 Solution

The recursive definitions of ordinal addition and ordinal multiplication arewhere the limit clauses apply to nonzero limit ordinals. Ordinal exponentiation has ${\alpha }^0=1$, ${\alpha }^{\beta +1}={\alpha }^{\beta }\cdot \alpha$, and for $\alpha >0$, ${\alpha }^{\lambda }={\sup }_{\beta <\lambda }{\alpha }^{\beta }$. Complete the zero-base convention by $0^{\beta }=0$ for $\beta >0$, with $0^0=1$. Stating the zero-base exception avoids including the value $0^0$ in a naive supremum at a positive limit.

For finite $n$, induction gives $1+n=n+1$. Now prove commutativity for finite $m,n$ by induction on $m$. The case $m=0$ is immediate. Associativity, the finite identity just obtained, and the induction hypothesis giveThus $\boxed{m+n=n+m\text{ for finite ordinals}}$.

It fails for general ordinals: $\boxed{1+\omega =\omega <\omega +1}$. The equality follows from the limit clause, since the finite sums $1+n$ are cofinal in $\omega$.