On 6/16/2010 5:37 AM, Aatu Koskensilta wrote: > Virgil<Virgil@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?
To Virgil: Also many classical mathematicians appreciate this as an example showing that the extensional notion 'Turing computable' is a slight distortion of the intuitive notion 'computable'.