Ordinal arithmetic

Remarks

Example: $\omega + 1$ is the successor of $\omega$, and $\omega \cdot 2 = \omega + \omega$ by clauses 2 and 1. Non-example of commutativity: $1 + \omega = \sup_{n < \omega} (1 + n) = \omega$, while $\omega + 1 \ne \omega$. Likewise $2 \cdot \omega = \sup_n 2n = \omega$, while $\omega \cdot 2 = \omega + \omega > \omega$. Equivalently, $\alpha + \beta$ is the order type of $\alpha$ followed by $\beta$. And $\alpha \cdot \beta$ is the order type of the pairs $(b, a)$ with $b < \beta$ and $a < \alpha$, compared first by $b$ and then by $a$. Addition and multiplication are associative, and $\alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma$, but the other distributive law fails. On finite ordinals the operations agree with those of the natural numbers. Ordinal exponentiation differs from cardinal exponentiation: $2^\omega = \omega$ as ordinals, while $2^{\aleph_0} > \aleph_0$ as cardinals.