Cantor Confusion
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From: David Marcus on 12 Oct 2006 14:15 Dik T. Winter wrote: > In article <452d14fe$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > It defines something. What do you call that? If the value up to and > > including every digit is finite, how can the string represetn anything > > but a finite value? > > I define it as a string of digits and it does not represent a number. It is > only when you give proper definitions of what strings extending infinitely > far away to the left represent, that you can talk about what it represents. > In common mathematics there is no such definition. > > That infinite strings to the right define real numbers is entirely due to > the *definition* of real numbers. Perhaps better to say it is due to the way that we define the meaning of infinite strings when used to represent real numbers. It is possible to construct the real numbers as infinite strings, but you can also construct them as Dedekind cuts or Cauchy sequences. -- David Marcus
From: David Marcus on 12 Oct 2006 14:26 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <452d14fe$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: > > > Dik T. Winter wrote: > > > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > ... > > > > > 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. What do you call that? If the value up to and > > > including every digit is finite, how can the string represetn anything > > > but a finite value? > > > > I define it as a string of digits and it does not represent a number. It is > > only when you give proper definitions of what strings extending infinitely > > far away to the left represent, that you can talk about what it represents. > > In common mathematics there is no such definition. > > When Peano defines the natural numbers, does he talk about what they > represent, or only how they are generated? If you are asking what Peano himself did, I don't know, since I'm not a historian. The Peano axioms for the natural numbers are a bunch of axioms, not a construction of the natural numbers. You can simply start with the axioms and go from there or you can start with something more basic (sets) and construct the natural numbers. If you do the latter, then you prove that what you constructed satisfies the Peano axioms. In this case, the axioms are theorems. Once you get to this point, you keep going as in the first case. (To those who've had a course in mathematical logic: I'm aware that the preceding ignores the question of what first-order language the Peano axioms are stated in. You can assume I'm just using whatever language I'm using for all my Mathematics.) -- David Marcus
From: David Marcus on 12 Oct 2006 14:27 Poker Joker wrote: > > "Virgil" <virgil(a)comcast.net> wrote in message > news:virgil-DE33FD.21001629092006(a)comcast.dca.giganews.com... > > > When Joker can prove > > "For ANY real number x there is a procedure to find > > a real number y, such that x/y=1." > > only then need we consider any special cases. > > Virgil spouts the most asinine things. We all know that there are > special cases that have nothing to do with this thread. If you wish to discuss Mathematics, please state what your mathematical point/idea is. -- David Marcus
From: David Marcus on 12 Oct 2006 14:31 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > I am sure you are able to translate brief notions like "to enter, to > > > > > escape" etc. by yourself into terms of increasing or decreasing values > > > > > of variables of sets, if this seems necessary to you. Here, without > > > > > being in possession of suitable symbols, it would become a bit tedious. > > > > > > > > Yes, I can translate it myself. However, that would only tell me how I > > > > interpret the problem. > > > > > > Hasn't it become clear by the discussion? > > > > > > I use two variables for sequences of sets. Further I use a function. I > > > use the natural numbers t to denote the index number. The balls are > > > simply the natural numbers. I speak of "balls" in order to not > > > intermingle these numbers with the index-numbers. > > > > > > The set of balls having entered the vase may be denoted by X(t). > > > So we have the mathematical definition: > > > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > > > There is a bijection between t and X(t). > > > > t is a number and X(t) is a set. If t = 1, then your sentence says, > > "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. > > So, I don't follow. What do you mean, please? > > That what is written. There is a bijection between the set of all > numbers t and the set of all sets X(t). 1 is mapped on X(1), 2 is > mapped on X(2), and so on. Is there anythng wrong? > > > > > The set of balls having left the vase is described by Y(t). So we have > > > the mathematical definition: > > > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > > > There is a bijection between t and Y(t). > > Here we have the same as above with Y instead of X. > > > > > > And the cardinal number of the set of balls remaining in the vase is > > > Z(t). So we have the mathematical definition: > > > Z(t) = 9t with Z(t) > 0 for every t > 0. > > > There is a bijection between t and Z(t). Sorry, but I'm not following. I asked you for a translation into Mathematics of the ball and vase problem. The problem in English ends with a question mark. I don't see the question mark in your translation above. Would you please state just the problem in both English and Mathematics? -- David Marcus
From: Virgil on 12 Oct 2006 14:34
In article <c663b$452dec28$82a1e228$15418(a)news2.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > How serious is mathematics? As serious as chess is, to those who take chess seriously. |