naturalArithmetic.tex

\documentclass{article}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{mathtools}
\usepackage{enumitem}

\newtheorem*{mythm*}{Theorem}
\newtheorem{mythm}{Theorem}
\newtheorem{mydef}{Definition}

\begin{document}
\begin{mythm}
	For all natural numbers $m$ and $n$,
	\begin{align*}
		m + 0 &= m \\
		m + n^+ &= (m + n)^+
	\end{align*}
\end{mythm}

\begin{mythm}
	For all $m$ in $\omega$, $m + 1 = m^+$.
\end{mythm}

\begin{mythm}
	For all natural numbers $m$ and $n$,
	\begin{align*}
		m \cdot 0 &= 0 \\
		m \cdot n^+ &= m \cdot n + m
	\end{align*}
\end{mythm}

\begin{mythm}
	(Associative Law for Addition). For all $m$, $n$, and $p$ in $\omega$,
	\begin{equation*}
		m + (n + p) = (m + n) + p
	\end{equation*}
\end{mythm}

\begin{mythm}
	For all natural numbers $m \in \omega$ and $n \in \omega$, we have $m^+ + n &= (m + n)^+$.
\end{mythm}

\begin{mythm}
	(Commutative Law for Addition). For all $m$, $n$ in $\omega$,
	\begin{equation*}
		m + n = n + m 
	\end{equation*}
\end{mythm}

\begin{mythm}
	(Distributive Law). For all natural numbers $m$, $n$, and $p$ in $\omega$,
	\begin{equation*}
		m \cdot (n + p) = m \cdot n + m \cdot p
	\end{equation*}
\end{mythm}

\begin{mythm}
	(Associative Law for Multiplication). For all natural numbers $m$, $n$, and $p$ in $\omega$,
	\begin{equation*}
		m \cdot (n \cdot p) = (m \cdot n) \cdot p
	\end{equation*}
\end{mythm}

\begin{mythm}
	(Commutative Law for Multiplication). For all natural numbers $m$, $n$, and $p$,
	\begin{equation*}
		m \cdot n = n \cdot m
	\end{equation*}
\end{mythm}

\begin{mydef}
	Let $E \colon \omega \times \omega \to \omega$ be the unique function satisfying
	\[
	E(m, n) =
	\begin{cases}
		E(m, 0) = 1 \\
		E(m, n^+) = E(m, n) \cdot m
	\end{cases}
	\]
\end{mydef}

\end{document}