Exercise 15, § 6

Let $\gamma$ be an ordinal $> 1$, and for each ordinal $\alpha$ let $L(\alpha)$ denote the set of exponents $\lambda_i$ in the expression for $\alpha$ given in Exercise 12 (a).

(a) Show that $\lambda_i \leqslant \alpha$ for each $\lambda_i \in L(\alpha)$, and that $\lambda_i = \alpha$ for one of these ordinals only if $\alpha = 0$ or if $\alpha$ is a critical ordinal with respect to $\xi^\eta$ (Exercise 14 (c)).

(b) Define $L_n(\alpha)$ by induction on $n$ as follows : $L_1(\alpha) = L(\alpha)$, and $L_n(\alpha)$ is the union of the sets $L(\beta)$ as $\beta$ runs through $L_{n-1}(\alpha)$. Show that there exists an integer $n_0$ such that $L_{n+1}(\alpha) = L_n(\alpha)$ whenever $n \geqslant n_0$, and that the elements of $L_n(\alpha)$ are then either 0 or critical ordinals with respect to $\xi^\eta$. (Argue by contradiction: for each $n$, consider the set $M_n(\alpha)$ of elements $\beta \in L_n(\alpha)$ such that $\beta \notin L(\beta)$, and assume that $M_n(\alpha)$ is not empty for any $n$; use (a) to obtain a contradiction.)