What is the general approach to oridinal product of $(\omega^2 + \omega...) \cdot \omega$?

Question

If we have some kind of ordinal in Cantor normal form, and there are at least two non-trivial parts of it, what is the general approach to calculating it? Only way I heard of is building an explicit isomorphic function, but this is getting quite complex once you take something more sophisticated than $(\omega + k) \cdot \omega$.

Answer 1

Note that $\alpha (\beta+\gamma) = \alpha \beta + \alpha \gamma$ (though it's not distributive in the other direction), so your case of $(\omega^2 + \omega) \omega$ is structurally as complicated as it gets.

We define $$(\omega^2 + \omega) \omega = \sup \{ (\omega^2 + \omega) n : n < \omega\}$$ But $$(\omega^2 + \omega)(n+1) = (\omega^2 + \omega)n + (\omega^2 + \omega)$$ and $$(\omega^2+\omega)0 = 0$$ so inductively $$(\omega^2+\omega)n = \omega^2n + \omega$$ whence $$(\omega^2+\omega) \omega = \sup_{n < \omega} (\omega^2n + \omega) = \omega^3$$

This same technique works pretty much always, though perhaps it gets a bit more complicated to apply. You might like to prove several generalisations.