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Alternativamente, si può dimostrare + la tesi per induzione transfinita, come fatto per il punto (ii.). + + \item Si mostra la tesi per induzione transfinita su $\beta$. + \begin{enumerate}[(i.)] + \item[$\boxed{\beta = 0}$] $\alpha + \beta = \alpha + 0 = \alpha$, banalmente. + \item[$\boxed{\beta + 1}$] $\alpha + (\beta + 1)$ è per definizione $(\alpha + \beta) + 1$. + Impiegando l'ipotesi induttiva si ottiene allora $\alpha + \beta \geq \beta$, e + dunque $(\alpha + \beta) + 1 \geq \beta + 1$. + \item[$\boxed{\beta = \lambda \text{ limite}}$] Per definizione: + $\alpha + \lambda = \sup_{\gamma < \lambda} \alpha + \gamma$. Allora, per + ipotesi induttiva, $\sup_{\gamma < \lambda} \alpha + \gamma \geq \sup_{\gamma < \lambda} \gamma = \lambda$, + da cui la tesi. + \end{enumerate} + + \item Deriva immediatamente dal punto (ii.): $\alpha + \beta + 1 > \alpha + \beta \geq \beta$. + + \item Si mostra per induzione transfinita in modo del tutto simile a come fatto per il punto (ii.), + sfruttando l'identità $\alpha \cdot (\beta + 1) = \alpha \cdot \beta + \alpha$ e sapendo che + $\alpha \cdot \lambda = \sup_{\gamma < \lambda} \alpha \cdot \gamma$ per $\lambda$ limite. + \end{enumerate} +\end{solution} + \begin{problem}{Disuguaglianze strette sugli ordinali con ordinale fisso a sinistro}{problem-52} Siano $\alpha$ e $\gamma < \gamma'$ ordinali. Allora si mostri che: @@ -1433,6 +1505,21 @@ originale in una formula esplicita: \end{enumerate} \end{problem} +\begin{solution} + Poiché $\gamma < \gamma'$, per il teorema della differenza a destra, esiste + $\rho > 0$ tale per cui $\gamma + \rho = \gamma'$. Mostriamo ora i due + risultati separatamente. + + \begin{enumerate}[(i.)] + \item $\alpha + \gamma' = (\alpha + \gamma) + \rho > \alpha + \gamma$, dove + si è usato la tesi del \textit{Problema 51}. + + \item $\alpha \cdot \gamma' = \alpha \cdot (\gamma + \rho) = \alpha \cdot \gamma + \alpha \cdot \rho$. + Poiché $\rho$ e $\alpha$ sono diversi da zero, $\alpha \cdot \rho > \alpha > 0$ per il \textit{Problema 51}. Allora + si conclude ancora utilizzando la tesi del \textit{Problema 51}, analogamente al punto (i.). + \end{enumerate} +\end{solution} + \begin{problem}{Distributività a destra degli ordinali}{problem-53} Siano $\alpha$, $\beta$ e $\gamma$ ordinali. Si mostri allora che: \[ \alpha \cdot (\beta + \gamma) = \alpha \cdot \beta + \alpha \cdot \gamma. \] @@ -1441,7 +1528,7 @@ originale in una formula esplicita: \begin{solution} La tesi è immediatamente implicata dal fatto che per gli ordinali $\alpha \cong \beta \implies \alpha = \beta$ e che il prodotto e la somma tra ordinali - corrispondono al prodotto (vd. \textit{Problema 49}) e alla somma tra insiemi ben ordinati, per i quali vale: + corrispondono al prodotto (vd.~\textit{Problema 49}) e alla somma tra insiemi ben ordinati, per i quali vale: \[ A \times (B + C) \cong (A \times B) + (A \times C), \] per il \textit{Problema 43}. \end{solution} @@ -1534,7 +1621,7 @@ originale in una formula esplicita: $n$ a $S(n) = n+1$. Assumiamo ora la tesi per $n \geq 1$ e mostriamola per $n+1$: \[ (n + 1) + \omega = n + (1 + \omega) = n + \omega = \omega, \] - dove si è usata l'associatività della somma (vd. \textit{Problema 48}). + dove si è usata l'associatività della somma (vd.~\textit{Problema 48}). \end{solution} \begin{problem}{Caratterizzazione degli ordinali che rispettano \underline{sulla somma} la proprietà di assorbimento a sinistra per ordinali più piccoli}{problem-57} @@ -1563,7 +1650,7 @@ originale in una formula esplicita: \item[\fbox{(iii.) $\implies$ (i.)}] 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 $\delta = 0$, allora $\alpha = 1$, per il quale ovviamente $0 + 1 = 1$ - (vd. \textit{Esercizio 47}). + (vd.~\textit{Esercizio 47}). \item[$\boxed{\delta + 1}$] Se $\delta = \delta' + 1$ è un successore, allora, dato $\beta < \alpha = \omega^{\delta'} \cdot \omega$, si deve avere $\beta < \omega^{\delta'} \cdot n$ per qualche $n \in \omega$. Pertanto: \[ \omega^{\delta' + 1} \leq \beta + \alpha \leq \omega^{\delta'} (n + \omega) = \omega^{\delta' + 1}, \] @@ -1607,7 +1694,7 @@ originale in una formula esplicita: \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}). + 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: @@ -1626,7 +1713,7 @@ originale in una formula esplicita: $\omega^\delta$. $n$ 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}). + 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} @@ -1956,7 +2043,7 @@ originale in una formula esplicita: \item Supponiamo che $\alpha = \lambda$ sia limite. Sia $X \in V_\lambda \setminus \{\emptyset\}$. In ZFC esiste dunque per scelta una funzione di scelta $f : \PP(X) \setminus \{\emptyset\} \to X$, appartenente a $\PP((\PP(X) \setminus \{\emptyset\}) \times X)$, - che è un elemento di $V_\lambda$ dal momento che soddisfa l'assioma delle parti (vd. (ii.)), di unione e di separazione (essendo transitivo), + che è un elemento di $V_\lambda$ dal momento che soddisfa l'assioma delle parti (vd.~(ii.)), di unione e di separazione (essendo transitivo), e dunque $V_\lambda$ soddisfa l'assioma di scelta. \medskip Viceversa supponiamo che $V_\alpha$ soddisfi scelta. Allora esiste una funzione di scelta $f : \PP(X) \setminus \{\emptyset\} \to X$ per