Definition 2.2.1 (Recursively enumerable). We say that $X\subseteq \mathbb{N}$ is recursively enumerable if any of the following are true:

• (1)
  $X$ is the image of some partial recursive $f:\mathbb{N}\to \mathbb{N}$;
• (2)
  $X$ is the image of some total recursive $f:\mathbb{N}\to \mathbb{N}$;
• (3)
  $X=\mathrm{dom}f$, for $f$ a partial recursive $f:\mathbb{N}\to \mathbb{N}$.