Set of Non-Zero Natural Numbers is Primitive Recursive

Theorem
Let $\N^*$ be defined as $\N^* = \N - \left\{{0}\right\}$.

The subset $\N^* \subset \N$ is primitive recursive.

Proof
We have that the characteristic function $\chi_{\left\{{0}\right\}}$ of $\left\{{0}\right\}$ is primitive recursive.

We note that:


 * If $n = 0$ then $\chi_{\left\{{0}\right\}} \left({n}\right) = 1$ therefore $\chi_{\left\{{0}\right\}} \left({\chi_{\left\{{0}\right\}} \left({n}\right)}\right) = 0$.


 * If $n > 0$ then $\chi_{\left\{{0}\right\}} \left({n}\right) = 0$ therefore $\chi_{\left\{{0}\right\}} \left({\chi_{\left\{{0}\right\}} \left({n}\right)}\right) = 1$.

Thus $\chi_{\left\{{0}\right\}} \left({\chi_{\left\{{0}\right\}}\left({n}\right)}\right) = \chi_{\N^*} \left({n}\right)$.

So $\chi_{\N^*}$ is obtained by substitution from the primitive recursive function $\chi_{\left\{{0}\right\}}$.

Hence the result.