I had come across this question when revising for an upcoming exam in Set Theory:
Put these ordinals in ascending order:
$\omega^3 + \omega^2 + \omega, \\ \omega + \omega^2 + \omega^3, \\\omega^3 + \omega + \omega^2$
When trying to expand these by the definition of $+$, I get in a mess taking unions of a limit ordinal. For example:
$\omega^3 + \omega^2 + \omega = \bigcup_{n < \omega} (w^3 + w^2 + n) = \bigcup_{n<\omega} ((\bigcup_{m<\omega}(\omega^3 + \omega.m))+n)$ and so on, where I cannot see a way to further simplify.
I had always struggled with these types of questions before so I would be very grateful if anyone could provide any hints.
There is no way (that I'm aware of) to further simplify $\omega^3+\omega^2+\omega$. But the other expressions can be simplified.
Remember the definition of ordinal addition. You're simply juxtaposing the first ordinal with the second, with the first ordinal coming first. So, for example, $5+ \omega=\omega$ because if you precede the order type of $\omega$ with $5$ (or any finite number of) elements, you still end up with order type $\omega$.
Does that help you see how to order these expressions?