 Well-order code

A well-order code represents a pair $(D,R)$ with $D\subseteq \omega$ and $R\subseteq D\times D$ such that $R$ is a strict well-order of $D$. Its rank is the unique countable ordinal isomorphic to $(D,R)$. Allowing a subset $D$ is essential: $D=\varnothing$ codes zero, a finite $D$ codes a finite ordinal, and infinite subsets code infinite countable ordinals.

Both the domain and relation can be recorded by bits in a real number, using a fixed enumeration of pairs of natural numbers. Different well-order codes can represent the same ordinal; changing the labels of the elements changes the code while preserving its order type.