Lectures 18

iii/forcing/03_forcing.tex

@@ -0,0 +1,91 @@
+\subsection{Introduction}
+The idea behind forcing is to widen a given model of \( \mathsf{ZFC} \) to `add lots of reals'.
+But if we work over \( \mathrm{V} \), we already have added all of the sets, so there is nothing left to add.
+Instead, we will work over countable transitive set models of \( \mathsf{ZFC} \).
+However, this means that we will not immediately get \( \Con(\mathsf{ZF}) \to \Con(\mathsf{ZFC} + \neg\mathsf{CH}) \).
+We will then use the reflection theorem to obtain this result.
+
+If \( M \) is such a countable transitive model, we want to add \( \omega_2^M \)-many reals to \( M \).
+We will try to do this a `minimal way'; for example, we do not want to add any ordinals.
+This gives us much more control over the model that we build.
+
+Recall the argument that the sentence \( \varphi(x) \equiv \exists x.\, x^2 = 2 \) is independent of the axioms of fields: we began with a field in which the sentence failed, namely \( \mathbb Q \), and then extended it in a minimal way to \( \mathbb Q\qty[\sqrt{2}] \).
+The model \( \mathbb Q\qty[\sqrt{2}] \) does not just contain \( \mathbb Q \cup \qty{\sqrt{2}} \), it also contains everything that can be built from \( \mathbb Q \) and \( \sqrt{2} \) using the axioms of fields.
+The field \( \mathbb Q\qty[\sqrt{2}] \) is the minimal field extension of \( \mathbb Q \) satisfying \( \varphi \).
+
+We may encounter some difficulties when adding arbitrary reals to our model.
+Suppose that \( M \) is of the form \( \mathrm{L}_\gamma \), where \( \gamma \) is a countable ordinal.
+Then \( \gamma \) can be coded as a subset \( c \) of \( \omega \), which can be viewed as a real.
+If we added \( c \) to \( M \), we could decode it to form \( \gamma = \mathrm{Ord} \cap M \).
+This would violate the principle of not adding any new ordinals.
+
+Suppose we enumerate all formulas as \( \qty{\varphi_n \mid n \in \omega} \).
+Let \( r = \qty{n \mid M \vDash \varphi_n} \).
+If we added \( r \) to \( M \), we could then build a truth predicate for \( M \).
+This would cause problems due to Tarski.
+
+The main issues we must overcome are the following.
+\begin{enumerate}
+    \item We need a method to choose the \( \omega_2^M \)-many subsets of \( M \) to be added.
+    \item Given these, we need to ensure that the extension satisfies \( \mathsf{ZFC} \).
+    \item We must ensure that \( \omega_1^M \) and \( \omega_2^M \) are still cardinals in the extension.
+\end{enumerate}
+We will build these reals from within \( M \) itself.
+Note that if \( r \) is a real, then each of its finite decimal approximations is already in \( M \).
+The issue is that from within \( M \), we do not know what the real we want to add is.
+So we may not know from within \( M \) which reals we will add.
+Instead, we will add a \emph{generic} real.
+To be generic, we will not specify any particular digits, but its decimal expansion will contain every finite sequence.
+We will call a specification \emph{dense} if any finite approximation can be extended to one satisfying the specification.
+For example, `beginning with a \( 7 \)' is not dense, but `containing the subsequence \( 746 \)' is dense.
+It turns out that a real is generic precisely when it meets every dense specification.
+
+The axiom of choice is not needed in the basic machinery of forcing, so we will work primarily over \( \mathsf{ZF} \) and state explicitly where choice is used.
+
+\subsection{Forcing posets}
+\begin{definition}
+    A \emph{preorder} is a pair \( (\mathbb P, \leq) \) such that
+    \begin{itemize}
+        \item \( \mathbb P \) is nonempty;
+        \item \( \leq \) is a binary relation on \( \mathbb P \);
+        \item \( \leq \) is transitive, so \( p \leq q \) and \( q \leq r \) implies \( p \leq r \);
+        \item \( \leq \) is reflexive, so \( p \leq p \).
