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Sia $X \subsetneq A$ un segmento iniziale proprio di $A$. Poiché $X$ è proprio, $A \setminus X$ \underline{non} è vuoto, quindi ne esiste il minimo, detto $a$. Mostriamo che $X = A_a$. Se $x \in A_a$, allora $x < a$, dunque $x \in X$; altrimenti $x$ apparterrebbe a - $X \setminus A$ e sarebbe contemporaneamente il suo minimo, $\Lightning$. Viceversa, se $x \in X$, - se fosse $a \leq x$, si avrebbe $a \in X$, essendo $A$ un segmento iniziale, $\Lightning$. Dunque + $X \setminus A$ e sarebbe contemporaneamente il suo minimo, \Lightning. Viceversa, se $x \in X$, + se fosse $a \leq x$, si avrebbe $a \in X$, essendo $A$ un segmento iniziale, \Lightning. Dunque $x < a$, ossia $x \in A_a$; si conclude allora che $X = A_a$. \medskip Si supponga che ogni segmento iniziale proprio è generato da un elemento in $A$. Sia $X$ un sottinsieme \underline{non} vuoto di $A$. I minoranti stretti di $X$ sono allora un segmento iniziale proprio di $A$, e dunque sono generati da un elemento $a \in A$. Mostriamo che $a = \min X$. Sia $x \in X$. Se fosse - $x < a$, allora $x$ apparterrebbe ad $A_a$, ovverosia sarebbe un minorante stretto di $X$, $\Lightning$. Dunque + $x < a$, allora $x$ apparterrebbe ad $A_a$, ovverosia sarebbe un minorante stretto di $X$, \Lightning. Dunque $a \leq x$ per ogni $x \in X$. Se $a$ non appartenesse ad $X$, allora sarebbe $a < x$ per ogni $x \in X$, e dunque - si avrebbe $a \in A_a$, $\Lightning$. Dunque $x$ appartiene ad $X$ e ne è minorante debole, ovverosia è il minimo + si avrebbe $a \in A_a$, \Lightning. Dunque $x$ appartiene ad $X$ e ne è minorante debole, ovverosia è il minimo di $X$. \end{solution} @@ -1640,7 +1640,7 @@ originale in una formula esplicita: \end{solution} \begin{problem}{Condizioni sull'assorbimento della somma}{problem-61} - Siano $\alpha$, $\beta \neq 0$. Allora vale una e una sola delle seguenti affermazioni: + Siano $\alpha$, $\beta \neq 0$ distinti. Allora vale una e una sola delle seguenti affermazioni: \begin{enumerate}[(i.)] \item $\alpha + \beta = \beta$. @@ -1649,6 +1649,39 @@ originale in una formula esplicita: \end{enumerate} \end{problem} +\begin{solution} + Mostriamo innanzitutto che al più una affermazione è vera. Se $\alpha + \beta = \beta$, + sicuramente non può essere $\beta + \alpha = \alpha$, e viceversa, altrimenti: + \[ \beta + \beta = \beta + \alpha + \beta = \beta + \alpha. \] + Per il teorema della differenza a destra, allora si avrebbe $\alpha = \beta$, \Lightning. + Supponiamo ora senza perdita di generalità che valga $\alpha + \beta = \beta$. Non può esistere + $\delta$ tale per cui $\omega^\delta \leq \alpha$, $\beta < \omega^{\delta + 1}$: se così fosse + si avrebbe: + \[ \alpha + \beta \] + + Supponiamo ora esista $\delta$ tale per cui $\omega^\delta \leq \alpha$, $\beta < \omega^{\delta + 1}$, e + mostriamo che né (i.), né (ii.) sono possibili. Poiché $\alpha$, $\beta$ sono compresi tra le + stesse potenze successive di $\omega$, + esistono $m$, $n \in \omega \setminus \{0\}$ e $\rho$, $\rho' < \omega^\delta$ tali per cui: + \[ \alpha = \omega^\delta \cdot m + \rho, \quad \beta = \omega^\delta \cdot n + \rho'. \] + Allora: + \[ \alpha + \beta = \omega^\delta \cdot m + \rho + \omega^\delta \cdot n + \rho' = + \omega^\delta \cdot (m + n) + \rho', \] + dove si è usato l'assorbimento di $\omega^\delta$ e che $n > 0$. Allora $\alpha + \beta$ non può + coincidere con $\beta$, infatti si avrebbe: + \[ \omega^\delta \cdot n + \omega^\delta \cdot m + \rho' = \omega^\delta \cdot n + \rho', \] + e quindi per il teorema della differenza a destra i avrebbe $\omega^\delta \cdot m + \rho' = \rho'$, che è + assurdo, \Lightning. Analogamente $\beta + \alpha$ non può coincidere con $\alpha$. \medskip + + Mostriamo che una delle tre affermazioni è sempre vera. Se $\exists \, \delta$ + con $\omega^\delta \leq \alpha$, $\beta < \omega^{\delta + 1}$, la tesi è vera. Altrimenti, + esiste $\delta$ tale per cui $\alpha < \omega^\delta \leq \beta$ o + $\beta < \omega^\delta \leq \alpha$. Nel primo caso, allora $\beta$ assorbe $\alpha$; nel secondo + $\alpha$ assorbe $\beta$. Per vederlo è sufficiente scrivere il maggiore dei due come $\omega^\delta \cdot n + \rho$, + come prima, e calcolare la somma direttamente, impiegando l'assorbimento di $\omega^\delta$ nei confronti + del