In article <firstname.lastname@example.org>, WM <email@example.com> wrote: > Am Sonntag, 8. Dezember 2013 02:28:10 UTC+1 schrieb Virgil: > > Note that for every n in |N there will be infinitely many rationals > > whose first n digits agree with d and only differ later.
> Of course. This is a fact since every decimal representation of a number is > nothing but the set of FIS.
For many reals it is not merely a collection of FISs, butthe entire infinite sequence, as given by a rule.
> So up to every d_n there is a FIS. And a FIS is a rational number. > > So that there can never be any n such that. by digit n. d will differ > > from all rationals.
> Simply because d has no complete decimal representation.
While for many sequence of reals the antidiagonal of such a sequence may not be finitely describable, it is not true fr all, since there are many irrationals whose infinite decimal representataions can be finitely described. E.g., x = sum 1/10^(n!) > > Nevertheless any irrational, of which even WM admits there are > > infinitely many, will differ from every rational in infinitely many > > places. > No. If WM claims otherwise, he must provide a specific rational for which this does not hold.
> But most rationals cannot be > given, because only the tiny set of the first rationals is available to > supply "given" rationals. Most rationals belong to the inaccessible set > beyond any "given" one.
Then they must be a part of the set of inaccessible reals that WM denies theexistence of. --