From ede97ee42612a702f6079653efd6c062a0601d9b Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 7 Dec 2023 15:54:40 +0100 Subject: [PATCH] lecture 15 --- inputs/lecture_14.tex | 3 +- inputs/lecture_15.tex | 229 ++++++++++++++++++++++++++++++++++++++++++ logic2.tex | 1 + 3 files changed, 232 insertions(+), 1 deletion(-) create mode 100644 inputs/lecture_15.tex diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 9435d2f..250793c 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -130,11 +130,12 @@ We have shown (assuming \AxC to choose contained clubs): \] \end{definition} \begin{lemma} + \label{lem:diagiclub} Let $\kappa$ be a regular, uncountable cardinal. If $\langle C_{\beta} : \beta < \kappa \rangle$ is a sequence of club subsets of $\kappa$, then $\diagi_{\beta < \kappa} C_{\beta}$ - contains a club. + contains a club. % TODO: contains or is? \end{lemma} \begin{proof} Let us fix $\langle C_{\beta} : \beta < \alpha \rangle$. diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex new file mode 100644 index 0000000..23acc5c --- /dev/null +++ b/inputs/lecture_15.tex @@ -0,0 +1,229 @@ +\lecture{15}{2023-12-07}{} +% RECAP + +% Let $\kappa$ be uncountable and regular. +% If $\alpha < \kappa$ and $\{C_{\xi} : \xi < \alpha\}$ +% is a family of sets which are club in $\kappa$ +% then $\bigcap_{\xi < \alpha} C_\xi$ is a club. +% +% If $\{C_\xi : \xi < \kappa\}$ is a family of sets which are club in $\kappa$, +% then $\diagi_{\xi < \kappa} = \{\alpha < \kappa : \alpha \in \bigcap_{\xi < \alpha} C_\xi\}$ +% is a club. + +% END RECAP + + + + +\begin{theorem}[Fodor] + \yalabel{Fodor's Theorem}{Fodor}{thm:fodor} + Let $\kappa$ be a regular and uncountable cardinal. + Let $S \subseteq \kappa$ be stationary and + let $f\colon S \to \kappa$ be \vocab{regressive} + in the following sense: + $f(\alpha) < \alpha$ for all $\alpha \in S$. + + Then there exists a stationary subset $T \subseteq S$ + and some $\nu < \kappa$ such that + $f(\alpha) = \nu$ for all $\alpha \in T$. +\end{theorem} +\begin{proof} + Let $S, f$ be given. + For $\nu < \kappa$ set $S_\nu \coloneqq \{\alpha \in S : f(\alpha) = \nu\}$. % f^{-1}(\nu)$. + We aim to show that one of the $S_\nu$ is stationary. + Suppose otherwise. + Then for every $\nu$ there exists a club $C_\nu$ + such that $S_\nu \cap C_\nu = \emptyset$.\footnote{Here we use \AxC to choose the $C_\nu$ uniformly.} + Let $C = \diagi_{\nu < \alpha} C_\nu$. + By \yaref{lem:diagiclub} $C$ is a club.\todo{Show that it \emph{is} a club not just contains one} + So we may pick some $\alpha \in C \cap S$. + In particular $\alpha \in C_\nu$ for all $\nu < \alpha$. + Hence $f(\alpha) \neq \nu$ for all $\nu < \alpha$, + so $f(\alpha) \ge \alpha$. + But $f$ is regressive $\lightning$ +\end{proof} + +\subsection{Some model theory and a second proof of Fodor's Theorem} + + +Recall the following: + +\begin{definition} + A substructure $X \subseteq V_\theta$\todo{make this more general. Explain why $V_\theta$ is a model} + is an \vocab{elementary substructure} + of $V_\theta$, + denoted $X \prec V_{\theta}$,\footnote{more formally $(X,\in ) \prec (V_{\theta})$} + iff for all formulae $\phi$ of the language of set theory + and for all $x_1,\ldots,x_k \in X$, + \[ + (X; \in\defon{X}) \models \phi(x_1,\ldots,x_k) + \iff(V_\theta; \in\defon{V_\theta}) \models\phi(x_1,\ldots,x_k). + \] +\end{definition} + +\begin{remark} + Löwenheim-Skolem allows us to find elementary substructures + of arbitrary sizes. + How do we do this? + Let $\phi$ be a formula. + A \vocab{Skolem-function} + over $V_\theta$ for $\phi$ + is a function + \[ + f\colon {}^k V_\theta \to V_\theta, + \] + where $k$ is the number of free variables of + $\exists v.~\phi$ + and for all $x_1,\ldots,x_k \in V_\theta$, + if $(V_\theta, \in) \models \exists v.~\phi(v,x_1,\ldots,x_k)$ + then $(V_\theta, \in) \models \phi(f(x_1,\ldots,x_k),x_1,\ldots,x_k)$. + + Using \AxC such Skolem-functions can be easily found for all formulae. +\end{remark} + +There is a sufficient criterion for $X \subseteq V_{\theta}$ +to be an elementary substructure of $V_\theta$. +\begin{lemma}[Tarski-Vaught Test] + Let $X \subseteq V_\theta$. + For each formula $\phi$, + let $f_\phi$ be a Skolem function over $V_\theta$ for $\phi$. + If for every $\phi$ and for all $x_1,\ldots, x_k \in X$ + (where $k$ is the number of free variables of $\exists v.