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.
If Cantor's diagonal proof results in any such a number, then it proves in effect the uncountability of a countable set.
Otherwise it proves nothing, because an infinite sequence of digits without other information does not determine anything.