 Another significant contribution of Gödel’s paper derives from the fact that after proving the incompleteness of \(\mathsf{P}\), he took several steps towards isolating features of axiomatic theories which are sufficient  to ensure that they satisfy analogous undecidability results. In addition to being sufficiently strong to satisfy (\ref{rep}), the other requirement which he identifies is that “the class of axioms and the rules of inference \(\ldots\) are recursively definable” (1931, 181). As he notes, these features hold both of Zermelo-Fraenkel set theory \([\mathsf{ZF}\)] and a first-order arithmetical system similar to what we now call first-order Peano arithmetic \([\mathsf{PA]}\), relative to an appropriate Gödel numbering of their axioms. In particular, while neither of these systems is finitely axiomatizable, they may be axiomatized by a finite number of schemes (e.g., of induction or comprehension) such that the relation \(\ulcorner \varphi \urcorner\) is the Gödel number of an axiom of T is primitive recursive.