Limit ordinal that cannot be written as $\alpha \cdot \omega$? I learned from this thread that every limit ordinal can be written as $\omega\cdot\alpha$ for some $\alpha$ but is this also true for $\alpha\cdot\omega$ even though ordinal multiplication is not commutative?
If it does not hold there must be a limit ordinal that cannot be written as $\alpha\cdot\omega$. I wouldn't even know how to write $\omega+\omega$ as $\alpha\cdot\omega$ but how can I prove this is genuinely impossible?

Think about the following, if $n<\omega$, what is $n\cdot\omega$? On the other hand, if $\alpha\geq\omega$, it is certainly the case that $\alpha\cdot\omega\geq\omega\cdot\omega$.
So where does $\omega+\omega$ fit in? (Hint: It doesn't.)
