Cantor Confusion
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From: Dik T. Winter on 16 Oct 2006 22:04 In article <1161004988.055723.235170(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Virgil schrieb: > > > > According to the ZFC system: The vase is empty at noon, because all > > > natural numbers left it before noon. > > > By means of the ZFC system we can formulate sequences and their limits > > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > > And from this it follows that the vase is not empty at noon. > > > > By what axiom do you conclude that the limit as t increases towards noon > > of any function and the value of that function at noon must be the same? > > By that or those axiom(s) which lead(s) to the result lim {t-->oo} 1/t > = 0. The Peano axioms and the definitions show that that limit is indeed 0. They do not show that 1/oo = 0 (this is just abuse of notation). In the same way, lim {x --> 0} f(x) can be 0, while f(0) != 0. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: cbrown on 16 Oct 2006 22:16 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. 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. > 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); 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. 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. Besides, it can't be true. What would Deepak Chopra write about if we removed indeterminancy?! Cheers - Chas
From: Dik T. Winter on 16 Oct 2006 22:10 In article <1161007554.513186.56640(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > The function f(t) = 9t is continuous, because the function 1/9t is > > > continuous. > > > > Yes, but that is not the number of balls in the vase. > > > For the t-th transaction 9t is the number of balls in the vase. Let me clarify. WM apparently asserts that if a function g(x) is continuous at some point, so is 1/g(x). That is (obviously) false. sin(x) is continuous at x = 0, while 1/sin(x) is not. Moreover, when we let t go to infinity, 1/9t is *not* continuous at infinity (whatever that may mean). We can only define the limit, not the function value. -- 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 16 Oct 2006 22:25 In article <1161008572.469763.93200(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > jpalecek(a)web.de schrieb: .... > > The fact that you cannot compute a list of all computable reals does > > not mean that there is no list of all computable numbers. There is one, > > and it is not computable. > > > The fact that you cannot compute a list of all reals does not mean that > there is no list of all reals. There is one, but it is not possible to > publish this list. You are seriously wrong. A computable number is a number that will be given by a Turing machine. More precise, when a Turing machine gives a result when asked for the n-th digit for arbitrary n, the number that is constructed from those digits is computable. The number of Turing machines is countable (because the are a finit sequence of symbols from a finite alphabet). However, that numbering will not give you a list of computable numbers as some Turing machines will not halt. Computing the list of computable numbers means, in essence, that you can determine which Turing machines do halt and which do not halt. So when you give a list of computable numbers (by providing their Turing machine), you can calculate an anti-diagonal number, but there is no way to prove that it is produced by a halting Turing machine or not. On the other hand, any sequence of decimal numbers is *by the very definition of the real numbers* a real number. So, given a list of decimal numbers, the diagonal argument gives a sequence of decimal digits that is different from all the sequences of decimal digits in the list, and so is not in the list, and is, by definition, a real number. > > Every list of reals can be shown incomplete in exactly the same way as > every list of computable reals can be shown incomplete. > > Regards, WM > -- 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 16 Oct 2006 22:28
In article <1eabb$453398c7$82a1e228$10949(a)news1.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > stephen(a)nomail.com wrote: > > > Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > > > >>Dik T. Winter wrote: > > > >>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> > >>>Han.deBruijn(a)DTO.TUDelft.NL writes: > >>> > Virgil schreef: > >>>... > >>> > > 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 (empirical) physics is an absolute guarantee for consistency? > >>> > >>>Can you prove that? > > > >>Is it possible to live in a (physical) world that is inconsistent? > > > > Perhaps. How could we know? > > How can we know, heh? Can things in the real world be true AND false > (: definition of inconsistency) at the same time? Weren't we talking about empirical physics? Or, as you said, the world of approximation, where sqrt(2) is rational? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |