On the Difficulty of Writing Out formal Proofs in Arithmetic

Let ℸ be the set of Gödel numbers Gn( f ) of function symbols f such that PRA ⊢ and let γ be the function such that\documentclass{article}\pagestyle{empty}\begin{document}$$\begin{array}{*{20}c} {\gamma (n) = \left\{ {\mathop G\limits_0 n({\rm PRA} - {\rm proof}\,{\rm of}\,\forall x(f(x) = 0))} \right.} & {\mathop {{\rm if}\,n = }\limits_{otherwise.} Gn(f) \in \Gamma,}\\\end{array} $$\end{document} We prove: (1) The r. e. set ℸ is m‐complete; (2) the function γ is not primitive recursive in any class of functions { f 1 , f 2 , ⃛} so long as each f i has a recursive upper bound. This implies that γ is not primitive recursive in ℸ although it is recursive in ℸ.