Cantor Confusion
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From: David Marcus on 13 Oct 2006 13:54 Han de Bruijn wrote: > David Marcus 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. > > > > Are you saying that "completed infinity" does not exist in standard > > mathematics? If so, please define "completed infinity". If not, then > > please define "exist". > > Completed infinity does "exist" in standard mathematics. It's embodied > by Cantor's Set Theory (: cardinals, ordinals, aleph_0 and some such). > > > Define "exist". > > Existence in physics is given by nature itself. And I am a physicist by > education. The mantra is: A little bit of Physics would be NO idleness > in Mathematics. (A bit cryptic - so it seems - but it will do.) So, "completed infinity" does exist (in standard mathematics, the topic of discussion). If you think that what mueckenh wrote is correct (in standard mathematics), please give a (mathematical) reason. -- David Marcus
From: Virgil on 13 Oct 2006 16:24 In article <14624$452f4c99$82a1e228$1582(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > 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 Scientific reasoning does not work in mathematical contexts.
From: Virgil on 13 Oct 2006 16:28 In article <ecef7$452f4eb8$82a1e228$2523(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > 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. All of which will be removed before noon, as will any others 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. I am assuming time is a real valued variable which can take any real value including 0 and positive values. > > >>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? And all of those, and their replacements and so on ad infinitum have been removed. > > Han de Bruijn
From: Virgil on 13 Oct 2006 16:34 In article <b2f47$452f51eb$82a1e228$2726(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > 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. Then as there is no limit, there is also no requirement for a 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. You seem to be. > > > 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 is not physical time, but the problem is not a physical problem. > > It's impossible in a renormalized mathematics that limits are different > from the actual values of functions at that place. The world of mathematics exists quite happily without any enforced renormalization. Those who insist on applying the "rules" of the physical world to situations in which those rules are nonsense, should not be surprised when their results are nonsense.
From: Virgil on 13 Oct 2006 16:35
In article <661f8$452f5232$82a1e228$2726(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > 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. Even scientists are allowed philosophies. Horatio himself was not a philosopher, either, but Hamlet conceded him a philosophy. |