Definition:Power (Algebra)/Natural Number

Theorem
Let $\N$ be the natural numbers, defined as the minimal infinite successor set $\omega$.

For each $m \in \N$, let $e_m: \N \to \N$ be the mapping:
 * $e_m \left({n}\right) = \begin{cases}

1 & : n = 0 \\ x \times e_m \left({x}\right) & : n = x^+ \end{cases}$ where:
 * $x^+$ is the successor set of $x$
 * $\times$ denotes natural number multiplication.

$e_m \left({n}\right)$ is then expressed as a binary operation in the form:
 * $n^m := e_m \left({n}\right)$