From 7afac869cda1108dfd7cc834f6d7e2e5c8a4e088 Mon Sep 17 00:00:00 2001 From: zeramorphic <50671761+zeramorphic@users.noreply.github.com> Date: Fri, 16 Feb 2024 12:59:23 +0000 Subject: [PATCH 1/2] Lectures 22 --- iii/forcing/03_forcing.tex | 167 ++++++++++++++++++++++++++++- iii/lc/02_measurable_cardinals.tex | 87 +++++++++++++++ 2 files changed, 253 insertions(+), 1 deletion(-) diff --git a/iii/forcing/03_forcing.tex b/iii/forcing/03_forcing.tex index 23088c3..112964b 100644 --- a/iii/forcing/03_forcing.tex +++ b/iii/forcing/03_forcing.tex @@ -71,7 +71,11 @@ \subsection{Forcing posets} The notation \( \mathbb P \in M \) abbreviates \( (\mathbb P, \leq_{\mathbb P}, \Bbbone_{\mathbb P}) \in M \). Note that by transitivity if \( \mathbb P \) is an element of \( M \), then \( \Bbbone_{\mathbb P} \in M \), but we do not necessarily have \( \leq_{\mathbb P} \in M \). \begin{definition} - A preorder is \emph{separative} if whenever \( q \nleq p \), there is \( r \leq q \) such that \( r \perp p \). + A preorder is \emph{separative} if whenever \( p \neq q \), exactly one of the following two cases holds: + \begin{enumerate} + \item \( q \leq p \) and \( p \nleq q \); or + \item there exists \( r \leq q \) such that \( r \perp p \). + \end{enumerate} \end{definition} \begin{proposition} If \( (\mathbb P, \leq) \) is a separative preorder, it is a partial order. @@ -223,3 +227,164 @@ \subsection{Dense sets and genericity} Finally, let \( G = \qty{r \in \mathbb P \mid \exists n.\, q_n \leq r} \). Then \( G \) is a filter as the \( q_n \) form a chain, and it is clearly generic. \end{proof} +\begin{definition} + A condition \( p \in \mathbb P \) is \emph{minimal} if whenever \( q \leq p \), we have \( q = p \). +\end{definition} +\begin{lemma} + Let \( M \) be a countable transitive model of \( \mathsf{ZF} \), and let \( \mathbb P \in M \) be a separative partial order. + Then either \( \mathbb P \) has a minimal element, or for every filter \( G \) which is \( \mathbb P \)-generic over \( M \), we have \( G \notin M \). +\end{lemma} +\begin{proof} + Suppose \( \mathbb P \) has no minimal element. + Let \( G \) be a \( \mathbb P \)-generic filter over \( M \). + We show that if \( F \subseteq \mathbb P \) is a filter in \( M \), then the set \( D_F = \mathbb P \setminus F \in M \) is a dense set. + Then \( G \cap D_F \) is nonempty for all filters \( F \), so \( G \) cannot be equal to any filter \( F \in M \). + + Fix \( p \in \mathbb P \). + If \( p \notin F \), then \( p \in D_F \) as required. + Otherwise, suppose \( p \in F \). + As \( p \) is not minimal, we can fix some \( q \in F \) with \( q < p \). + Then \( p \nleq q \), so by separativity, there is \( r \leq p \) such that \( r \perp q \). + But all conditions in \( F \) are compatible, so one of \( r \) and \( q \) is not in \( F \). +\end{proof} +\begin{proposition} + For sets \( I, J \) such that \( \abs{I} \geq \omega \) and \( \abs{J} \geq 2 \), the forcing poset \( \operatorname{Fn}(I, J) \) is a separative partial order without a minimal element. +\end{proposition} +\begin{proposition} + (\( \mathsf{ZFC} \)) + Let \( \mathbb P \in M \) be a forcing poset, and let \( G \subseteq \mathbb P \). + Then the following are equivalent. + \begin{enumerate} + \item \( G \) is \( \mathbb P \)-generic over \( M \), that is, for all dense sets \( D \in M \), we have \( G \cap D \neq \varnothing \); + \item for all \( p \in G \) and \( D \in M \), if \( D \) is dense below \( p \) in \( \mathbb P \), then \( G \cap D \neq \varnothing \); + \item for all open dense sets \( D \in M \), we have \( G \cap D \neq \varnothing \); + \item for all \( D \in M \) that are maximal