# Set of Natural Numbers is Primitive Recursive

## Theorem

The set of natural numbers $\N$ is primitive recursive.

## Proof

The characteristic function $\chi_\N: \N \to \N$ is defined as:

$\forall n \in \N: \chi_\N \left({n}\right) = 1$.

So:

$\chi_\N \left({n}\right) = f^1_1 \left({n}\right)$

Hence the result.

$\blacksquare$