$\{0,1,2,...\}$ is $\mathbb{N}$, and $\{1,2,3,...\}$ is $\mathbb{N}^+$ if I'm not wrong.
Does that mean this author does not want to include $0$ in the function?
Why some authors include $0$ in Natural numbers $(\mathbb{N})$ when Whole numbers $(\mathbb{W})$ are defined to do exactly the same.