antichains in \( \mathbb P \), we have \( G \cap D \neq \varnothing \). + \end{enumerate} +\end{proposition} + +\subsection{Names} +\begin{definition} + Let \( \mathbb P \) be a forcing poset. + We define the class of \emph{\( \mathbb P \)-names} \( M^{\mathbb P} \) recursively as follows. + \begin{enumerate} + \item \( M_0^{\mathbb P} = \varnothing \); + \item \( M_{\alpha + 1}^{\mathbb P} = \mathcal P^M(\mathbb P \times M_\alpha^{\mathbb P}) \); + \item at limit stages \( \lambda \), \( M_\lambda^{\mathbb P} = \bigcup_{\alpha < \lambda} M_\alpha^{\mathbb P} \); + \item \( M^{\mathbb P} = \bigcup_{\alpha \in \mathrm{Ord}} \mathbb M_\alpha^{\mathbb P} \). + \end{enumerate} +\end{definition} +Being a \( \mathbb P \)-name is absolute for transitive models. +\( \mathbb P \)-names are denoted with overdots, such as in \( \dot x \). +\begin{definition} + The \emph{range} of a \( \mathbb P \)-name \( \dot x \) is + \[ \operatorname{ran}(\dot x) = \qty{\dot y \mid \exists p \in \mathbb P \mid \langle p, \dot y \rangle \in \dot x} \] +\end{definition} +\begin{remark} + Alternatively, by transfinite recursion on rank, we could define the class of \( \mathbb P \)-names over \( \mathrm{V} \) in the following way. + If \( \rank x = \alpha \), then \( x \) is a \( \mathbb P \)-name if and only if it is a relation such that for all \( \langle p, \dot y \rangle \in x \), we have \( p \in \mathbb P \) and \( \dot y \) is a \( \mathbb P \)-name in \( \mathrm{V}_\alpha \). + Finally, \( M^{\mathbb P} = \mathrm{V}^{\mathbb P} \cap M \). +\end{remark} +\begin{definition} + The \emph{\( \mathbb P \)-rank} of a name \( \dot x \), written \( \rank_{\mathbb P} \dot x \), is the least \( \alpha \) such that \( \dot x \subseteq \mathbb P \times M_\alpha^{\mathbb P} \). +\end{definition} +\begin{definition} + Let \( \dot x \) be a \( \mathbb P \)-name and \( G \) be an arbitrary subset of \( \mathbb P \). + We define the \emph{interpretation of \( \dot x \) by \( G \)} recursively by + \[ \dot x^G = \qty{\dot y^G \mid \exists p \in G.\, \langle p, \dot y \rangle \in \dot x} \] +\end{definition} +\begin{definition} + The \emph{forcing extension of \( M \) by \( G \)}, written \( M[G] \), is + \[ M[G] = \qty{\dot x^G \mid \dot x \in M^{\mathbb P}} \] +\end{definition} +\begin{example} + \begin{enumerate} + \item If \( \varnothing \in M \), then \( \varnothing^G = \varnothing \). + \item Let + \[ \dot x = \qty{\langle p, \varnothing \rangle, \langle r, \qty{\langle q, \varnothing \rangle} \rangle} \] + If \( p, q, r \in G \), then + \begin{align*} + \dot x^G &= \qty{(\langle p, \varnothing \rangle)^G, \qty(\langle r, \qty{\langle q, \varnothing \rangle} \rangle)^G} \\ + &= \qty{\varnothing, \qty{(\langle q, \varnothing \rangle)^G}} \\ + &= \qty{\varnothing, \qty{\varnothing}} + \end{align*} + If \( p, r \notin G \), then + \[ \dot x^G = \varnothing \] + If \( r \in G \) but \( p, q \notin G \), then + \[ \dot x^G = \qty{(\langle q, \varnothing \rangle)^G} = \qty{\varnothing} \] + Finally, if \( p \in G \) but \( r \notin G \), then + \[ \dot x^G = \qty{\varnothing} \] + \end{enumerate} +\end{example} +We aim to show the following major theorem. +\begin{theorem}[generic model theorem] + Let \( M \) be a countable transitive model of \( \mathsf{ZF} \), let \( \mathbb P \) be a forcing poset, and let \( G \) be a \( \mathbb P \)-generic filter. + Then + \begin{enumerate} + \item \( M[G] \) is a transitive set; + \item \( \abs{M[G]} = \aleph_0 \); + \item \( M[G] \vDash \mathsf{ZF} \), and if \( M \vDash \mathsf{AC} \) then \( M[G] \vDash \mathsf{AC} \); + \item \( \mathrm{Ord}^M = \mathrm{Ord}^{M[G]} \); + \item \( M \subseteq M[G] \); + \item \( M[G] \) is the smallest countable transitive model of \( \mathsf{ZF} \) such that \( M \subseteq M[G] \) and \( G \) is a set in \( M[G] \). + \end{enumerate} +\end{theorem} +Countability is only needed to show the existence of a generic filter, so parts (i) and (ii)--(vi) of this theorem hold without this assumption. + +\subsection{Canonical names} +We can prove some parts of the generic model theorem by introducing the notion of \emph{canonical names}. +\begin{definition} + Given a forcing poset \( (\mathbb P, \leq, \Bbbone) \) and a set \( x \in M \), we define the \emph{canonical} name of \( x \) by + \[ \check x = \qty{\langle \Bbbone, \check y \rangle \mid y \in x} \] +\end{definition} +The symbol \( \check x \) is pronounced \emph{\( x \)-check}. +\begin{lemma} + If \( M \) is a transitive model of \( \mathsf{ZF} \), \( \mathbb P \in M \), and \( 1 \in G \subseteq \mathbb P \), then + \begin{itemize} + \item for all \( x \in M \), \( \check x \in M^{\mathbb P} \) and \( \check x^G = x \); + \item \( M \subseteq M[G] \); + \item \( M[G] \) is transitive. + \end{itemize} +\end{lemma} +\begin{proof} + \emph{Part (i).} + We show \( \check x \in M^{\mathbb P} \) by induction, using the definition of \( \mathbb P \)-names by transfinite recursion. + Hence + \[ \check x^G = \qty{\check y^G \mid y \in x} = \qty{y \mid y \in x} = x \] + Part (ii) follows directly from part (i). + + \emph{Part (iii).} + Suppose that \( x \in y \) and \( y \in M[G] \). + By definition, \( y = \dot y^G \) for some \( \mathbb P \)-name \( \dot y \). + By construction, any element of \( y \) is of the form \( \dot z^G \), so in particular, \( x = \dot x^G \) for some \( \mathbb P \)-name \( \dot x \in M^{\mathbb P} \). +\end{proof} +\begin{remark} + Even if \( G \notin M \), we can still define a name for \( G \) in \( M \). + From this, it follows that if \( G \notin M \), then \( M[G] \neq M \). +\end{remark} +\begin{proposition} + Let + \[ \dot G = \qty{\langle p, \check p \rangle \mid p \in \mathbb P} \] + Then \( \dot G^G = G \). +\end{proposition} +\begin{proof} + \[ \dot G^G = \qty{\check p^G \mid p \in G} = \qty{p \mid p \in G} = G \] +\end{proof} + +\subsection{Pairing} +We can define unordered and ordered pairs of names, with sensible interpretations. +\begin{definition} + Given \( \mathbb P \)-names \( \dot x, \dot y \), let + \[ \operatorname{up}(\dot x, \dot y) = \qty{\langle \Bbbone, \dot x \rangle, \langle Bbbone, \dot y \rangle} \] + and + \[ \operatorname{op}(\dot x, \dot y) = \operatorname{up}(\operatorname{up}(\dot x, \dot x), \operatorname{up}(\dot x, \dot y)) \] +\end{definition} +\begin{proposition} + For \( \dot x, \dot y \in M^{\mathbb P} \) and \( \Bbbone \in G \subseteq \mathbb P \), + \[ (\operatorname{up}(\dot x, \dot y))^G = \qty{\dot x^G, \dot y^G} \] + and + \[ (\operatorname{op}(\dot x, \dot y))^G = \langle \dot x^G, \dot y^G \rangle \] +\end{proposition} +\begin{lemma} + Suppose \( M \) is a transitive model of \( \mathsf{ZF} \) and \( \mathbb P \in M \) is a forcing poset. + If \( \Bbbone 1 \in G \subseteq \mathbb P \), then \( M[G] \) is a transitive model of extensionality, empty set, foundation, and pairing. +\end{lemma} diff --git a/iii/lc/02_measurable_cardinals.tex b/iii/lc/02_measurable_cardinals.tex index 7113f2f..9f7a183 100644 --- a/iii/lc/02_measurable_cardinals.tex +++ b/iii/lc/02_measurable_cardinals.tex @@ -336,3 +336,90 @@ \subsection{Strongly compact cardinals} This is a \( \kappa \)-complete filter on \( \kappa \). If \( U \) extends \( F \) then \( U \) must be nonprincipal, so by the Keisler--Tarski theorem, \( F \) can be extended to a \( \kappa \)-complete nonprincipal ultrafilter on \( \kappa \) as required. \end{proof} + +\subsection{Reflection} +\begin{definition} + A cardinal \( \kappa \) has the \emph{Keisler extension property}, written \( \mathsf{KEP}(\kappa) \), if there is \( \kappa \in X \supsetneq \mathrm{V}_\kappa \) transitive such that \( \mathrm{V}_\kappa \preceq X \). +\end{definition} +\begin{proposition} + If \( \kappa \) is inaccessible and satisfies the Keisler extension property, there is an inaccessible cardinal \( \lambda < \kappa \). +\end{proposition} +\begin{proof} + Fix \( X \) as in the Keisler extension property. + As \( \kappa \) is inaccessible, \( X \vDash \mathsf{I}(\kappa) \) because \( \kappa \in X \) and inaccessibility is downwards absolute for transitive models. + Also, \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} \), so \( X \vDash \mathsf{ZFC} \) as it is an elementary superstructure. + Therefore, \( X \vDash \mathsf{ZFC} + \mathsf{IC} \), so \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} + \mathsf{IC} \). + So as inaccessibility is absolute between \( \mathrm{V}_\kappa \) and \( \mathrm{V} \), there is an inaccessible \( \lambda < \kappa \). +\end{proof} +The phenomenon that properties of \( X \) occur below \( \kappa \) is called \emph{reflection}. +This argument can be improved in the following sense. +For a given \( \alpha < \kappa \), +\[ X \vDash \exists \lambda > \alpha.\, \mathsf{I}(\lambda) \] +But as \( \alpha \in \mathrm{V}_\kappa \), elementarity gives +\[ \mathrm{V}_\kappa \vDash \exists \lambda > \alpha.\, \mathsf{I}(\lambda) \] +So the set +\[ \qty{\lambda < \kappa \mid \mathsf{I}(\lambda)} \] +is not only nonempty, but cofinal in \( \kappa \). +\begin{corollary} + Let \( \mathsf{A} \) be the axiom + \[ \exists \kappa.\, \mathsf{I}(\kappa) \wedge \mathsf{KEP}(\kappa) \] + Then + \[ \mathsf{ZFC} + \mathsf{IC} <_{\Con} \mathsf{ZFC} + \mathsf{A} \] +\end{corollary} +\begin{proof} + It suffices to show that \( \mathsf{ZFC} + \mathsf{A} \vDash \Con(\mathsf{ZFC} + \mathsf{IC}) \). + We have seen that \( \mathsf{ZFC} + \mathsf{A} \) proves the existence of (at least) two inaccesible cardinals below \( \kappa \), and in particular the larger of the two is a model of \( \mathsf{ZFC} + \mathsf{IC} \). +\end{proof} +\begin{remark} + This is the main technique for proving strict inequalities of consistency strength. + Given two large cardinal properties \( \Phi, \Psi \) with the appropriate amount of absoluteness properties, we show that \( \mathsf{ZFC} + \Phi(\kappa) \) proves that the set + \[ \qty{\lambda < \kappa \mid \Psi(\lambda)} \] + is cofinal in \( \kappa \). + Then \( \mathsf{ZFC} + \Phi\mathsf{C} \vDash \Con(\mathsf{ZFC} + \Psi\mathsf{C}) \). +\end{remark} +\begin{example} + Consider the proof that every inaccessible cardinal has a worldly cardinal below it. + In the construction, we produce a sequence of ordinals \( (\alpha_i)_{i \in \omega} \), and the worldly cardinal is \( \sup \alpha_i \). + But we can set \( \alpha_0 = \lambda + 1 \) for a given worldly cardinal \( \lambda < \kappa \), so this gives a cofinal sequence of worldly cardinals below every given inaccessible. +\end{example} +\begin{theorem} + Every strongly compact cardinal has the Keisler extension property. +\end{theorem} +\begin{proof} + We want to use the method of (elementary) diagrams to produce a model with \( \mathrm{V}_\kappa \) as a substructure. + However, we have no way to control whether such a model is well-founded using standard first-order model-theoretic techniques. + To bypass this issue, we will use infinitary operators. + + Let \( c_x \) be a constant symbol for each \( x \in \mathrm{V}_\kappa \), and let \( L \) be the language with \( \in \) and the \( c_x \). + Let + \[ \mathcal V = (\mathrm{V}_\kappa, \in, \qty{x \mid x \in \mathrm{V}_\kappa}) \] + In first-order logic, \( \mathrm{Th}(X) \) is the elementary diagram of \( \mathrm{V}_\kappa \), so if \( M \vDash \mathrm{Th}(X) \), then \( \mathrm{V}_\kappa \subseteq M \). + Let \( L_{\kappa} \) be the \( \mathcal L_{\kappa\kappa} \)-language with the same symbols. + Consider + \[ \psi \equiv \forall^\omega \vb v.