From 946da98a0465efe77a9e30ea11c5d7da01ff9711 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 18 Jan 2024 15:43:47 +0100 Subject: [PATCH] lecture 21 --- inputs/lecture_14.tex | 2 +- inputs/lecture_19.tex | 4 ++ inputs/lecture_21.tex | 161 ++++++++++++++++++++++++++++++++++++++++++ logic2.tex | 1 + 4 files changed, 167 insertions(+), 1 deletion(-) create mode 100644 inputs/lecture_21.tex diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index c91f05f..56711e7 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -225,7 +225,7 @@ We have shown (assuming \AxC to choose contained clubs): $\lambda \not\in D_\alpha$. Since $D_\alpha$ is closed, we get $\sup(D_{\alpha} \cap \lambda) < \lambda$. - In particular $\sup (\lambda \cap\diagi_{\beta < \kappa} D_{\beta}) \le \max (\alpha ,\sup(D_\alpha \cap \lambda) < \lambda$. + In particular $\sup (\lambda \cap\diagi_{\beta < \kappa} D_{\beta}) \le \alpha \cup \sup(D_\alpha \cap \lambda) < \lambda$. \end{remark} \begin{definition} diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 6a22304..8e49054 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -23,9 +23,13 @@ and $\forall x \in y.~\phi$ abbreviates $\forall x.~x \in y \to \phi$. If $\phi(x_0,\ldots,x_m) \in \Sigma_n$, then $(\forall x_0.~\ldots\forall x_m.~\phi(x_0,\ldots,x_m)) \in \Pi_{n+1}$. + If $\ZFC \models \phi \leftrightarrow \psi$ + and $\phi \in \Sigma_n$, then $\psi \in \Sigma_n$. If $\phi(x_0,\ldots,x_m) \in \Pi_n$, then $(\exists x_0.~\ldots\exists x_m.~\phi(x_0,\ldots,x_m) \in \Sigma_{n+1}$. + If $\ZFC \models \phi \leftrightarrow \psi$ + and $\phi \in \Pi_n$, then $\psi \in \Pi_n$. $\Delta_n \coloneqq \Sigma_n \cap \Pi_n$. \end{definition} diff --git a/inputs/lecture_21.tex b/inputs/lecture_21.tex new file mode 100644 index 0000000..e337b5f --- /dev/null +++ b/inputs/lecture_21.tex @@ -0,0 +1,161 @@ +\lecture{21}{2024-01-18}{} + +\begin{goal} + We want to show that certain statements are consistent with $\ZFC$ + (or $\ZF$), for instance $\CH$. + + We start with a model $M$ of $\ZFC$. + Usually we want $M$ to be transitive. + + We want to enlarge $M$ to get a bigger model, + where our desired statement holds, + i.e.~add more reals to violate $\CH$. + + However we need to do this in a somewhat controlled way, + so we can't just do it the way one builds field extensions. + In particular, when trying to violate $\CH$ we need to make sure that + we don't collapse cardinals. +\end{goal} +\begin{remark} + The idea behind forcing is clever. + Unfortunately an easy ``how could I have come up with this myself''-approach + does not seem to exist. +\end{remark} +\begin{remark} + How can a countable transitive model $M$ even exist? + + $M$ believes some statements that are wrong from the outside perspective. + For example there exists $\aleph_1^M \in M$ + such that $M \models x = \aleph_1$. + $\aleph_1^M$ is indeed an ordinal (since being an ordinal is a $\Sigma_0$-statement). + However $\aleph_1^M$ is countable, since $M$ is countable + and transitive. + This is fine. + (Note that ``$\aleph_1^M$ is uncountable'' is a $\Pi_1$-statement.) +\end{remark} + +\begin{idea}[The method of \vocab{forcing}] + Start with $M$, a countable transitive model of $\ZFC$ + and let $\mathbb{P} \in M$ be a partial order, + where $p \le q$ means that $p$ has ``more information'' + than $q$. + + A filter $g \subseteq \mathbb{P}$ is $\mathbb{P}$-generic over $M$ + iff $g \cap D \neq \emptyset$ for all dense $D \subseteq \mathbb{P}$, + $D \in M$. + + Next steps: + \begin{enumerate}[(1)] + \item Define the \vocab{forcing extension} $M[g]$. + \item Show that $M[g] \models \ZFC$. + \item Determine other facts