In article <79d04ee8-00f7-4aad-a499-2b9241ff3b6d@y11g2000yqm.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> On 18 Jun., 05:25, Sylvia Else <syl...@not.here.invalid> wrote: > > On 18/06/2010 4:27 AM, WM wrote: > > > > > > > > > > > > > On 17 Jun., 15:56, Sylvia Else<syl...@not.here.invalid> wrote: > > >> On 15/06/2010 2:13 PM, |-|ercules wrote: > > > > >>> Consider the list of increasing lengths of finite prefixes of pi > > > > >>> 3 > > >>> 31 > > >>> 314 > > >>> 3141 > > >>> .... > > > > >>> Everyone agrees that: > > >>> this list contains every digit of pi (1) > > > > >>> as pi is an infinite digit sequence, this means > > > > >>> this list contains every digit of an infinite digit sequence (2) > > > > >>> similarly, as computable digit sequences contain increasing lengths of > > >>> ALL possible finite prefixes > > > > >>> the list of computable reals contain every digit of ALL possible > > >>> infinite sequences (3) > > > > >> Obviously not - the diagonal argument shows that it doesn't. > > > > > There is no diagonal element for a list of finite lines. > > > > The list of computable reals is not a list of finite lines. > > It is. Every real number that is defined is defined by a finite word > (definition or formula). It is impossible to define a number by an > infinite sequence, because the sequence never ends and the definition > is never known.
F:N -> R: n -> 3/10^n is an infinite sequence defining a real number.
So as one can easily see it is possible to have an infinite sequence defining a real number.
In real mathematics, an infinite sequence is merely a function whose domain is the set of natural numbers, and there are lots of them which define real numbers.
