Set Containing Only Zero is Primitive Recursive
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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.
$\blacksquare$