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.
A finite word W can define an infinite sequence S up to every desired n. W ==> S But the reversal of the implication is not true (as usual). S ==> W is impossible and wrong.