about (the theory of) $M[g]$. + This depends on the partial order $\mathbb{P}$ we chose + in the beginning (and maybe $M$). + \end{enumerate} +\end{idea} + +\begin{example}[Prototypical example] + Let $\bP = 2^{< \omega}, p \le q \mathop{:\iff} p \supseteq q$ be Cohen forcing, + often denoted $\bC$. + + Let $M$ be a countable transitive model of $\ZFC$. + Since the definition of $\bC$ is simple enough, + $\bC \in M$. + Let $g$ be $\bC$-generic over $M$. + + \begin{claim} + \label{ex:cohen:c1} + For each $n \in \omega$, + the set $D_n \coloneqq \{ p \in \bC : n \in \dom(p)\}$ + is dense. + \end{claim} + \begin{subproof} + This is trivial. + \end{subproof} + \begin{claim} + \label{ex:cohen:c2} + $D_n \in M$. + \end{claim} + \begin{subproof} + The definition of $D_n$ is absolute. + \end{subproof} + \begin{claim} + \label{ex:cohen:c3} + If $p,q \in g \cap D_n$, + then $p(n) = q(n)$. + \end{claim} + \begin{subproof} + $g$ is a filter, so $p$ and $q$ are compatible. + $p,q \in D_n$ makes sure that $p(n)$ and $q(n)$ are defined. + \end{subproof} + Let $x = \bigcup g$. + By \yaref{ex:cohen:c3}, $x \in 2^{\le \omega}$. + By \yaref{ex:cohen:c1} and \yaref{ex:cohen:c2}, + we have $g \cap D_n \neq \emptyset$ for all $n < \omega$, + hence $n \in \dom(x)$ for all $n < \omega$. + So $x \in 2^{\omega}$. + + \begin{claim} + Let $z \in 2^{\omega}$, $z \in M$. + Then $D^z = \{p \in \bC : \exists n \in \dom(p) .~p(n) \neq z(n)\} $ + is dense. + \end{claim} + \begin{subproof} + Trivial. + \end{subproof} + \begin{claim} + $D^z \in M$ for all $z \in 2^{< \omega}$ with $z \in M$. + Therefore, $g \cap D^z \neq \emptyset$ for all $z \in M$, + $z\colon 2^{<\omega}$. + Hence $x \neq z$ for all $z \in M$, $z \in 2^{< \omega}$. + In other words $x \not\in M$. + \end{claim} + + The new real $x$ does not do too much damage to $M$ + when adding it.\footnote{We still need to make this precise.} + (Some reals would completely kill the model.) + + Now let $\alpha$ be an ordinal in $M$. + Let + \begin{IEEEeqnarray*}{rCll} + \bC(\alpha) &\coloneqq& \{p \colon &\text{$p$ is a function with domain $\alpha$,}\\ + &&&\text{$p(\xi) \in \bC$ for all $\xi < \alpha$,}\\ + &&&\text{$\{\xi < \alpha : p(\xi) \neq \emptyset\}$ is finite}\} + \end{IEEEeqnarray*} + ($\alpha$ many copies of $\bC$ with \vocab{finite support}). + + For $p, q \in \bC(\alpha)$ define $p \le q :\iff \forall \xi < \alpha .~p(\xi) \supseteq q(\xi)$. + We have $\bC(\alpha) \in M$ + + Let $g$ be $\bC(\alpha)$-generic over $M$. + Let $x_\xi = \bigcup \{p(\xi) : p \in g\}$ + for $\xi < \alpha$. + $x_\xi \in 2^{ \omega}$: + For each $n < \omega$ and $\xi < \alpha$, + \[ + D_{n,\xi} \coloneqq \{ p \in \bC(\alpha) : n \in \dom(p(\xi))\} \in M + \] + and $D_{n,\xi}$ is dense. + + \begin{claim} + For all $\xi, \eta < \alpha$, $\xi \neq \eta$, + \[ + D^{\xi, \eta} \coloneqq \{ p \in \bC(\alpha) : \exists n \in \dom(p(\xi)) \cap \dom(p(\eta)) .~ + p(\xi)(n) \neq p(\eta)(n)\} + \] + we have that $D^{\xi, \eta} \in M$ + and is $D^{\xi, \eta}$ dense. + \end{claim} + Therefore if $\xi \neq \eta$, $x_\xi \neq x_\eta$. + + Currently this is not very exciting, + since we only showed that for a countable transitive model $M$, + there is a countable set of reals not contained in $M$. + The interesting point will be, that we can actually add these reals + to $M$. +\end{example} + +Next steps: +\begin{itemize} + \item Make sense of $M[g]$. +\end{itemize} diff --git a/logic2.tex b/logic2.tex index cabf756..4686286 100644 --- a/logic2.tex +++ b/logic2.tex @@ -44,6 +44,7 @@ \input{inputs/lecture_18} \input{inputs/lecture_19} \input{inputs/lecture_20} +\input{inputs/lecture_21} \cleardoublepage