 Gödel’s arithmetization of syntax provides a means of assigning to each primitive symbol, term, formula, and proof \(\alpha\) of \(\mathsf{P}\) a unique Gödel number \(\ulcorner \alpha \urcorner \in \mathbb{N}\) according to its syntactic structure. This technique takes advantage of the familiar observation that a finite sequence of numbers \(n_1,\ldots,n_k\) can be encoded as a product of prime powers \(2^{n_1} \cdot 3^{n_2} \cdot \ldots p_k^{n_k}\) so that various correlative operations on sequences can be shown to be primitive recursive—e.g., the operation which takes two numbers \(x\) and \(y\) encoding sequences and returns the code \(x * y\) of the result of concatenating \(x\) followed by \(y\). Gödel proceeded on this basis to show that a sequence of notions about the syntax and proof theory of \(\mathsf{P}\) are primitive recursive—e.g., the function \(\textrm{Neg}(x)\) which returns the Gödel number of the negation of the formula coded by \(x\) can be defined as \(\ulcorner \neg \urcorner * x\). The availability of the relevant recursive definitions thus falls out naturally since the inductive definitions of syntactic notions like well-formed formula generalize the “construction and deconstruction of number signs” in the sense described by  Hilbert.[8] 