Set Containing Only Zero is Primitive Recursive

Theorem
The subset $$\left\{{0}\right\} \subset \N$$ is primitive recursive.

Proof
We note that:
 * $$1 \, \dot - \, n = \begin{cases}

1 & : n = 0 \\ 0 & : n > 0 \end{cases}$$ and so the characteristic function $$\chi_{\left\{{0}\right\}}$$ is given by $$\chi_{\left\{{0}\right\}} \left({n}\right) = 1 \, \dot - \, n$$.

So $$\chi_{\left\{{0}\right\}}$$ is obtained by substitution from the primitive recursive function $1 \, \dot - \, n$ using constants, which are primitive recursive.

Hence the result.