From 5804440eb21e64b4976f1597eedf849114205a00 Mon Sep 17 00:00:00 2001 From: Hearot Date: Wed, 2 Jul 2025 17:43:41 +0200 Subject: [PATCH] feat(eti): aggiunge altri risultati --- .../commands.tex | 1 + .../Elementi di Teoria degli Insiemi/main.pdf | Bin 520944 -> 523736 bytes .../Elementi di Teoria degli Insiemi/main.tex | 26 ++++++++++++++++++ 3 files changed, 27 insertions(+) diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/commands.tex b/Terzo anno/Elementi di Teoria degli Insiemi/commands.tex index 6ff14e4..4af8d87 100644 --- a/Terzo anno/Elementi di Teoria degli Insiemi/commands.tex +++ b/Terzo anno/Elementi di Teoria degli Insiemi/commands.tex @@ -11,6 +11,7 @@ \newcommand{\Rint}[4]{\int\limits_{#1}^{#2}{#3}\dd{#4}} +\newcommand{\HH}{\mathbb{H}} \DeclareMathOperator{\ot}{ot} \DeclareMathOperator{\cof}{cof} \DeclareMathOperator{\TC}{TC} diff --git a/Terzo anno/Elementi di Teoria degli Insiemi/main.pdf b/Terzo anno/Elementi di Teoria degli Insiemi/main.pdf index 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((a, 0), 0) \mapsto (a, 0), \] + \[ ((b, 1), 0) \mapsto ((b, 0), 1), \] + \[ (c, 1) \mapsto ((c, 1), 1). \] + \end{enumerate} +\end{solution} + \begin{problem}{$\aleph_\alpha \leq \beth_\alpha$ per ogni ordinale $\alpha$}{problem-63} Sia $\alpha$ un ordinale. Si mostri che $\aleph_\alpha \leq \beth_\alpha$. \end{problem} +\begin{solution} + Dimostriamo la tesi per induzione transfinita su $\alpha$. + + \begin{enumerate}[(i.)] + \item[$\boxed{\alpha = 0}$] Per definizione $\aleph_0 = \beth_0 = \omega$, dunque la tesi è vera. + \item[$\boxed{\alpha + 1}$] Per definizione $\aleph_\alpha \leq \beth_\alpha$, e dunque + $\aleph_{\alpha + 1} = \HH(\aleph_\alpha) \leq \HH(\beth_\alpha)$. Poiché $\beth_{\alpha + 1} = 2^{\beth_\alpha} > \beth_\alpha$, + essendo il numero di Hartogs di $\beth_\alpha$ il più piccolo cardinale maggiore di $\beth_\alpha$, si ottiene infine: + \[ \aleph_{\alpha + 1} = \HH(\aleph_\alpha) \leq \HH(\beth_\alpha) \leq \beth_{\alpha + 1}. \] + \item[$\boxed{\alpha = \lambda \text{ limite}}$] Per definizione $\aleph_\lambda = \sup_{\beta < \lambda} \aleph_\beta$ e $\beth_\lambda = \sup_{\beta < \lambda} \beth_\beta$. La tesi + segue allora immediatamente dall'ipotesi induttiva. + \end{enumerate} +\end{solution} + \begin{problem}{Proprietà fondamentali della somma infinita di cardinali}{problem-64} Sia $I$ un insieme. Si mostri allora che: