 By formalizing his informal characterization of recursiveness via  Definition 1.2,  Gödel succeeded in formulating a definition which subsumes the primitive recursion scheme (\ref{gprimrec}), the definition of the Ackermann-Péter function, as well as several other schemes considered by Hilbert. Gödel’s definition of general recursiveness thus also defined a class \(\mathbf{GR}\) of functions of type \(\mathbb{N}^k \rightarrow \mathbb{N}\) which properly subsumes the primitive recursive functions  \(\mathbf{PR}\).  Moreover, we now know that the class of functions representable in \(\mathsf{P}\) (and in fact in far weaker arithmetical systems) corresponds not to the primitive recursive functions, but rather to the general recursive functions. Weakening the hypothesis that the set of (Gödel numbers) of the axioms of a formal system to the requirement that they be general recursive rather than primitive recursive thus indeed provides a generalization of the First Incompleteness Theorem the manner in which Gödel envisioned.