Past exam of the logic and set theory course of the University of Oxford / 2008 / B / 5 / 3 / Solution

False. Use $\alpha =2$ and $\beta =\omega$. The recursive definition of ordinal multiplication givesThe first product concatenates countably many finite two-point blocks. The second concatenates two whole copies of the order type $\omega$, producing a strictly larger ordinal. Hence ordinal multiplication is not commutative.