Cantor Confusion
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From: William Hughes on 13 Oct 2006 07:47 Han de Bruijn 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. > > 0.11111111111111111111111111111111111111111111111 = 1 (binary) Okay. > > 0.000000000000000000000000000000000000000000000001 : error accepted. > If you want to claim that there is a largest integer, knock youself out. However, your claims to be doing a mathematics based on "common sense" have just gone poof. - William Hughes
From: Randy Poe on 13 Oct 2006 08:42 Han de Bruijn wrote: > > 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. Wrong. That is a flat out incorrect statement showing a fundamental misunderstanding about what limits mean. A CONTINUOUS function at x0 has the property that the limit of f(x) as x->x0 is f(x0). But not all functions are continuous. - Randy
From: georgie on 13 Oct 2006 08:43 Virgil wrote: > In article <452e5bbb$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > Virgil wrote: > > > In article <452d14fe$1(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >> Dik T. Winter wrote: > > >>> 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. > > >> It defines something. > > > > > > An infinite string of digits. but every standard natural number is > > > defined by a finite string of digits, given a base, so those infinite > > > string define nothing at all. Besides themselves. > > > > > > > > > > > > What do you call that? > > > > > > An infinite string. > > > > > >> If the value up to and > > >> including every digit is finite, how can the string represetn anything > > >> but a finite value? > > > > > > If a binary string s:N --> {01} is such that s(n) = 1 for all n in N, > > > then its "value" is sum_{n in N} 2^n, which diverges. > > > > Of course it diverges, which means it attains an infinite value for > > infinite n. But, for all finite n, sum(x=0->n: 2^x) is finite. You have > > no infinite n in N. > But the "value" of that string, if it is to exist at all, cannot be > finite even thought all its digit positions are finite. > Thus contradicting TO's claims. > > > > > > > > But all the partial sums, sum_{n =1..k}2^n are all finite. > > > > Right, and that's all there is in N. There is nothing in N that is > > infinite in value or in index. > > > > > > > > So the value up to and including every digit is finite and the string > > > itself cannot represent any finite value. > > > > It cannot represent any infinite value for that very reason. > > So is TO claiming to have created/discovered values between finite and > infinite? Is Virgil so stupid that he can't understand what others post?
From: georgie on 13 Oct 2006 08:48 Virgil wrote: > In article <1160675643.344464.88130(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > With the diagonal proof you cannot show anything for infinite sets. > > Maybe "Mueckenh" can't but may others can. Only a very very small group of self-proclaimed experts better known as the mathematics community think they can. But they do so with circular arguments as this thread shows. The only explanation to the OP from the math community so far: #2 is not self-referential because #1 says ANY. #1 is correct in saying ANY because #2 holds.
From: Dik T. Winter on 13 Oct 2006 08:40
In article <virgil-9C34B2.13522312102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > 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. "Why, sometimes I've belieevd as many as six impossible things before breakfast." -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |