diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index 0a39244..c1a9012 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -130,7 +130,7 @@ The next goal is to show the following: (However the method might be more interesting than the result) \begin{theorem}[Silver] - \yalabel{Silver's Theorem}{Silver}{thm:silver} + \yalabel{Silver's Theorem}{Silver}{thm:silver1} If $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for all $\alpha < \omega_1$, then $2^{\aleph_{\omega_1}} = \aleph_{\omega_1 + 1}$. diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index cacde8b..120c3ee 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -100,7 +100,7 @@ $j(\alpha)$ is an ordinal such that $j(\alpha) \ge \alpha$. \end{claim} \begin{subproof} - $\alpha \in \Ord$ + $\alpha \in \OR$ can be written as \[\forall x \in \alpha .~\forall y \in x.~y \in \alpha \land \forall x \in \alpha .~ \forall y \in \alpha.~(x \in y \lor x = y \lor y \in x). diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex new file mode 100644 index 0000000..6d5fe9d --- /dev/null +++ b/inputs/lecture_19.tex @@ -0,0 +1,222 @@ +\lecture{19}{2024-01-11}{Forcing} + +Recall that $\exists x \in y.~ \phi$ abbreviates +$\exists x.~ x \in y \land \phi$ +and $\forall x \in y.~\phi$ abbreviates $\forall x.~x \in y \to \phi$. + +\begin{definition}[Arithmetical Hierarchy] + Let $\phi$ be a $\cL_{\in}$-formula. + We say that $\phi$ is \vocab{$\Delta_0$} (or \vocab{$\Sigma_0$} + or \vocab{$\Pi_0$}) + iff it is in the smallest set $\Gamma$ of formulas + such that + \begin{enumerate}[(1)] + \item $\Gamma$ contains all \vocab[Formula!atomic]{atomic} formulas ($x \in y$,$x=y$). + \item If $\phi, \psi \in \Gamma$, + then so are $\lnot \phi$ + and $\phi \land \psi$.% + \footnote{It follows that $\phi \lor \psi$, $\phi \to \psi$ and $\phi \leftrightarrow \psi$ + are also in $\Gamma$.} + \item If $\phi \in \Gamma$, + then $(\exists x \in y.~\phi), (\forall x \in y.~\phi) \in \Gamma$. + \end{enumerate} + + 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 $\phi(x_0,\ldots,x_n) \in \Pi_n$, + then $(\exists x_0.~\ldots\exists x_n.~\phi(x_0,\ldots,x_n) \in \Sigma_{n+1}$. + + $\Delta_n \coloneqq \Sigma_n \cap \Pi_n$. +\end{definition} + +\begin{notation} + Assume that $M$ is transitive + and $\phi$ is sentence. + Then + \[ + M \models \phi + \] + means that $(M, \in\defon{M}) \models \phi$. + + If $ a_0,\ldots,a_n \in M$ + and $\phi(x_0,\ldots,x_n)$ is a $\cL_\in$-formula, + then we say $M \models \phi(a_0,\ldots,a_n)$ + iff $M$ satisfies $\phi(x_0,\ldots,x_n)$ + for the assignment $x_i \mapsto a_i$. +\end{notation} + +\begin{lemma} + Let $M$ be transitive, $\phi \in \Delta_0$ + and $a_0,\ldots, a_n \in M$. + Then $M \models\phi(a_0,\ldots,a_n)$ + iff $V \models\phi(a_0,\ldots,a_n)$. +\end{lemma} +\begin{proof} + Clearly $M \models a_i \in a_j \iff V \models a_i \in a_j$ + and $M \models a_i = a_j \iff V \models a_i = a_j$, + i.e.~the lemma holds for atomic $\phi$. + % TODO transitivity needed? + + It is clear that if $M \models \phi_i \iff V \models \phi_i, i = 1,2$, + then also $M \models \lnot \phi_i \iff V \models \lnot \phi_i$ + and $M \models \phi_1 \land \phi_2 \iff V \models \phi_1 \land \phi_2$. + + Assume that the lemma holds for $\phi$. + Then it also holds for $\exists a_i \in a_j.~\phi$: + We have that $a_i \in a_j$ is atomic and by the assumption that the lemma holds for $\phi$ + so since $M$ is transitive, a witness can be transferred from $V$ to $M$ + and vice versa. + The case of $\forall a_i \in a_j.~\phi$ can be treated similarly. +\end{proof} +A similar arguments yields \vocab{upwards absoluteness} for $\Sigma_1$-formulas +and \vocab{downwards absoluteness} for $\Pi_1$-formulas: +\begin{lemma} + Let $M$ be transitive. + Let $\phi(x_0,\ldots,x_n) \in \cL_\in$ and $a_0,\ldots,a_n \in M$. + Then + \begin{itemize} + \item If $\phi$ is $\Sigma_1$, then + \[ + M \models\phi(a_0,\ldots,a_n) \implies V \models \phi(a_0,\ldots,a_n). + \] + \item If $\phi$ is $\Pi_1$, then + \[ + V \models\phi(a_0,\ldots,a_n) \implies M \models \phi(a_0,\ldots,a_n). + \] + \end{itemize} +\end{lemma} +\begin{definition} + Assume that $T$ is a theory and $\phi \in \cL_\in $ a formula + We say that $\phi$ is \vocab{$\Delta_1^T$} + iff there are formulas $\psi$, $\tau$ + such that $\psi \in \Sigma_1$, $\tau \in \Pi_1$ + and + \[ + T \vdash\phi \leftrightarrow \psi \leftrightarrow\tau. + \] +\end{definition} +Again by a similar argument we get: +\begin{lemma} + Let $M$ be a transitive model of a theory $T$. + Let $\phi$ be a $\Delta_1^T$ formula + and $a_0,\ldots,a_n \in M$. + Then $M \models \phi(a_0,\ldots,a_n) \iff V \models \phi(a_0,\ldots,a_n)$. +\end{lemma} +\begin{lemma} + Let $\phi$ denote the statement ``$R$ is a well-founded relation''. + Then $\phi \in \Delta_1^{\ZFC^-}$. +\end{lemma} +\begin{proof} + $\phi$ is equivalent to + \begin{itemize} + \item $R$ is a relation ($\Delta_0$) and + \item $\forall b.~b \cap \ran(R) = \emptyset \lor \exists x \in b.~\text{``$x$ is $R$-minimal''}$. + \end{itemize} + We only need to care about the second point. + This is equivalent (using \AxC!) to + the statement that there is no + \[ + f\colon \omega \to \dom(R) \cup \ran(R) \text{ such that } \forall n < \omega.~f(n+1)Rf(n), + \] + which can be written as a $\Pi_1$-formula. + With the help of ranks, we can also write it as a $\Sigma_1$-formula: + \[ + \exists r\colon \OR \to \dom(R) \cup \ran(R).~ + \forall x \in \dom(R) \cup \ran(R).~ r(x) = \{\sup(r(y) +1) : y R x\}. + \] + + So $\phi \in \Delta_1^{\ZFC^-}$. +\end{proof} +\begin{lemma} + Assume that $M$ is transitive. Then + \begin{enumerate}[(1)] + \item $M \models \AxExt$. + \item $M \models \AxFund$. + \item If $\omega \in M$, then $M \models \AxInf$. + \item If $M$ is closed under $(x,y) \mapsto \{x,y\}$, + then $M \models \AxPair$. + \item If $M$ is closed under $x \mapsto \bigcup x$, + then $M \models \AxUnion$. + \end{enumerate} +\end{lemma} +\begin{proof} + \begin{enumerate}[(1)] + \item Let $x, y \in M$ + such that $M \models \forall t .~t \in x \iff t \in y$. + Since $M$ is transitive $V \models \forall t.~t \in x \iff t \in y$. + Since $V \models \Ext$, we can apply + $V \models x = y \iff M \models x = y$. + \item We need to show $M \models \forall y \neq \emptyset.