In article <email@example.com>, firstname.lastname@example.org wrote:
> On Tuesday, 19 August 2014 17:48:23 UTC+2, Zeit Geist wrote: > > > > > If the list has not a finite definition, then every line has to be used > > > to determine the corresponding digit. That means every digit has to be > > > determined separately. > > > > > > > > If the Particular List has Not a Finite Definition, then it does Matter. > > Of course. > > For the Diagonal Proof is Valid for any List of Real Numbers. All that is > Required is that there is a Unique Decimal Sequence Associated with each > Natural Number. > > Numbers are defined by digits. > > > Since each Decimal Sequence can be Shown to Represent Real Number, EVEN IF > WE DON'T KNOW WHICH REAL IS ASSOCIATED TO WHICH NATURAL, we Find a Real > Number Not Enumerated by any Natural Number. Therefore it is Not on the > List. Again, ABSTRACTION!
The definition of a set being countable requires that its members be listable. Cantor's proof that no list of reals can list all reals proves that the set of reals does not satisfy the definition of countability and is therefore not countable.
> > The Set of Finitely Definable Real Numbers is NOT COUNTABLE! > > > > Proof has been Given, > > No. Not in mathematics.
Maybe not in WM's worthless world of WMytheology but definitely so in all proper mathematics outside the corruption of that WMytheology. > > If we define the real numbers in a strictly formal system, where only finite > derivations and fixed symbols are permitted, then these real numbers can > certainly be enumerated because the formulas and derivations on the basis of > their constructive definition are countable.
But if every infinite string of decimal digits ( other than those ending in all 9's) corresponds to a real number in [0,1), then that limited set of reals in [0,1) is not even countable, much less all of |R. -- Virgil "Mit der Dummheit kampfen Gotter selbst vergebens." (Schiller)