Exercise 12, § 6

(a) Show that, for each ordinal $\alpha$ and each ordinal $\gamma > 1$, there exist two finite sequences of ordinals $(\lambda_i)$ and $(\mu_i)$ $(1 \leqslant i \leqslant k)$ such that

where $0 < \mu_i < \gamma$ for each $i$, and $\lambda_i > \lambda_{i+1}$ for $1 \leqslant i \leqslant k - 1$ (use Exercise 18 (d) of § 2 and Exercise 3 of § 4). Moreover, the sequences $(\lambda_i)$, $(\mu_i)$ are uniquely determined by these conditions. In particular, there exists a unique finite decreasing sequence $(\beta_j)_{1 \leqslant j \leqslant m}$ such that

Let $\varphi(\alpha)$ denote the greatest ordinal $\omega^{\beta_1}$ in this decomposition.

(b) For each integer $n$ let $f(n) \leqslant n!$ be the greatest number of elements in the set of ordinals of the form $\alpha_{\sigma(1)} + \alpha_{\sigma(2)} + \cdots + \alpha_{\sigma(n)}$, where $(\alpha_i)_{1 \leqslant i \leqslant n}$ is an arbitrary sequence of $n$ ordinals, and $\sigma$ runs through the set of permutations of the interval $[1, n]$. Show that

(1)

(Consider first the case where all the $\varphi(\alpha_i)$ are equal and show that the largest possible number of distinct ordinals of the desired form is equal to $n$, by using Exercise 16 (a) of § 2. Then use induction on the number of ordinals $\alpha_i$ for which $\varphi(\alpha_i)$ takes the least possible value among the set of ordinals $\varphi(\alpha_j)$ $(1 \leqslant j \leqslant n)$.) Deduce from (1) that for $n \geqslant 20$ we have $f(n) = 81 f(n-5)$.

(c) Show that the $n!$ ordinals $(\omega + \sigma(1))(\omega + \sigma(2)) \ldots (\omega + \sigma(n))$, where $\sigma$ runs through the set of all permutations of $[1, n]$, are all distinct.