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Poiché $\gamma < \alpha$ e $\beta \neq 0$, allora $\beta \cdot \gamma < \beta \cdot \alpha$. Inoltre + $\beta < \alpha$, dunque per assorbimento $\beta \cdot \alpha = \alpha$, da cui $\beta \cdot \gamma < \alpha$. + \item[\fbox{(ii.) $\implies$ (iii.)}] Mostriamo la contronominale, ovverosia che se $\alpha$ non è un $\omega^{(\omega^\delta)}$, + allora $\alpha$ non è moltiplicativamente chiuso. Esiste comunque un $\delta$ ordinale per cui + $\omega^{(\omega^\delta)} < \alpha < \omega^{(\omega^{\delta + 1})}$. Dal momento che $\omega^{\delta + 1} = \omega^\delta \cdot \omega$, + allora $\omega^{(\omega^{\delta + 1})} = \omega^{(\omega^\delta \omega)} = {(\omega^{(\omega^\delta)})}^\omega$. + Esiste dunque $n \in \omega \setminus \{0\}$ tale per cui ${(\omega^{(\omega^\delta)})}^n < \alpha \leq {(\omega^{(\omega^\delta)})}^{n+1}$. Ma allora + ${(\omega^{(\omega^\delta)})}^n \omega^{(\omega^\delta)} = {(\omega^{(\omega^\delta)})}^{n+1} \geq \alpha$, mentre + $\omega^{(\omega^\delta)}$, ${(\omega^{(\omega^\delta)})}^n < \alpha$, dunque $\alpha$ non è moltiplicativamente chiuso. + \item[\fbox{(iii.) $\implies$ (i.)}] Se $\beta = 0$, la tesi è banale. Supponiamo ora $\beta > 0$. Mostriamo la validità della tesi nel caso in cui $\delta$ è nullo, è successore o è limite non nullo. + \begin{enumerate} + \item[$\boxed{\delta = 0}$] Se $\delta = 0$, la tesi è banale dato che non ha minoranti stretti non nulli. + \item[$\boxed{\delta + 1}$] Se $\delta = \delta' + 1$ è un successore, allora, dato $\beta < \alpha = {(\omega^{(\omega^{\delta'})})}^\omega$, + si deve avere $\beta < {(\omega^{(\omega^{\delta'})})}^n$ per qualche $n \in \omega$. Pertanto: + \[ \alpha = {(\omega^{(\omega^{\delta'})})}^\omega \leq \beta \cdot \alpha \leq {(\omega^{(\omega^{\delta'})})}^n \cdot {(\omega^{(\omega^{\delta'})})}^\omega = {(\omega^{(\omega^{\delta'})})}^{n+\omega} = \alpha, \] + da cui $\beta + \alpha = \omega^\delta$ (si è usato che $n + \omega = \omega$, vd. \textit{Esercizio 56}). + \item[$\boxed{\delta = \lambda \text{ limite}}$] Se invece $\delta = \lambda$ è un limite, dacché $\omega^{(\omega^\lambda)} = \sup_{\gamma < \lambda} \omega^{(\omega^\gamma)}$, $\beta < \omega^{(\omega^\lambda)}$ implica che esiste + $\gamma < \lambda$ per cui $\beta < \omega^{(\omega^\gamma)}$. Per il teorema della differenza a destra, esiste inoltre + $\varepsilon < \lambda$ per cui $\gamma + \varepsilon =\lambda$. Pertanto si deduce che: + \[ \omega^{(\omega^\lambda)} \leq \beta \cdot \omega^{(\omega^\lambda)} \leq \omega^{(\omega^\gamma)} \cdot {(\omega^{(\omega^\gamma)})}^{\omega^\varepsilon} = {(\omega^{(\omega^\gamma)})}^{1 + \omega^\varepsilon} = \omega^{(\omega^\lambda)}, \] + dove si è usato che $1 + \omega^\varepsilon = \omega^\varepsilon$, dal momento che $\omega^\varepsilon$ è infinito. + \end{enumerate} + \end{enumerate} +\end{solution} + \begin{problem}{Moltiplicare a destra per $\omega$ trasforma l'ordinale nella più piccola potenza di $\omega$ che lo maggiora}{problem-59} Sia $\alpha$ un ordinale per cui $\omega^\delta \leq \alpha < \omega^{\delta+1}$. Allora $\alpha \cdot \omega = \omega^{\delta + 1}$. \end{problem} +\begin{solution} + Sia $\alpha = \omega^\delta \cdot n + \rho$, dove $n$ e $\rho < \omega^\delta$ sono rispettivamente il quoziente e il resto di $\alpha$ diviso + $\omega^\delta$. $n$ è necessariamente un naturale, altrimenti $\alpha$ maggiorerebbe debolmente $\omega^{\delta+1}$. + Allora: + \[ \omega^{\delta + 1} = \omega^\delta \omega \leq \alpha \cdot \omega = (\omega^\delta \cdot n + \rho) \omega \leq (\omega^\delta \cdot (n + 1)) \omega = \omega^\delta \omega = \omega^{\delta + 1}, \] + dove si è usato che $(n+1) \cdot \omega = \omega$ (vd. \textit{Esercizio 58}). +\end{solution} + \begin{problem}{$\alpha \cdot \omega = \beta \cdot \omega$ se e solo se $\alpha$ e $\beta$ sono contenuti tra due stesse potenze successive di $\omega$}{problem-60} Siano $\alpha$, $\beta$ ordinali. Se $\alpha \cdot \omega = \beta \cdot \omega$, allora esiste $\delta$ ordinale per cui $\omega^\delta \leq \alpha$, $\beta < \omega^{\delta + 1}$.