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Teoria degli Insiemi/main.tex index e80f7f0..1961c4f 100644 --- a/Terzo anno/Elementi di Teoria degli Insiemi/main.tex +++ b/Terzo anno/Elementi di Teoria degli Insiemi/main.tex @@ -1220,6 +1220,13 @@ originale in una formula esplicita: Si dimostri che $(\omega, \in)$ è un insieme ben ordinato. \end{problem} +\begin{solution} + La tesi deriva immediatamente dal fatto che $\omega$ è un insieme + totalmente ordinato per il quale ogni elemento in $\omega \setminus {0}$ è + un successore. Poiché su $\omega$ vale il principio di induzione (debole), + allora $\omega$ è ben ordinato (vd. \textit{Problema 35}). +\end{solution} + \begin{problem}{Equivalenza tra l'induzione forte e il buon ordinamento}{problem-34} Sia $(A, <)$ un insieme totalmente ordinato avente minimo $0 := \min A$. Si mostri che sono equivalenti: @@ -1231,7 +1238,7 @@ originale in una formula esplicita: \end{enumerate} \end{problem} -\begin{problem}{Equivalenza tra l'induzione (debole) e il buon ordinamento su insiemi di successori}{problem-35} +\begin{problem}{Equivalenza tra l'induzione (debole) e il buon ordinamento su insiemi con tutti gli elementi eccetto il minimo successori}{problem-35} Sia $(A, <)$ un insieme totalmente ordinato avente minimo $0 := \min A$ tale per cui ogni elemento in $A \setminus \{0\}$ è successore. Si mostri che sono equivalenti: @@ -1536,6 +1543,11 @@ originale in una formula esplicita: $\omega^\delta \leq \alpha$, $\beta < \omega^{\delta + 1}$. \end{problem} +\begin{solution} + La tesi deriva immediatamente dal \textit{Problema 59}, dal momento che $\alpha \cdot \omega$ è la più piccola potenza di $\omega$ + che maggiora $\alpha$, e che lo stesso succede con $\beta \cdot \omega$. +\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: @@ -1586,12 +1598,21 @@ originale in una formula esplicita: \end{enumerate} \end{problem} -\begin{problem}{Le cofinalità sono regolari: $\cof(\cof(A)) = \cof(A)$}{problem-66} +\begin{problem}{Le cofinalità sono cardinali regolari: $\cof(\cof(A)) = \cof(A)$}{problem-66} Sia $(A, <)$ un insieme totalmente ordinato. Si mostri allora che: \[ \cof(\cof(A)) = \cof(A). \] \end{problem} +\begin{solution} + 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 + $\cof(\cof(A)) \geq \cof(A)$, da cui la tesi, osservando che vale $\cof(A) \leq \abs{A}$ per + ogni insieme $A$. +\end{solution} + \begin{problem}{$\cof(\alpha + \beta) = \cof(\beta)$}{problem-67} Siano $\alpha$ e $\beta$ ordinali. Allora $\cof(\alpha + \beta) = \cof(\beta)$. \end{problem} @@ -1619,6 +1640,14 @@ originale in una formula esplicita: Sia $V_* := \bigcup_{\alpha \in \ORD} V_\alpha$. Si mostri che $V_*$ è una classe propria. \end{problem} +\begin{solution} % TODO: motivazione sul perché c'è \alpha in V_{\alpha + 1} + Osserviamo innanzitutto che per ogni ordinale $\alpha$, $\alpha \subseteq V_\alpha$, e quindi + $\alpha \in V_{\alpha + 1}$, + 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. +\end{solution} + \begin{problem}{Caratterizzazione della validità degli assiomi di coppia, delle parti, dell'infinito e della scelta per i livelli della gerarchia di von Neumann} Sia $\alpha$ un ordinale. Si mostri che: