Set of Non-Zero Natural Numbers is Primitive Recursive

Theorem

Let $\N^*$ be defined as $\N^* = \N \setminus \set 0$.

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

Proof

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

We note that:

If $n = 0$ then $\map {\chi_{\set 0} } n = 1$ therefore $\map {\chi_{\set 0} } {\map {\chi_{\set 0} } n} = 0$.

If $n > 0$ then $\map {\chi_{\set 0} } n = 0$ therefore $\map {\chi_{\set 0} } {\map {\chi_{\set 0} } n} = 1$.

Thus $\map {\chi_{\set 0} } {\map {\chi_{\set 0} } n} = \map {\chi_{\N^*} } n$.

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

Hence the result.

$\blacksquare$