Definition 2.2.20 (Canonically coded). A subset $S\subseteq \mathbb{N}$ is canonically coded in a model $M$ of $\mathrm{PA}$ if there is $c\in M$ such that

$$S=\{n\in \mathbb{N}:\exists y.({\pi }_{\underline{n}}\times y=c)\}$$

where $\underline{n}$ denotes the standard number $n$ in the model.