In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> > > Whether or not something like d is in the rationals-complete list can > > > only be judged by means of the d_n - at least in mathematics.
Without any need to consider d itself? How unusual!
> > > No unsuitable analogies please. The difference of d and all entries of the > list, if existing, does not fall down from heaven. It can be proven by digits > which belong to FIS as well as to d or it cannot be proven.
And given that each d_n is a real number, there is always a real number d different from all of the d_n's.
> > > > > What is the reason for this strange behaviour? > > > > > > > > > > > Because d is not one of the d_n, it can have properties that none of > > > > > > them can have, like not being one of them. > > > > > > > > > > The property of differing or not depends on the d_n only.
In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor. The elements of a countable set can be counted one at a time and although the counting may never finish, every element of the set will eventually be associated with a natural number.
So a shorter definition is that a set is countable if and only if its elements can be entirely listed.
Thus we need to ask whether the reals can be entirely listed.
Can a list of real numbers contain every real number?
Cantor showed that for EVERY list of real numbers there was a least one real number missing.
Thus the reals cannot be COMPLETELY listed.
Thus they cannot be counted, and the set of them is uncountable. --