Exercise 19, § 6

(a) Let $\alpha$ and $\beta$ be two ordinals such that $\alpha$ has no predecessor, and let $\xi \to \sigma_\xi$ be a strictly increasing mapping of the ordinal $\omega_\beta$ into the ordinal $\alpha$ such that $\sup_{\xi < \omega_\beta} \sigma_\xi = \alpha$. Show that

(With each mapping $f$ of the ordinal $\omega_\beta$ into the ordinal $\omega_\alpha$ associate an injective mapping $\bar{f}$ of $\omega_\beta$ into the set of all $\omega_{\sigma_\xi}$ ($\xi < \omega_\beta$) such that $f(\xi) \leq \bar f(\xi)$ for all $\xi < \omega_\beta$. Calculate the cardinal of the set of mappings $f$ associated with the same $\bar f$ and observe that

and $m\geq\aleph_\alpha$ (cf. § 3, Exercise 3).)

(b) Let $\bar\alpha$ be the ordinal such that $\omega_{\bar\alpha}$ is the final character of $\omega_\alpha$. Show that $\aleph_\alpha^{\aleph_{\bar\alpha}}>\aleph_\alpha$ and that if there exists $n$ such that $\aleph_\alpha=n^{\aleph_\gamma}$, then $\gamma<\bar\alpha$ (use (a) and Exercise 3 of § 3).

(c) Show that, if $\lambda<\bar\alpha$, then

(argue as in Exercise 18 (a)).