Exercise 13, § 6

(a) Let $w(\xi)$ be an ordinal functional symbol (§2, Exercise 17), defined for $\xi \geqslant \alpha_0$ and such that the relation $\alpha_0 \leqslant \xi < \xi'$ implies $w(\xi) < w(\xi')$. Show that, if $\xi \geqslant \alpha_0$, then $w(\xi + \eta) \geqslant w(\xi) + \eta$ for every ordinal $\eta$ (argue by contradiction). Deduce that there exists $\alpha$ such that $w(\xi) \geqslant \xi$ for all $\xi \geqslant \alpha$ (take $\alpha$ to be the least indecomposable ordinal $\geqslant \alpha_0$; cf. Exercise 11 (a)).

(b) Let $f(\xi, \eta)$ be the ordinal functional symbol defined in §2, Exercise 17 (b). Suppose that the relations $\alpha_0 \leqslant \xi \leqslant \xi'$ and $\alpha_0 \leqslant \eta \leqslant \eta'$ imply $g(\xi, \eta) \leqslant g(\xi', \eta')$, so that the relations $\alpha_0 \leqslant \xi \leqslant \xi'$ and $1 \leqslant \eta \leqslant \eta'$ imply $f(\xi, \eta) \leqslant f(\xi', \eta')$ (§ 2, Exercise 17 (d)). Show that for each ordinal $\beta$ there exists at most a finite number of ordinals $\eta$ for which the equation $f(\xi, \eta) = \beta$ has at least one solution. (Note that if $\xi_1$ is the least solution of $f(\xi, \eta_1) = \beta$ and if $\xi_2$ is the least solution of $f(\xi, \eta_2) = \beta$, then the relation $\eta_1 < \eta_2$ implies $\xi_1 > \xi_2$.)

(c) A critical ordinal with respect to $f$ is any infinite ordinal $\gamma > \alpha_0$ such that $f(\xi, \gamma) = \gamma$ for all $\xi$ such that $\alpha_0 \leqslant \xi < \gamma$. Show that a critical ordinal (with respect to $f$) has no predecessor. If there exists a set A of ordinals such that $f(\xi, \gamma) = \gamma$ for all $\xi \in A$, and if $\gamma$ is the least upper bound of A, show that $\gamma$ is a critical ordinal.

(d) Let $h(\xi) = f(\xi, \xi)$ (defined for $\xi \geqslant \alpha_0$). Define inductively $\alpha_1 = \alpha_0 + 2$, $\alpha_{n+1} = h(\alpha_n)$ for $n \geqslant 1$. Show that the least upper bound of the sequence $(\alpha_n)$ is a critical ordinal with respect to $f$.

(e) Show that the least upper bound of every set of critical ordinals with respect to $f$ is again a critical ordinal, and that every critical ordinal is indecomposable (note that $f(\xi, \eta + 1) \geqslant \omega(\xi) + \eta \geqslant \xi + \eta$ for all $\xi \geqslant \alpha_0$).