Exercise 11, § 6

(a) Show that the ordinal $\omega$ is the least ordinal $> 0$ which has no predecessor, that $\omega$ is indecomposable (§ 2, Exercise 16), and that for each ordinal $\alpha > 0$, $\alpha\omega$ is the least indecomposable ordinal which is $> \alpha$ (note that $n\omega = \omega$ for each integer $n$). Deduce that

(b) Deduce from (a) that an ordinal is indecomposable if and only if it is of the form $\omega^\beta$ (use Exercise 18 (d) of § 2).