~\exists x \in y.~x \cap y = \emptyset$. + Let $y \in M$. Since $V \models \AxFund$, + $V \models \exists x \in y .~x \cap y = \emptyset$. + Note that this is a $\Delta_0$-formula, + hence $M \models \exists x \in y.~x \cap y = \emptyset$. + \item By assumption $\omega \in M$. + Since $M$ is transitive, we get $\omega \subseteq M$. + Hence $\omega$ is a witness for $\AxInf$. + \item Trivial. + \item Trivial. + \end{enumerate} +\end{proof} + + +\section{Forcing} +Recall that a structure $\bP = (P, \le )$ +is a partially ordered set (\vocab{poset}) +if $\le $ is reflexive, symmetric nd transitive. + +\begin{definition} + A non-empty poset $\bP = (P, \le )$ is called a \vocab{forcing notion}. + The elements of $P$ are called \vocab{conditions}. + If $q \le p$ we say that $q$ is \vocab{stronger} than $p$.% + \footnote{i.e.~it carries more information.} + $D \subseteq P$ is called \vocab{dense} + iff $\forall p \in P.~\exists q \in D.~q \le p$. + + Let $p \in P, D \subseteq P$. Then $D$ is \vocab{dense below $p$} + iff $\forall P \ni q \le p.~\exists r \in D.~r \le q$. + + $G \subseteq P$ is called a \vocab{filter} + iff + \begin{enumerate}[(1)] + \item $\forall p,q \in G.~\exists r \in G.~r \le p \land r \le q$. + \item $(p \in G \land p \le q) \implies q \in G$. + \end{enumerate} + + For $p,q \in P$ we say that $p$ and $q$ are \vocab{compatible}, + $p || q$, + iff $\exists r \in P .~r \le p \land r \le q$. + Otherwise they are \vocab{incompatible}, $p \perp q$. + + Let $\cD$ be a family of dense subsets of $P$ + and $G$ a filter. + We say that $G$ is \vocab{$\cD$}-generic + iff $\forall D \in \cD.~G \cap D \neq \emptyset$. +\end{definition} + +\begin{lemma} + Let $\cP = (P, \le )$ be a poset, + $\cD$ a countable family of dense subsets of $P$ + and $p \in P$. + Then there exists a $\cD$-generic filter $G \subseteq P$ + such that $p \in G$. +\end{lemma} +\begin{proof} + Fix $p$ as above. + Let $\langle D_n : n < \omega \rangle$ be an enumeration of $\cD$. + Let $p_0 \le p$ be such that $p_0 \in D_0$. + If $p_n$ is given, let $p_{n+1} \le p_n$ be such that + $p_{n+1} \in D_{n+1}$. + This is possible since $\cD$ is a collection of dense sets. + Define $ G \coloneqq \{ q \in P : \exists n.~p_n \le q\}$. + + $G$ is a filter: + Let $r,q \in G$. Let $n_r, n_q < \omega$ such that $p_{n_r} \le r$ + and $p_{n_q} \le q$. + Let $m = \max \{n_r, n_q\}$. + Then $p_m$ is a common extension. + + Clearly $G$ is $\cD$-generic. +\end{proof} + diff --git a/logic.sty b/logic.sty index 7c3a7af..1484949 100644 --- a/logic.sty +++ b/logic.sty @@ -129,7 +129,7 @@ \DeclareSimpleMathOperator{CH} \DeclareSimpleMathOperator{GCH} \DeclareSimpleMathOperator{DC} -\DeclareSimpleMathOperator{Ord} +%\DeclareSimpleMathOperator{Ord} \DeclareSimpleMathOperator{OR} % Ordinals \DeclareSimpleMathOperator{trcl} \DeclareSimpleMathOperator{tcl} diff --git a/logic2.tex b/logic2.tex index 7bc043d..da74ac4 100644 --- a/logic2.tex +++ b/logic2.tex @@ -42,6 +42,7 @@ \input{inputs/lecture_16} \input{inputs/lecture_17} \input{inputs/lecture_18} +\input{inputs/lecture_19} \cleardoublepage