+    \end{itemize}
+    A preorder is called a \emph{partial order} if \( \leq \) is antisymmetric, so \( p \leq q \) and \( q \leq p \) implies \( p = q \).
+\end{definition}
+\begin{definition}
+    A \emph{forcing poset} is a triple \( (\mathbb P, \leq_{\mathbb P}, \Bbbone_{\mathbb P}) \), where \( (\mathbb P, \leq_{\mathbb P}) \) is a preorder and \( \Bbbone_{\mathbb P} \) is a maximal element.
+    Elements of \( \mathbb P \) are called \emph{conditions}, and we say \( q \) is \emph{stronger} than \( p \) or an \emph{extension} of \( p \) if \( q \leq p \).
+    We say that \( p, q \) are \emph{compatible}, written \( p \|_{\mathbb P} q \), if there exists \( r \) such that \( r \leq_{\mathbb P} p, q \).
+    Otherwise, we say they are \emph{incompatible}, written \( p \perp q \).
+\end{definition}
+\begin{remark}
+    In some texts, the partial order is reversed.
+    This is called \emph{Jerusalem notation}.
+\end{remark}
+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 \( r \perp p \).
+\end{definition}
+\begin{proposition}
+    If \( (\mathbb P, \leq) \) is a separative preorder, it is a partial order.
+\end{proposition}
+\begin{proposition}
+    Suppose that \( (\mathbb P, \leq) \) is a preorder.
+    Define \( p \sim q \) by
+    \[ p \sim q \leftrightarrow \forall r \in P.\, (r \| p \leftrightarrow r \| q) \]
+    Then there is a separative preorder on \( \faktor{\mathbb P}{\sim} \) such that
+    \[ [p] \perp [q] \leftrightarrow p \perp q \]
+    and if \( \mathbb P \) has a maximal element, so does \( \faktor{\mathbb P}{\sim} \).
+\end{proposition}
+\begin{example}
+    For sets \( I, J \), we let \( F_n(I, J) \) denote the set of all finite partial functions from \( I \) to \( J \).
+    \[ F_n(I, J) = \qty{p \mid \abs{p} < \omega \wedge p \text{ is a function} \wedge \dom p \subseteq I \wedge \operatorname{ran} p \subseteq J} \]
+    We let \( \leq \) be the reverse inclusion on \( F_n(I, J) \), so \( q \leq p \) if and only if \( q \supseteq p \).
+    The maximal element \( \Bbbone \) is the empty set.
+    Then \( (F_n(I, J), \leq, \varnothing) \) is a forcing poset, and moreover, the preorder is separative.
+\end{example}
+\begin{remark}
+    When \( \alpha \) is an ordinal, the forcing poset \( F_n(\alpha \times \omega, 2) \) is often written \( \operatorname{Add}(\omega, \alpha) \), denoting the idea that we are adding \( \alpha \)-many subsets of \( \omega \).
+\end{remark}

iii/forcing/main.tex

@@ -14,5 +14,7 @@ \section{Set theoretic preliminaries}
 \input{01_set_theory.tex}
 \section{Constructibility}
 \input{02_constructibility.tex}
+\section{Forcing}
+\input{03_forcing.tex}
 
 \end{document}

iii/lc/02_measurable_cardinals.tex

@@ -141,26 +141,28 @@ \subsection{Measurable cardinals}
         \item A cardinal \( \kappa \) is called \emph{Ulam measurable} if there is an \( \aleph_1 \)-complete nonprincipal ultrafilter on \( \kappa \).
         With this definition, the least Ulam measurable cardinal is measurable.
         So the existence of an Ulam measurable cardinal is equivalent to the existence of a measurable cardinal.
+        \item The theories \( \mathsf{ZFC} + \mathsf{MC} \) and \( \mathsf{ZFC} + \mathsf{RVMC} \) are equiconsistent.
+        This can be shown analogously to inaccessible and weakly inaccessible cardinals, this time using a variant of G\"odel's constructible universe.
     \end{enumerate}
 \end{remark}
 \begin{theorem}
     Every measurable cardinal is inaccessible.
