Cantor Confusion
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From: mueckenh on 16 Oct 2006 15:48 Dik T. Winter schrieb: > In article <1160857922.727908.74990(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > ... > > > > We know that a set of numbers consisting altogether of 100 bits cannot > > > > contain more than 100 numbers. Therefore the set is finite. The largest > > > > number of such a set cannot be determined, as far as I know. > > > > > > That set is indeterminate. Just use Ascii notation. The string > > > "Graham's number" fits in 100 bits. > > > > > > > Could you determine it? Or would you prefer to define that such ideas > > > > do not belong to mathematics? Then I would not be interested in that > > > > definition. > > > > > > It is an indeterminate set. > > > > The set of bits is determined: exactly 100. What you can build from 100 > > bits belongs to the power set of this set. It is probably large but > > certainly not infinite. > > The set of bits is determined. The set of numbers you can build from it is > indeterminate. But whatever way you build your set of numbers, the size is > certainly <= 2^100, The cardinal number of any such set is =< 100 > and whatever way you build your set of numbers, there > is a largest one that can be determined. But nobody knows it. That is just the case with N, only a bit more sophisticated. >> "...classical logic was abstracted from the mathematics of finite sets >> and their subsets...Forgetful of this limited origin, one afterwards >> mistook that logic for something above and prior to all mathematics, >> and finally applied it, without justification, to the mathematics of >> infinite sets. This is the Fall and original sin of [Cantor's] set >> theory ..." (Weyl) > Oh, perhaps. What is the relevance to mathematics? Nothing, as soon as we withdraw to call set theory mathematics. Regards, WM
From: William Hughes on 16 Oct 2006 15:51 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > One cannot compute a list of all computable numbers. By this > > > definition, > > > (1) the computable numbers are uncountable. > > > (2) There is no question, that the computable numbers form a countable > > > set. > > > This is a contradiction. It is not necessary to come up with a list of > > > all computable numbers. > > > > Nope. You are mixing two approaches.. > > > > A set X is countable if there exists a surjective function,f, > > from the natural numbers to X > > > > There are two possibilities > > No. > > > > A: you allow arbitrary functions f > > > > B: you allow only computable functions f > > What is a function which is not computable? > > > > In case A, the computable reals are countable [ but you > > also have arbitrary reals, and the set of reals (computable > > and arbitrary) is uncountable.] > > > > In case B. the computable reals are not countable > > (i.e. there is no list of computable reals) > > > > You cannot arrive at a contradiction by taking one result from > > case B (the computable reals are uncountable) > > and one result from case A > > (the computable reals are countable). > > I take not at all results from your mysterious cases. > I see: Every list of computable numbers supplies a diagonal number > which is computable but not contained in the list. > And I see: Every list of real numbers supplies a diagonal number which > is not contained in the list. > There is absolutely no difference. > If real numbers are also computable numbers there is absoluetly no difference. If real numbers contain both computable real numbers and non-computable real numbers then there is a difference. But in either case Every list of real numbers supplies a diagonal number which is not contained in the list. so any way you cut it there is no complete list of real numbers. - William Hughes
From: Han.deBruijn on 16 Oct 2006 15:51 cbrown(a)cbrownsystems.com schreef: > Han de Bruijn wrote: > > 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? > > The cat in the box is dead; and the cat in the box is not dead. 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 :-( Han de Bruijn
From: Han.deBruijn on 16 Oct 2006 15:52 Virgil schreef: > In article <78866$45333919$82a1e228$8559(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Virgil wrote: > > > > > In article <1160933229.072292.316580(a)e3g2000cwe.googlegroups.com>, > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > >>Dik T. Winter schreef: > > >> > > >>>In article <1160856895.115824.134080(a)b28g2000cwb.googlegroups.com> > > >>>Han.deBruijn(a)DTO.TUDelft.NL writes: > > >>> > stephen(a)nomail.com schreef: > > >>> > > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > > >>>... > > >>> > > > Pray warn me when 2 has changed sufficiently to be the square of a > > >>> > > > rational. > > >>> > > > I would not like to miss that moment. > > >>> > > > > >>> > > The day the circle is squared cannot be to far behind. > > >>> > > > >>> > Come on, guys! You all know that, in the world of approximations, > > >>> > 2 _is_ the square of a rational and the circle _is_ squared. > > >>> > > >>>I thought you were talking mathematics? > > >> > > >>I thought approximations were a part of mathematics? > > > > > > But approximations are not all of mathematics in the way that HdB > > > preaches, and "approximately equal" is still mathematically > > > distinguishable from "exactly equal". > > > > How? > > By allowing a possibly non-zero distance between approximately equal > values, but not allowing it for exactly equal values, of course! > > Note that, starting with any rational number, x_0, the sequence of > rationals defined recursively by x_{n+1} = (x_n + 2/x_n)/2 eventually > approximates sqrt(2) to any desired degree of accuracy short of > exactness, but no member of that sequence is ever exactly equal to > sqrt(2). What is "equal"? Han de Bruijn
From: Han.deBruijn on 16 Oct 2006 15:54
Virgil schreef: > In article <18bda$45333c50$82a1e228$8972(a)news2.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Sigh! Start digging into my website. I've said more about mathematics > > than anybody else in 'sci.math'. > > About as much as JSH, perhaps. Uh, no. Much more! Han de Bruijn |