A set $X$ with an endomorphism $f \colon X \to X$ and a distinguished element $x_0$ is called a discrete dynamical system. This data allows one to consider the discrete-time evolution of the initial element $x_0$, a sequence defined by $x_{n+1} \coloneqq f(x_n)$. The principle of mathematical recursion asserts that the natural numbers $\mathbb{N}$, the successor function $s \colon \mathbb{N} \to \mathbb{N}$, and the element $0 \in \mathbb{N}$ define the universal discrete dynamical system: which is to say, there is a unique function $r \colon \mathbb{N} \to X$ so that $r(n) = x_n$ for each $n$, i.e., so that $r(0)=x_0$ and so that the diagram \(\vcenter{\xymatrix{ \NN \ar[r]^s \ar[d]_r & \NN \ar[d]^r \\ X \ar[r]_f & X}}\) commutes.