minore. +\end{solution} + \begin{problem}{Proprietà fondamentali dell'algebra cardinale}{problem-62} Siano $\kappa$, $\mu$, $\ni$ cardinali. Allora: @@ -1668,10 +1701,25 @@ originale in una formula esplicita: deriva immediatamente. \begin{enumerate}[(i.)] - \item $\abs{(A \sqcup B) \sqcup C} = \abs{A \sqcup (B \sqcup C)}$, dove la bigezione è data dalle seguenti relazioni: + \item $\abs{(A \sqcup B) \sqcup C} = \abs{A \sqcup (B \sqcup C)}$, dove una bigezione è data dalle seguenti relazioni: \[ ((a, 0), 0) \mapsto (a, 0), \] \[ ((b, 1), 0) \mapsto ((b, 0), 1), \] \[ (c, 1) \mapsto ((c, 1), 1). \] + + \item $\abs{(A \times B) \times c} = \abs{A \times (B \times C)}$, dove una bigezione è data da $((a, b), c) \mapsto (a, (b, c))$. + + \item $\abs{A \sqcup B} = \abs{B \sqcup A}$, dove una bigezione è data da $(x, i) \mapsto (x, 1-i)$. + + \item $\abs{A \times B} = \abs{B \times A}$, dove una bigezione è data da $(a, b) \mapsto (b, a)$. + + \item $\abs{A \times (B \sqcup C)} = \abs{(A \times B) \sqcup (A \times C)}$, dove una bigezione è data da + $(a, (x, i)) \mapsto ((a, x), i)$. + + \item $\abs{A^C \times B^C} = \abs{(A \times B)^C}$, dove una bigezione è data da + $(f, g) \mapsto [c \mapsto (f(c), g(c))]$. + + \item $\abs{{(A^B)}^C} = \abs{A^{B \times C}}$, dove una bigezione è data da + $f \mapsto [(b, c) \mapsto (f(c)(b))]$ (\textit{currying}). \end{enumerate} \end{solution} @@ -1759,7 +1807,7 @@ originale in una formula esplicita: Se fosse $\cof(\cof(A)) < \cof(A)$, esisterebbe un sottinsieme cofinale in $\cof(A)$ di cardinalità strettamente minore di quella di $\cof(A)$. Questo sottinsieme sarebbe cofinale anche in $A$, essendo $\cof(A)$ cofinale in $A$. Allora $A$ ammetterebbe un sottinsieme - cofinale di cardinalità strettamente minore di $\cof(A)$, $\Lightning$. Dunque + cofinale di cardinalità strettamente minore di $\cof(A)$, \Lightning. Dunque $\cof(\cof(A)) \geq \cof(A)$, da cui la tesi, osservando che vale $\cof(A) \leq \abs{A}$ per ogni insieme $A$. \end{solution} @@ -1858,7 +1906,7 @@ originale in una formula esplicita: $\alpha \in V_{\alpha + 1}$, Sia dunque $F : \ORD \to V_*$ la funzione-classe che associa un ordinale $\alpha$ a sé stesso in $V_{\alpha + 1} \subseteq V_*$. $F$ è iniettiva, e dunque se $V_*$ fosse un insieme, - lo sarebbe anche $\ORD$, $\Lightning$. Dunque $V_*$ è una classe propria. + lo sarebbe anche $\ORD$, \Lightning. Dunque $V_*$ è una classe propria. \end{solution} \begin{problem}{Caratterizzazione della validità degli assiomi di coppia, delle parti, dell'infinito @@ -1886,7 +1934,7 @@ originale in una formula esplicita: $\alpha$ fosse un successore $\delta + 1$, allora, dacché $0$, $\delta \in V_{\delta + 1} = V_\alpha$, si avrebbe $\{0, \delta\} \in V_\alpha = \PP(V_\delta)$, e quindi $\delta \in \{0, \delta\} \subseteq V_\delta$, che è assurdo dal momento che un ordinale non può appartenere al livello della gerarchia di von Neumann che - indicizza, $\Lightning$. Dunque $\alpha$ è limite. + indicizza, \Lightning. Dunque $\alpha$ è limite. \item Supponiamo che $\alpha = \lambda$ sia limite. Allora dato $a \in V_\lambda$, esiste sicuramente $\mu < \lambda$ per cui $a \in V_\mu$. Poiché $V_\mu$ è transitivo, $a \subseteq V_\mu$, dunque @@ -1897,12 +1945,12 @@ originale in una formula esplicita: $\alpha$ fosse un successore $\delta + 1$, allora, dacché $\delta \in V_{\delta + 1} = V_\alpha$, si avrebbe $\PP(\delta) \in V_\alpha$, e quindi $\delta \in \PP(\delta) \subseteq V_\delta$, che è assurdo dal momento che un ordinale non può appartenere al livello della gerarchia di von Neumann che - indicizza, $\Lightning$. Dunque $\alpha$ è limite. + indicizza, \Lightning. Dunque $\alpha$ è limite. \item Poiché $V_n = n$ per ogni $n$ naturale, $V_\omega = \bigcup_{n \in \omega} V_n$ è esattamente $\omega$. Dunque per $\alpha \leq \omega$ non può essere vero l'assioma dell'infinito in $V_\alpha$: per $\alpha = \omega$, perché altrimenti si avrebbe $\omega \in V_\omega$, assurdo dal momento che un ordinale non può - appartenere al livello della gerarchia di von Neumann che indicizza, $\Lightning$; per $\alpha < \omega$, + appartenere al livello della gerarchia di von Neumann che indicizza, \Lightning; per $\alpha < \omega$, $V_\alpha$ è finito, e dunque non può ammettere per transitività un elemento infinito. Per $\alpha > \omega$, invece $\omega = V_\omega \subseteq V_\alpha$.