\, \bigvee_{i \in \omega} v_{i+1} \notin v_i \] + This expresses well-foundedness (assuming \( \mathsf{AC} \)). + Writing \( \Phi = \mathrm{Th}_{L_\kappa}(\mathcal V) \) for the \( L_\kappa \)-theory of \( \mathcal V \), we must have \( \psi \in \Phi \) since \( \mathrm{V}_\kappa \) is well-founded. + Thus, if \( M \vDash \Phi \), then \( M \) is a well-founded model containing \( \mathrm{V}_\kappa \). + By taking the Mostowski collapse, we may also assume that any such \( M \) is transitive. + + Extend \( L_\kappa \) to \( L_\kappa^+ \) with one extra constant \( c \), and let + \[ \Phi^+ = \Phi \cup \qty{c \text{ is an ordinal}} \cup \qty{c \neq c_x \mid x \in \mathrm{V}_\kappa} \] + Any model of \( \Phi^+ \) induces a transitive elementary superstructure of \( \mathrm{V}_\kappa \) that contains an ordinal at least \( \kappa \), so by transitivity, \( \kappa \) is in this model. + + We show that \( \Phi^+ \) is satisfiable by showing that it is \( \kappa \)-satisfiable, using the fact that \( \kappa \) is strongly compact. + Let \( \Phi^0 \subseteq \Phi^+ \) be a subset of size less than \( \kappa \). + Then we can interpret \( c \) as some ordinal \( \alpha \) greater than all ordinals \( \beta \) occurring in the sentences \( c \neq c_\beta \) in \( \Phi^+ \). + Then \( \mathcal V \), together with this interpretation of \( c \), is a model of \( \Phi_0 \). +\end{proof} +\begin{corollary} + \[ \mathsf{ZFC} + \mathsf{IC} <_{\Con} \mathsf{ZFC} + \mathsf{SCC} \] +\end{corollary} +The proof above only used languages with at most \( \kappa \)-many symbols. +Let \( \mathsf{WC}(\kappa) \) be the axiom that every \( \mathcal L_{\kappa\kappa} \)-language with at most \( \kappa \)-many symbols satisfies \( \kappa \)-compactness. +Then we have shown that \( \mathsf{WC}(\kappa) \) implies the Keisler extension property. +One can show that +\[ \mathsf{W}(\kappa) \leftrightarrow \mathsf{WC}(\kappa) \] +So the cardinals \( \kappa \) that satisfy \( \mathsf{WC}(\kappa) \) are precisely the weakly compact cardinals. +In particular, +\[ \mathsf{ZFC} + \mathsf{IC} <_{\Con} \mathsf{ZFC} + \mathsf{WCC} \] +Note that in the proof that strongly compact cardinals are measurable, we used a language with \( 2^\kappa \)-many symbols. From 50fc6a55e9c581d99708ecd948c84faee388a708 Mon Sep 17 00:00:00 2001 From: zeramorphic <50671761+zeramorphic@users.noreply.github.com> Date: Fri, 16 Feb 2024 13:00:36 +0000 Subject: [PATCH 2/2] Fix typo --- iii/forcing/03_forcing.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/iii/forcing/03_forcing.tex b/iii/forcing/03_forcing.tex index 112964b..046c28d 100644 --- a/iii/forcing/03_forcing.tex +++ b/iii/forcing/03_forcing.tex @@ -374,7 +374,7 @@ \subsection{Pairing} We can define unordered and ordered pairs of names, with sensible interpretations. \begin{definition} Given \( \mathbb P \)-names \( \dot x, \dot y \), let - \[ \operatorname{up}(\dot x, \dot y) = \qty{\langle \Bbbone, \dot x \rangle, \langle Bbbone, \dot y \rangle} \] + \[ \operatorname{up}(\dot x, \dot y) = \qty{\langle \Bbbone, \dot x \rangle, \langle \Bbbone, \dot y \rangle} \] and \[ \operatorname{op}(\dot x, \dot y) = \operatorname{up}(\operatorname{up}(\dot x, \dot x), \operatorname{up}(\dot x, \dot y)) \] \end{definition}