Natural Powers

Why

We want to repeatedly multiply.

For each natural number $m$, there exists a function $e_m: \omega \to \omega $ which satisfies

\[ e_m(0) = 1 \quad \text{ and } \quad e_m(\ssuc{n}) = \ssuc{(e_m(n))} \cdot m \]

for every natural number $n$.

The proof uses the recursion theorem (see Recursion Theorem).¹

Let $m$ and $n$ be natural numbers. The value $p_m(n)$ is the power of $m$ with $n$. Or the $n$th power of $m$

We denote the $n$th power of $m$ by $m^n$.

Here are some basic properties of powers.

Let $k$, $m$, and $n$ be natural numbers. Then

\[ m^{n}m^{k} = m^{k + k}. \]

Let $k$, $m$, and $n$ be natural numbers. Then

\[ (m^{n})^k = m^{nk}. \]