~\phi$) + $f_\phi(x_1,\ldots,x_k) \in X$, + then $X \prec V_{\theta}$. + +\end{lemma} + +Let's do a second proof of \yaref{thm:fodor}. +\begin{refproof}{thm:fodor} + Fix $\theta > \kappa$ and look at $V_{\theta}$. + + Fix $S \subseteq \kappa$ stationary + and $f\colon S \to \kappa$ regressive. + + For each formula $\phi$ + fix a Skolem function $f_\phi$ + over $V_\theta$ for $\phi$. + Let $(X_\xi: \xi \le \kappa)$ + be a sequence of elementary substructures + of $V_\theta$ + defined as follows: + Let $X_0$ be the least $X$ such that + $S, f \in X$ and $X$ is closed under $f_\phi$. + Note that $X_0$ is countable. + + For $\xi < \kappa$ let + $X_{\xi + 1}$ + be the least $X \subseteq V_\theta$ + such that $X_\xi \subseteq X$, + $\min(\kappa \setminus X_\xi) \in X$ and $X$ is closed under + all $f_\phi$. + For limits $\lambda \le \kappa$ + let + \[ + X_\lambda \coloneqq \bigcup_{\xi < \lambda} X_\xi. + \] + + Note that $|X_{\xi}| = |X_{\xi + 1}|$ + but the size is increased at limits. + It is easy to see inductively that $|X_{\xi}| < \kappa$ + for every $\xi < \kappa$, + while $X_\xi \subsetneq X_{\xi'}$ + for all $\xi < \xi' \le \kappa$. + + Also $\xi \subseteq X_\xi$ for all $\xi \le \kappa$. + + + \begin{claim} + \label{thm:fodor:p2:c1} + There is a club $C \subseteq \kappa$ + such that $X_\xi \cap \kappa = \xi$ + for all $\xi \in C$. + \end{claim} + \begin{refproof}{thm:fodor:p2:c1} + Write $C = \{\xi < \kappa:X_\xi \cap \kappa = \xi\}$. + Trivially $C$ is closed. + Let us show that $C$ is unbounded in $\kappa$. + Let $\zeta < \kappa$. + Let us define a strictly increasing sequence + $ \langle \xi_n n < \omega \rangle$ + a follows. + Set $\xi_0 \coloneqq \zeta$. + Suppose $\xi_n$ has been chosen. + Look at $X_{\xi_n} \cap \kappa$. + Since $|X_{\xi_n} \cap \kappa| < \kappa$, + $\sup (X_{\xi_n \cap \kappa}) < \kappa$. + Set $\xi_{n+1} \coloneqq \sup(X_{\xi_n} \cap \kappa) + 1$. + Set $\xi \coloneqq \sup_{n<\omega} \xi_n$. + Clearly $\zeta < \xi$. + \begin{claim} + \label{thm:fodor:p2:c1.1} + $\xi \in C$, i.e.~$X_\xi \cap \kappa = \xi$. + \end{claim} + \begin{refproof}{thm:fodor:p2:c1.1} + If $\eta < \xi$, + then $\eta < \xi_n$ for some $n$ + and then $\eta \in \xi_n \subseteq X_{\xi_n} \subseteq X_{\xi}$. + + Now let $\eta \in X_\xi \cap \kappa$. + Then $\eta \in X_{\xi_n}$ for some $n < \omega$, + so $\eta < \xi_{n+1} < \xi$, + hence $X_{\xi} \cap \kappa \subseteq \xi$. + \end{refproof} + \end{refproof} + Now let $\alpha \in S \cap C$, + i.e.~$X_\alpha \prec V_{\theta}$ + and $\alpha = X_{\alpha} \cap \kappa$. + $f \in X_{\alpha}$ + and $f$ is regressive, so $f(\alpha) < \alpha$. + Write $\nu = f(\alpha)$. + Let $T = \{\xi \in S: f(\xi) = \nu\}$. + We have $T \in X_{\alpha}$, + as $T$ is definable from $S,f,\nu \in X_\alpha$. + + \begin{claim} + $T$ is stationary. + \end{claim} + \begin{subproof} + Otherwise there is a club $D \subseteq \kappa$ + such that $D \cap T = \emptyset$, + i.e.~ + \[V_\theta \models \exists D .~ D\text{ club in $\kappa$} \land D \cap T = \emptyset\] + hence + \[X_\alpha\models \exists D .~ D\text{ club in $\kappa$} \land D \cap T = \emptyset.\] + So there is $D \in X_\alpha$ such that + \[ + X_\alpha \models D \text{ is club in $\kappa$} \land D \cap T = \emptyset, + \] + hence + \[ + V_\theta \models D \text{ is club in $\kappa$} \land D \cap T = \emptyset. + \] + In other words, + there is some club + $D \in X_\alpha$ with $D \cap T = \emptyset$. + + + + We have $\alpha \in T$ as $\alpha \in S$ and $f(\alpha) = \nu$. + Let us show that $\alpha \in D$, which gives a contradiction. + For $\alpha \in D$ it suffices to show that $D \cap \alpha$ + is unbounded in $\alpha$. + Let $\xi < \alpha$. + As $D$ is unbounded in $\kappa$, + $\exists \eta > \xi .~ \eta \in D$, + so + \[ + V_{\theta} \models \exists \eta > \xi .~ \eta \in D, + \] + hence + \[ + X_\alpha \models \exists \eta > \xi .~ \eta \in D. + \] + Hence there is some $\eta \in X_\alpha$ with $\eta \in D$. + This means that + $\xi < \underbrace{\eta}_{\in D} < \alpha$.. + \end{subproof} +\end{refproof} + + diff --git a/logic2.tex b/logic2.tex index 1e132f3..70c2c44 100644 --- a/logic2.tex +++ b/logic2.tex @@ -38,6 +38,7 @@ \input{inputs/lecture_12} \input{inputs/lecture_13} \input{inputs/lecture_14} +\input{inputs/lecture_15} \cleardoublepage