We are given 2 ordinals: $\alpha$ and $\beta$ where $\beta$ does not have a maximal number (So it's transfinite, right?)
We are asked to find $\alpha,\beta$ such that:
$\alpha+\beta > \beta+\alpha$
My problem is, that I don't think such a solution exists when alpha is a finite ordinal (lets say that $\beta=\omega$). if it is, then $\alpha+\beta = \{0,1,2,...,\alpha-1,0^*,1^*,2^*,...\} = \omega$
and $\beta+\alpha = \{0,1,2,...,0^*,1^*,2^*,...(\alpha-1)^*\} > \omega$
So I think $\alpha$ also has to be a transfinite ordinals. But I can't for the life of me think of such an ordinal that this will be true.
HINT: Think of a case where $\beta+\alpha=\alpha$. For example when $\alpha$ is much bigger than $\beta$.
(Also, ordinal is without a maximum if and only if it is not a successor ordinal, which means either zero or a limit ordinal - which has to be infinite.)