= Solution
\b[False.] Take $\alpha=1$ and $\beta=\omega$. By the limit clause for <ordinal addition>,
$$
1+\omega=\sup_{n<\omega}(1+n)=\omega,
\qquad \omega+1>\omega.
$$
The first <order type> is one point followed by a countably infinite <sequence>, which is still of type $\omega$. The second has a greatest element after that infinite <sequence>, so its <order type> is the <successor ordinal> $\omega+1$. Thus <ordinal addition> is not commutative, even though the two resulting <sets> have equal <cardinality>.
Back to article page
