Some Recursive Function Theory

Abstract

We defined a function f to be computable just in case the relation j(x ₁,… x _n) = z is n.r. in Q. This implies that for some number e

$$ \begin{array}{*{20}{c}} {f\left( {{n_1}, \ldots {n_k}} \right) = miff \vdash \underline {\left( {\exists a} \right)T\left( {e,{n_1}, \ldots {n_k},m,a} \right)} } \\ {f\left( {{n_1}, \ldots {n_k}} \right) \ne miff \vdash \underline { - T\left( {e,{n_1}, \ldots {n_k},m,j} \right)} forallj.} \end{array} $$

$$ f\left( {{n_1},...{n_k}} \right) \ne m\;iff \vdash \;\underline { - T\left( {e,{n_1},...{n_k},m,j} \right)} \;for\,all\,j $$

. It will be useful to introduce the notation {e}(n ₁,...n _k ) for f(n ₁,...n _k ) where e is the number assigned to the function f. Recall that we defined a set S to be weakly n.r. just in case there is an Aa such that n ∈ S iff ⊢_Q An. We will now show the equivalence of several concepts of semi-effectiveness. Df S is recursively enumerable (r.e.) iff there is a total computable f such that n ∈ S iff (∃ z)f(z) = n.