Cantor Confusion
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From: Virgil on 13 Oct 2006 04:19 In article <b8869$452f4a39$82a1e228$32738(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > William Hughes wrote: > > > William Hughes wrote: > > > >>Han de Bruijn wrote: > >> > >>>Dik T. Winter wrote: > >>> > >>>>In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> > >>>>mueckenh(a)rz.fh-augsburg.de writes: > >>>>... > >>>> > If every digit position is well defined, then 0.111... is covered "up > >>>> > to every position" by the list numbers, which are simply the natural > >>>> > indizes. I claim that covering "up to every" implies covering "every". > >>>> > >>>>Yes, you claim. Without proof. You state it is true for each finite > >>>>sequence, so it is also true for the infinite sequence. That conclusion > >>>>is simply wrong. > >>> > >>>That conclusion is simply right. And yours is wrong. Completed infinity > >>>does not exist. So _each_ finite sequence "means" the infinite sequence. > >> > >>If you insist that '_each_ finite sequence "means" the infinite > >>sequence' > >>you have the empty conclusion that if each finite sequence 0.1,0.11, > >>0.111, ... > >>is covered by a list of numbers then each finite sequence > >>0.1,0.11, 0.111, ... is covered by a list of numbers. > >>You do not have, there exists one element of the list of numbers > >>which covers 0.1,0.11, 0.111, ... . > > > > To ellaborate: > > > > To get the result you want (there is a single number which covers > > all finite sequences) it is not enough to deny completed infinity. You > > must > > also insist that 0.111... has an end (i.e you must also deny potential > > infinity). You cannot say that the 0.111... has an end but it is > > arbitrary > > because "arbitrary" is not a property ends can have. > > How about: I'm not interested in "the end". I don't know where the end > is. And I don't care as well. As long as the end is somewhere where it > causes a uncertainity which is acceptable for my purpose. But what if "the end" isn't anywhere because there isn't one? As soon as you posit an end, you run into problems. You would be much better off saying that all such questions about an end to the naturals are unanswerable, and stick to what you can explicitly construct.
From: Han de Bruijn on 13 Oct 2006 04:21 Virgil wrote: > In article <1160640069.503756.100380(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >>Dik T. Winter schrieb: >> >>>In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> >>>"Albrecht" <albstorz(a)gmx.de> writes: >>> > David Marcus schrieb: >>>... >>> > > I don't follow. How do you know that the procedure that you gave >>> > > actually "defines/constructs" a natural number d? It seems that you >>> > > keep >>> > > adding more and more digits to the number that you are constructing. >>> > >>> > What is the difference to the diagonal argument by Cantor? >>> >>>That a (to the right after a decimal point) infinite string of decimal >>>digits defines a real number, but that a (to the left) infinite string >>>of decimal digits does not define a natural number. >> >>And why is this so? Because an infinite string of digits is not at all >>defined. Only by the factors 10^(-n) this is veiled. The due digits >>become more and more unimportant because their contributions to the >>number size are pulled down by the increasing exponents. But this has >>been forgotten by Cantor whose diagonal proof attaches the same weight >>to every digit. That is obviously wrong. > > Cantor merely assumes, as do most mathematicians, that in mathematics, > as contrasted with physics, there need not ever be a last significant > digit in a decimal expansion. > > HdB assumes otherwise, but has not the power to impose his assumptions > on others. I thought you were responding to Wolfgang Mueckenheim. But indeed, HdB has not the "power" to impose his assumptions on WM, neither has WM the "power" to impose his assumptions on HdB. But nevertheless WM and HdB do agree with each other on many issues, as an act of scientific reasoning and free will. Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 04:30 Virgil wrote: > In article <1160648741.707624.62340(a)m7g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > >>Apply your knowledge to the balls of the vase. > > Which knowledge tells me that at noon each and every ball has been > removed from the vase. Not AT noon but BEFORE noon. True then. And ten other balls have been inserted. > those times do not include noon or go past noon. True. Because those times are a FAKE. > You are assuming properties not given. No. YOU are assuming properties not given: _you_ are assuming that your fake parameter falsely called time has the properties of physical time, such that it can pass through noon, which it cannot. >>Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. Precisely! > Your assumption that some ball that has been removed has not been > removed is even more absurd. That ball has been replaced by nine others. So what? Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 04:44 Virgil wrote: > In article <463c0$452e265e$82a1e228$30902(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>mueckenh(a)rz.fh-augsburg.de wrote in response to Virgil: >> >>>For the vase problem with the number n(t) of balls in the vase after t >>>transactions we can find always a positive eps such that for t > t_0: >>>1/n(t) < eps, hence n(t) larger than an arbitrary positive number. >>> >>>Therefore, your assumption of lim {t-->oo} n(t) = 0 is absurd. >> >>Precisely! Virgil doesn't know how to handle limits. > > I merely note that there is no requirement in the problem that > the limit be the value at noon. The limit at noon - iff it existed - would be the value at noon. > We have an increasing sequence of times t_n which converge to noon > so lim_{n in N} t_n = noon Right. Iff given alone this sequence of (no)times: lim_{n in N} t_n = 0. > We also have a number of balls b(t_n) = 9*n Correct. > You are claiming , essentially that all limits exist and > lim_{n in N} b( t_n ) = b( lim_{n in N} t_n ) No! I'm NOT claiming that. > That would require that b(t) be continuous at t = noon, but nothing in > the problem either requires nor even specifically allows this. b(t) diverges at noon. Thus b(noon) is undefined. Thus t is not time. It's impossible in a renormalized mathematics that limits are different from the actual values of functions at that place. See: http://hdebruijn.soo.dto.tudelft.nl/QED/ http://groups.google.nl/group/sci.math/msg/c83b58832ecdff3f?hl=en& Han de Bruijn
From: Han de Bruijn on 13 Oct 2006 04:45
Virgil wrote: > In article <990aa$452e542e$82a1e228$16180(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > >>Completed infinity >>does not exist. > > There are more things in heaven and earth, Horatio, > Than are dreamt of in your philosophy. I'm not a philosopher. I'm a scientist. Han de Bruijn |