 \end{theorem}
 \begin{proof}
     We have already shown regularity in the real-valued measurable cardinal case.
-    Let \( \kappa \) be measurable with ultrafilter \( \mathcal U \).
+    Let \( \kappa \) be measurable with ultrafilter \( U \).
     Suppose it is not a strong limit, so there is \( \lambda < \kappa \) such that \( 2^\lambda \geq \kappa \).
     Then there is an injection \( f : \kappa \to B_\lambda \), where \( B_\lambda \) is the set of functions \( \lambda \to 2 \).
     Fix some \( \alpha < \lambda \), then for each \( \gamma < \kappa \), either
     \[ f(\gamma)(\alpha) = 0 \text{ or } f(\gamma)(\alpha) = 1 \]
     Let
     \[ A_0^\alpha = \qty{\gamma \mid f(\gamma)(\alpha) = 0};\quad A_1^\alpha = \qty{\gamma \mid f(\gamma)(\alpha) = 1} \]
     These two sets are disjoint and have union \( \kappa \).
-    So there is exactly one number \( b \in \qty{0,1} \) such that \( A^\alpha_b \in \mathcal U \).
+    So there is exactly one number \( b \in \qty{0,1} \) such that \( A^\alpha_b \in U \).
     Define \( c \in B_\lambda \) by \( c(\alpha) = b \).
     Then
-    \[ X_\alpha = A^\alpha_{c(\alpha)} \in \mathcal U \]
-    This is a collection of \( \lambda \)-many sets that are all in \( \mathcal U \), so by \( \kappa \)-completeness, their intersection \( \bigcap_{\alpha < \lambda} X_\alpha \) also lies in \( \mathcal U \).
+    \[ X_\alpha = A^\alpha_{c(\alpha)} \in U \]
+    This is a collection of \( \lambda \)-many sets that are all in \( U \), so by \( \kappa \)-completeness, their intersection \( \bigcap_{\alpha < \lambda} X_\alpha \) also lies in \( U \).
     Suppose \( \gamma \in \bigcap_{\alpha < \lambda} X_\alpha \), so for all \( \alpha < \lambda \), we have \( \gamma \in A^\alpha_{c(\alpha)} \).
     Equivalently, for all \( \alpha < \lambda \), we have \( f(\gamma)(\alpha) = c(\alpha) \).
     So \( \gamma \) lies in this intersection if and only if \( f(\gamma) \) is precisely the function \( c \).
@@ -225,3 +227,112 @@ \subsection{Weakly compact cardinals}
 \begin{theorem}
     Every measurable cardinal is weakly compact.
 \end{theorem}
+\begin{proof}
+    Let \( f : [\kappa]^2 \to 2 \) be a colouring of a measurable cardinal \( \kappa \).
+    Let
+    \[ X_0^\alpha = \qty{\beta \mid f(\qty{\alpha, \beta}) = 0};\quad X_1^\alpha = \qty{\beta \mid f(\qty{\alpha, \beta}) = 1} \]
+    For a given \( \alpha \), these are disjoint, and \( X_0^\alpha \cup X_1^\alpha = \kappa \setminus \qty{\alpha} \), so precisely one of them lies in the ultrafilter \( U \).
+    Define \( c : \kappa \to 2 \) be such that \( X_{c(\alpha)}^\alpha \in U \).
+    Now, let
+    \[ X_0 = \qty{\alpha \mid c(\alpha) = 0};\quad X_1 = \qty{\alpha \mid c(\alpha) = 1} \]
+    Precisely one of these two sets lies in \( U \).
+
+    We claim that if \( X_i \in U \), then there is a monochromatic set \( H \) for colour \( i \) with \( \abs{H} = \kappa \).
+    Without loss of generality, we may assume \( i = 0 \).
+    Define
+    \[ Z_\alpha = \begin{cases}
+        X^\alpha_0 \mid c(\alpha) = 0 & \text{if } c(\alpha) = 0 \\
+        \kappa & \text{if } c(\alpha) = 1
+    \end{cases} \]
+    Each of the \( Z_\alpha \) lie in the ultrafilter \( U \).
+    As we may assume \( U \) is normal, the diagonal intersection of the \( Z_\alpha \) also lies in \( U \).
+    So we can define
+    \[ H = X_0 \cap \operatorname*{\scalerel*{\mupDelta}{\textstyle\sum}}_{\alpha \leq \kappa} Z_\alpha \in U \]
+    and \( \abs{H} = \kappa \).
+    Let \( \gamma < \delta \) with \( \gamma, \delta \in H \).
+    Then \( \gamma, \delta \in X_0 \), so \( c(\gamma) = 0 = c(\delta) \).
+    Hence \( Z_\gamma = X^\gamma_0 \) and \( Z_\delta = X^\delta_0 \).
+    In particular,
+    \[ \delta \in \operatorname*{\scalerel*{\mupDelta}{\textstyle\sum}}_{\alpha \leq \kappa} Z_\alpha \subseteq \bigcap_{\xi < \delta} Z_\xi \subseteq Z_\gamma = X^\gamma_0 \]
+    Hence \( f(\qty{\gamma,\delta}) = 0 \).
+\end{proof}
+% TODO: Move this rk
+The large cardinal axioms discussed so far fall into a linear hierarchy of consistency strength.
+This is known as the \emph{linearity phenomenon}.
+
+\subsection{Strongly compact cardinals}
+The compactness theorem for first-order logic says that for any first-order language \( L_S \) and set of axioms \( \Phi \subseteq L_S \),
+\[ \Phi \text{ is satisfiable} \leftrightarrow \qty(\forall \Phi_0 \subseteq \Phi.\, \abs{\Phi_0} < \aleph_0 \to \Phi_0 \text{ is satisfiable}) \]
+This result cannot work for languages with infinitary conjunctions and disjunctions.
+Indeed, if we write
+\[ \varphi_F \equiv \bigvee_{i \in \mathbb N} \varphi_{=n};\quad \varphi_{=n} \equiv \text{there are precisely } n \text{ elements};\quad \varphi_{\geq n} \equiv \text{there are at least } n \text{ elements} \]
+then
+\[ \qty{\varphi_{\geq n} \mid n \in \mathbb N} \cup \qty{\varphi_F} \]
+is finitely satisfiable but not satisfiable.
+\begin{definition}
+    An \emph{\( \mathcal L_{\kappa\kappa} \)-language} is defined by
+    \begin{itemize}
+        \item a set of variables;
+        \item a set \( S \) of function, relation, and constant symbols of finite arity;
+        \item the logical symbols \( \wedge, \vee, \neg, \exists, \forall \); and
+        \item the infinitary logical symbols \( \bigwedge_{\alpha < \lambda}, \bigvee_{\alpha < \lambda}, \exists^\lambda, \forall^\lambda \) for \( \lambda < \kappa \).
+    \end{itemize}
+\end{definition}
+We define the new syntactic rules as follows.
+If \( \varphi_\alpha \) are \( L_S \)-formulas for \( \alpha < \lambda \), then so are \( \bigwedge_{\alpha < \lambda} \varphi_\alpha \) and \( \bigvee_{\alpha < \lambda} \varphi_\alpha \).
+If \( \vb v \) is a sequence of variables of length \( \lambda \) and \( \varphi \) is an \( L_S \)-formula, then \( \exists^\lambda \vb v.\, \varphi \) and \( \forall^\lambda \vb v.\, \varphi \) are \( L_S \)-formulas.
+
+We say that \( M \) is a model of \( \bigvee_{\alpha < \lambda} \varphi_\alpha \) if \( M \vDash \varphi_\alpha \) for all \( \alpha < \lambda \).
+Similarly, \( M \) models \( \exists^\lambda \vb v.\, \varphi \) if there is a function \( a : \lambda \to M \) such that
+\[ M\qty[\frac{a(0)a(1) \dots a(\xi)\dots}{v_0 v_1 \dots v_\xi \dots}] \vDash \varphi \]
+\begin{definition}
+    An \( \mathcal L_{\kappa\kappa} \)-language \( L_S \) \emph{satisfies compactness} if for all \( \Phi \subseteq L_S \),
+    \[ \Phi \text{ is satisfiable} \leftrightarrow \qty(\forall \Phi_0 \subseteq \Phi.\, \abs{\Phi_0} < \kappa \to \Phi_0 \text{ is satisfiable}) \]
+\end{definition}
+Note that if \( \kappa = \omega \), we recover the standard notion of a first-order language, so all \( \mathcal L_{\omega\omega} \)-languages satisfy compactness.
+\begin{definition}
+    An uncountable cardinal \( \kappa \) is called \emph{strongly compact}, denoted \( \mathsf{SC}(\kappa) \), if every \( \mathcal L_{\kappa\kappa} \)-language satisfies compactness.
+\end{definition}
+\begin{theorem}[Keisler--Tarski theorem]
+    Suppose \( \kappa \) is a strongly compact cardinal.
+    Then every \( \kappa \)-complete filter on \( \kappa \) can be extended to a \( \kappa \)-complete ultrafilter.
+\end{theorem}
+\begin{proof}
+    We define a language \( L \) by creating a constant symbol \( c_A \) for each \( A \subseteq \kappa \), giving \( 2^\kappa \)-many symbols.
+    Now let \( L^\star \) be \( L \) with an extra constant symbol \( c \).
+    Let
+    \[ M = (\mathcal P(\kappa), \in, \qty{A \mid A \subseteq \kappa}) \]
+    so \( c_A \) is interpreted by \( A \).
+    Let \( \Phi = \operatorname{Th}_L(M) \) be the \( L \)-theory of \( M \).
+    In particular,
+    \[ M \vDash \forall x.\, x \in c_A \to x \text{ is an ordinal} \]
+    and
+    \[ M \vDash \forall x.\, x \text{ is an ordinal} \to x \in c_A \vee x \in c_{\kappa \setminus A} \]
+    Now let
+    \[ \Phi^\star = \Phi \cup \qty{c \in c_A \mid A \in F} \]
+    This is a subset of \( L^\star \).
+    We show that \( \Phi^\star \) is \( \kappa \)-satisfiable.
+    If \( (A_\alpha)_{\alpha < \lambda} \) are subsets of \( \kappa \) such that \( c \in c_{A_\alpha} \) occurs in a \( \kappa \)-small subset of \( \Phi^\star \), then any element \( \eta \in \bigcap_{\alpha < \lambda} A_\alpha \) can be chosen as the interpretation of \( c \).
+    As \( F \) is \( \kappa \)-complete, this intersection lies in \( F \) and so is nonempty as required.
+
+    Hence, by strong compactness of \( \kappa \), the theory \( \Phi^\star \) is satisfiable.
+    Let \( M \) be a model of \( \Phi^\star \).
+    Define
+    \[ U = \qty{A \mid A \vDash c \in c_A} \]
+    We claim that this is a \( \kappa \)-complete ultrafilter extending \( F \).
+    The fact that \( U \) extends \( F \) holds by construction of \( \Phi^\star \).
+    It is an ultrafilter because \( M \) believes that \( c \in c_A \) or \( c \in c_{\kappa \setminus A} \).
+    It is \( \kappa \)-complete because if \( \qty{A_\alpha \mid \alpha < \lambda} \subseteq U \), let \( A = \bigcap_{\alpha < \lambda} A_\alpha \), then
+    \[ M \vDash \forall x.\, \qty(x \in c_A \leftrightarrow \bigwedge_{\alpha < \lambda} x \in c_{A_\alpha}) \]
+    As this holds in particular for \( c \), we obtain \( A \in U \).
+\end{proof}
+\begin{corollary}
+    Every strongly compact cardinal is measurable.
+\end{corollary}
+\begin{proof}
+    Let
+    \[ F = \qty{A \subseteq \kappa \mid \abs{\kappa\setminus A} < \kappa} \]
+    In the case \( \kappa = \omega \), this is known as the Fr\'echet filter.
+    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}