From 326cca0e5495efa24a198abfe54694c1c1c80ff2 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 25 Jan 2024 15:35:02 +0100 Subject: [PATCH] lecture 23 --- inputs/intro.tex | 2 + inputs/lecture_23.tex | 130 ++++++++++++++++++++++++++++++++++++++++++ logic2.tex | 1 + 3 files changed, 133 insertions(+) create mode 100644 inputs/lecture_23.tex diff --git a/inputs/intro.tex b/inputs/intro.tex index 04940f4..2f245fb 100644 --- a/inputs/intro.tex +++ b/inputs/intro.tex @@ -21,3 +21,5 @@ please send me a message:\\ These notes follow the way the material was presented in the lecture rather closely. Additions (e.g.~from exercise sheets) and slight modifications have been marked with $\dagger$. + +Cut off for the exam is Christmas. diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex new file mode 100644 index 0000000..8f275f3 --- /dev/null +++ b/inputs/lecture_23.tex @@ -0,0 +1,130 @@ +\lecture{23}{2024-01-25}{Negation of CH is consistent to ZFC} + +\begin{goal} + We want to construct a model of $\ZFC$ + such that $2^{\aleph_0} \ge \aleph_2$. + % = ? +\end{goal} + +Let $M$ be a countable transitive model of $\ZFC$. +Suppose that $M \models \CH$ +(otherwise we are done). + +Let $\alpha = \omega_2^M$. + +Let $\bC(\alpha) \coloneqq \{p : p\colon \alpha \to \bC \text{ is a function such that } \{\xi < \alpha : p(\xi) \neq \emptyset\} \text{ is finite} \}$, +ordered by $p \le_{\bC(\alpha)} q$ iff +$p(\xi) \le_{\bC} q(\xi)$ for all $\xi < \alpha$. + +Recall that $\bC$ is the set of finite sequences of natural numbers +ordered by $p \le_{\bC} q$ iff $p \supseteq q$. + +Let $g $ be $\bC(\alpha)$-generic over $M$. +For $\xi < \alpha$ let $x_\xi = \bigcup \{ p(\xi) : p \in g\}$. +We have already seen that $x_\xi\colon \omega \to \omega$ +is a function +and $x_\xi \neq x_\eta$ for $\xi \neq \eta$. + +We have $M[g] \models\ZFC$.\footnote{We only handwaved this step.} +As $g \in M[g]$, +we have $\langle x_\xi : \xi < \alpha\rangle \in M[g]$. +Therefore $M[g] \models \text{``$2^{\aleph_0} \ge \alpha$''}$. +Also $\alpha = \omega_2^M$. +However the proof is not finished yet, +since we need to make sure, +that $M[g]$ does not collapse cardinals. + +We only have $M[g] \models 2^{\aleph_0} \ge \aleph_2^M$, +i.e.~we need to see $\aleph_2^{M[g]} = \aleph_2^M$. + +\begin{claim} + Every cardinal of $M$ is still + a cardinal of $M[g]$. +\end{claim} +This suffices, because +then $\aleph_0^M = \aleph_0^{M[g]}$, +$\aleph_1^M = \aleph_1^{M[g]}$, +$\aleph_2^{M}= \aleph_2^{M[g]}$, $\ldots$ + +\begin{definition} + Let $(\bP, \le )$ be a partial order. + We say that $\bP$ has the + \vocab{countable chain condition} + (\vocab{c.c.c.})% + \footnote{it should really be the ``countable antichain condition''} + iff there is no uncountable + antichain, + i.e.~every uncountable $V \subseteq \bP$ + contains compatible $p \neq q$. +\end{definition} + +We shall prove: +\begin{claim} + \label{l23:c:1} + For all $\beta$, $\bC(\beta)$ has the c.c.c. +\end{claim} +\begin{claim} + \label{l23:c:2} + If $\bP \in M$ and + $M \models \text{`` $\bP$ has the c.c.c.''}$ + and $h$ is generic over $M$, + then all $M$-cardinals are still + $M[h]$ cardinals. + \footnote{Being a cardinal is $\Pi_1$, + so $M[h]$ cardinals are always $M$ cardinals.} +\end{claim} +\begin{refproof}{l23:c:2} + Suppose not. + Let $\kappa$ be minimal such that $M \models \text{``$\kappa$ is a cardinal''}$, + but $M[h] \models \text{``$\kappa$ is not a cardinal''}$. + Then $\kappa = (\lambda^+)^M$ for some unique $M$-cardinal $\lambda < \kappa$. + By minimality, $\lambda$ is also an $M[h]$-cardinal. + + Let $f \in M[h]$ be such that $M[h] \models \text{``$f$ is a surjection from $\lambda$ onto $\kappa$''}$. + There is a name $\tau \in M^{\bP}$ with $\tau^h = f$. + + We then have some $p \in h$ + with $p \Vdash_M^\bP \text{``$\tau$ is a surjection from $\check{\lambda}$ onto $\check{\kappa}$''}$. + + Let $\xi < \lambda$. + Consider $X_\xi \coloneqq \{\eta < \kappa: \exists q \le_{\bP} p .~q \Vdash \tau(\check{\xi}) = \check{\eta}\} \in M$. + + $X_\xi$ is countable in $M$ by the following argument (in $M$): + For every $\eta \in X_\xi$, + let $q_\eta \le p$ be such that $q_\xi \Vdash^{\bP}_M \tau(\check{\xi}) = \check{\eta}$. + The set $\{q_\eta : \eta \in X_\xi\}$ is an antichain + as for $\eta_1 \neq \eta_2$ we have that $q_{\eta_i} \Vdash \tau(\check{\xi}) = \check{\eta_i}$, + so they are not compatible. + So $\{q_\eta : \eta \in X_\xi\}$ is countable by the c.c.c. + Thus $X_\xi$ is countable. + + + + Therefore we may define a function in $M$ + \begin{IEEEeqnarray*}{rCl} + F\colon \lambda \times \omega &\longrightarrow & \kappa + \end{IEEEeqnarray*} + such that for all $\xi < \lambda$ + \[ + \{F(\xi,n) : n < \omega\} = X_\xi. + \] + + $F$ is surjective since $f$ is surjective: + For $\eta < \kappa$, + there is some $\xi < \lambda$ such that $M[h] \models \text{``$f(\xi) = \eta$''}$, + there is some $\overline{q} \in h$ with $\overline{q} \Vdash ^{\bP}_M \tau(\check{\xi}) = \check{\eta}$. + Pick $q \le \overline{q},p$. + This shows $\eta \in X_\xi$ + hence $\eta = F(\xi, n)$ for some $n$. + But $|\lambda \times \omega| = |\lambda| = \lambda$, + so in $M$ there is a surjection $F' \colon \lambda \to \kappa$, + but $\kappa$ is a cardinal in $M$ $\lightning$. +\end{refproof} + +\begin{refproof}{l23:c:1} + Omitted. + % TODO combinatorial argument +\end{refproof} + + + diff --git a/logic2.tex b/logic2.tex index 12b278c..047291b 100644 --- a/logic2.tex +++ b/logic2.tex @@ -46,6 +46,7 @@ \input{inputs/lecture_20} \input{inputs/lecture_21} \input{inputs/lecture_22} +\input{inputs/lecture_23} \cleardoublepage