Am Freitag, 6. Dezember 2013 03:00:28 UTC+1 schrieb Virgil: > 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! >
If d consists of nothing but the d_n it is true. Whether it may be usual or unusual is of no interest. > > > > > > > > > > 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. >
That shows the contradictory character of finished infinity: It is impossible to distinguish a real number d by its digits from all other real numbers. It is possible to distinguish a real number d by its digits from all other real numbers.
> In mathematics, a countable set is a set with the same cardinality > > (number of elements) as some subset of the set of natural numbers.
Therefore the set of numbers defined by finite definition which is a subset of the set of all finite words is a countable set, notwithstanding the way how finite definitions are defined and how they define numbers. We need only know that a finite definition is a finite word and that one finite word does never define more than one number (in fact not more than countably many numbers would be sufficient).