 Ordinal

An ordinal records the order type of a well-order. In the von Neumann representation, it is a transitive set well-ordered by membership. The first examples are $0=\varnothing$, $1=\{0\}$, $2=\{0,1\}$, and $\omega =\{0,1,2,\dots \}$.

Every well-order is isomorphic to exactly one ordinal. Ordinal order distinguishes arrangements with the same cardinality: $\omega$ and $\omega +1$ are both countably infinite, but only the latter has a greatest element. Transfinite induction and transfinite recursion extend finite reasoning through these order types. Their ordinal addition generally fails commutativity because it records concatenation order.