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.