variables of sets, if this seems necessary to you. Here, without > > > > > > > being in possession of suitable symbols, it would become a bit tedious. > > > > > > > > > > > > Yes, I can translate it myself. However, that would only tell me how I > > > > > > interpret the problem. > > > > > > > > > > Hasn't it become clear by the discussion? > > > > > > > > > > I use two variables for sequences of sets. Further I use a function. I > > > > > use the natural numbers t to denote the index number. The balls are > > > > > simply the natural numbers. I speak of "balls" in order to not > > > > > intermingle these numbers with the index-numbers. > > > > > > > > > > The set of balls having entered the vase may be denoted by X(t). > > > > > So we have the mathematical definition: > > > > > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > > > > > There is a bijection between t and X(t). > > > > > > > > t is a number and X(t) is a set. If t = 1, then your sentence says, > > > > "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. > > > > So, I don't follow. What do you mean, please? > > > > > > That what is written. There is a bijection between the set of all > > > numbers t and the set of all sets X(t). 1 is mapped on X(1), 2 is > > > mapped on X(2), and so on. Is there anythng wrong? > > > > > > > > > The set of balls having left the vase is described by Y(t). So we have > > > > > the mathematical definition: > > > > > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > > > > > There is a bijection between t and Y(t). > > > > > > Here we have the same as above with Y instead of X. > > > > > > > > > > And the cardinal number of the set of balls remaining in the vase is > > > > > Z(t). So we have the mathematical definition: > > > > > Z(t) = 9t with Z(t) > 0 for every t > 0. > > > > > There is a bijection between t and Z(t). > > > > Sorry, but I'm not following. I asked you for a translation into > > Mathematics of the ball and vase problem. The problem in English ends > > with a question mark. I don't see the question mark in your translation > > above. Would you please state just the problem in both English and > > Mathematics? > > Sorry, I don't know how to state the question " Hasn't it become clear > by the discussion?" in what you think is mathematics. > > By the way: Every means to draw conclusions and to calculate results is > mathematics. There is no need to prefer a certain language (unless > there is someone who cannot speak another one). Would you assert > Archimedes did not do mathematics, because he used only the Greek > language and had not yet special symbols but Greek letters to denote > numbers? The language of Mathematics has evolved over time. If you assert that the ball and vase problem shows that modern Mathematics contains a contradiction, then please state the problem using the language of modern Mathematics. "Balls" and "vases" are not part of Mathematics, although people may use such language to talk informally about Mathematics. So, please state the ball and vase problem using the language of Mathematics (e.g., "sets", "functions", "integers", "reals") so that we can see what mathematical problem you are talking about. What you wrote above (i.e., X(t), Y(t), Z(t)) uses the language of Mathematics, but (as I pointed out above) is not the statement of a problem since it doesn't end with a question. Please state just the problem without also including any part of the solution to the problem. -- David Marcus |