> 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?
-- Aatu Koskensilta (aatu.koskensilta@uta.fi)
"Wovon man nicht sprechan kann, darüber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
