Set of Strictly Positive Integers is Primitive Recursive

Theorem
Let $P \subseteq \N$ be the set of all $n \in \N$ such that:
 * $n$ codes an integer $k$ such that $k > 0$.

Then $P$ is a primitive recursive set.

Theorem
Let $p : \N \to \N$ be defined as:
 * $\map p n = \map {\operatorname{rem}} {n, 2}$

By:
 * Constant Function is Primitive Recursive
 * Remainder is Primitive Recursive

Suppose $n$ codes an integer $k$ such that $k > 0$.

Then:
 * $n = 2 k - 1$

Therefore:

Suppose $n$ codes an integer $k$ such that $k \le 0$.

Then:
 * $n = - 2 k$

Therefore:

By definition, $p$ is the characteristic function of $P$.

Thus, $P$ is primitive recursive by definition.