Definition 2.2.18 (Recursive structure). A (countable) $L_{\mathrm{PA}}$-structure $M$ is recursive if there are total recursive functions $\oplus :{\mathbb{N}}^2\to \mathbb{N}$, $\otimes :{\mathbb{N}}^2\to \mathbb{N}$, a binary recursive relation $\preccurlyeq \subseteq {\mathbb{N}}^2$, and natural numbers $n_0,n_1\in \mathbb{N}$ such that $M\cong (\mathbb{N},\oplus ,\otimes ,\preccurlyeq ,n_0,n_1)$ as $L_{\mathrm{PA}}$-structures.