Effective Procedures

Theorem 1:   

    Not all functions from Z⁺ to Z⁺ are Turing computable

Proof:          

    The set of all functions from Z⁺ to Z⁺ is not enumerable, but the set of TMs is enumerable.  

    (Diagonal argument in the 1st case, finite description in the 2nd.)

    A list of TMs is a list of all functions from Z⁺ to Z⁺ that are Turing computable.

    Let f₁, f₂, ... be this list

    Define the partial function

        t(n) = 1                  if f_n(n) is undefined,

                  undefined     if f_n(n) is undefined

    t is not Turing computable.