Corollary 2.2.11. A subset $A\subseteq {\mathbb{N}}^k$ is recursively enumerable if and only if there is a ${\mathrm{\Sigma }}_1$-formula $\psi (x_1,\dots ,x_k)$ such that, given $\overline{x}\in {\mathbb{N}}^k$, we have $\overline{x}\in A$ if and only if $\mathbb{N}\models \psi (x)$.