Set Containing Only Zero is Primitive Recursive

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

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

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

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

Hence the result.