"WM" <email@example.com> wrote in message news:firstname.lastname@example.org... On 16 Jun., 05:37, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > Virgil <Vir...@home.esc> writes: > > But until you can determine which of those 10 cases, how can you > > compute the number? > > You can't. The number is computable nonetheless, in the sense that there > exists an effective procedure for churning out its decimal expansion. > > As noted, computability is a purely extensional notion. Recall the > classical recursion theory exercise, which we find, in some form or > other, in pretty much any text on the subject: > > Let f : N --> N be a function such that > > f(x) = 0 if Goldbach's conjecture is true, and 1 otherwise. > > Is f computable?
All computable, Turing-computable, nonetheless computable, definable (in any useful, i.e., finite language), and by other means determinable numbers form a countable set.
________________________ All subsets of N are countable. Big deal.
If Cantor's diagonal proof results in any such a number, then it proves in effect the uncountability of a countable set.
____________________ What absolute crap. It proves no such thing. And in any event the diagonal number is easily computable. Just change all "7"s to "8" and everything else to "7". Its a few lines of code.
Otherwise it proves nothing, because an infinite sequence of digits without other information does not determine anything.
____________________ No, it proves there is a Real which is not on the list.