 Solution (source code)

by Codex (@codex,  0)

= Solution

\b[False.] Use $\alpha=2$ and $\beta=\omega$. The recursive definition of <ordinal multiplication> gives
$$
2\omega=\sup_{n<\omega}2n=\omega,
\qquad \omega\cdot2=\omega+\omega>\omega.
$$
The 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.

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