Minor edits

Signed-off-by: zeramorphic <[email protected]>

iii/forcing/03_forcing.tex

@@ -642,7 +642,7 @@ \subsection{The forcing theorem}
 \begin{corollary}
     Suppose that \( M \) is a countable transitive model of \( \mathsf{ZF} \), \( \mathbb P \in M \) is a forcing poset, and \( \varphi(u) \) is a formula.
     Then for any name \( \dot x \in M^{\mathbb P} \),
-    \[ (p \Vdash \varphi(\dot x))^m \leftrightarrow \text{for any \( \mathbb P \)-generic filter \( G \) with \( p \in G \), } M[G] \Vdash \varphi(\dot x)^G \]
+    \[ (p \Vdash \varphi(\dot x))^M \leftrightarrow \text{for any \( \mathbb P \)-generic filter \( G \) with \( p \in G \), } M[G] \Vdash \varphi(\dot x)^G \]
 \end{corollary}
 The only reason we need countability is so that every condition is contained in a generic filter.
 \begin{proof}
@@ -883,7 +883,7 @@ \subsection{\texorpdfstring{\( \mathsf{ZF} \)}{ZF} in forcing extensions}
     So in \( M[G] \),
     \[ \dot f^G = \qty{\langle \alpha, \dot x_\alpha^G \rangle \mid \alpha < \delta} \]
     Hence \( \dot f^G \) is a function with domain \( \delta \), and \( a \subseteq \ran \dot f^G \).
-    We can now define a well-order \( \triangleleft \) on \( a \) by defining that \( x \triangleleft y \) if and only if
+    We can now define a well-order \( \prec \) on \( a \) by defining that \( x \prec y \) if and only if
     \[ \min\qty{\alpha < \delta \mid \dot f^G(\alpha) = x} < \min\qty{\alpha < \delta \mid \dot f^G(\alpha) = y} \]
 \end{proof}
 \begin{remark}

iii/forcing/04_forcing_and_independence.tex

@@ -55,7 +55,7 @@ \subsection{Cohen forcing}
 Since \( G \) is a filter, if \( p, q \in G \) then \( p \cap q \in G \).
 Hence, if \( p, q \in G \), then \( p, q \) agree on the intersection of their domains.
 Let \( f_G = \bigcup G \).
-Then \( f_G \) is a function domain contained in \( I \) and range contained in \( J \).
+Then \( f_G \) is a function with domain contained in \( I \) and range contained in \( J \).
 Note that this function has name
 \[ \dot f = \qty{\langle p, \operatorname{op}(\check \imath, \check \jmath) \rangle \mid p \in \mathbb P, \langle i, j \rangle \in p} \]
 Since \( D_i, R_j \) are dense, we obtain \( G \cap D_i \neq \varnothing \), so we must have \( i \in \dom f_G \).
@@ -151,9 +151,9 @@ \subsection{Preservation of cardinals}
     \end{enumerate}
 \end{proof}
 \begin{lemma}[the approximation lemma]
-    Let \( A, B, \mathbb P \in M \), and suppose that \( (\mathbb P \text{ has the countable chain condition})^M \).
+    Let \( A, B, \mathbb P \in M \), and suppose that \( (\mathbb P \) has the countable chain condition\( )^M \).
     Let \( G \) be \( \mathbb P \)-generic over \( M \).
-    Then for any function \( f \in M[G] \) with \( f : A \to B \), there is a function \( F \in M \) with \( F : A \to \mathcal P^M(B) \) such that for all \( a \in A \), we have \( f(a) \in F(a) \) and \( \abs{F(a)} \leq \aleph_0 \).
+    Then for any function \( f \in M[G] \) with \( f : A \to B \), there is a function \( F \in M \) with \( F : A \to \mathcal P^M(B) \) such that for all \( a \in A \), we have \( f(a) \in F(a) \) and \( (\abs{F(a)} \leq \aleph_0)^M \).
 \end{lemma}
 This proof requires that \( M \) is countable.
 Note that the relativisation of the countable chain condition to \( M \) ensures that the hypothesis is non-vacuous, as any forcing poset in \( M \) is externally countable.
@@ -183,7 +183,7 @@ \subsection{Preservation of cardinals}
     \[ r \Vdash \dot f : \check A \to \check B \text{ is a function} \wedge \dot f(\check a) = \check b \wedge \dot f(\check a) = \check c \wedge \check b \neq \check c \]
     Let \( H \) be a generic filter with \( r \in H \); this exists by countability of \( M \).
     Then \( r \leq p \) and
-    \[ M[G] \vDash f : A \to B \text{ is a function} \wedge f(a) = b \wedge f(a) = c \wedge b \neq c \]
+    \[ M[H] \vDash f : A \to B \text{ is a function} \wedge f(a) = b \wedge f(a) = c \wedge b \neq c \]
     But \( M[H] \vDash \mathsf{ZFC} \), giving a contradiction.
 \end{proof}
 \begin{theorem}
@@ -199,8 +199,7 @@ \subsection{Preservation of cardinals}
     This is a union of countable sets indexed by \( \alpha < \beta \).
     So \( X \subseteq \beta \) and is a subset of less than \( \beta \)-many countable sets.
     Hence \( X \neq \beta \) as \( \beta \) is a regular cardinal in \( M \).
-    But \( f \) was cofinal, so \( \beta = \bigcup_{\alpha < \gamma} f(\gamma) \subseteq X \), giving a contradiction.
-    % TODO: Why this last \subseteq?
+    But \( f \) was cofinal, so \( \beta = \bigcup_{\gamma < \alpha} f(\gamma) \subseteq X \), giving a contradiction.
 \end{proof}
 
 \subsection{The failure of the continuum hypothesis}
@@ -392,15 +391,15 @@ \subsection{Larger chain conditions}
     Then
     \begin{enumerate}
         \item \( \mathbb P \) preserves cofinalities above \( \kappa \) if and only if for all \( \mathbb P \)-generic filters \( G \) and all limit ordinals \( \beta \) with \( \kappa \leq \beta \in \mathrm{Ord} \cap M \), we have \( (\beta \text{ is regular})^M \to (\beta \text{ is regular})^{M[G]} \);
-        \item \( \mathbb P \) preserves cofinalities above \( \kappa \) if and only if \( \mathbb P \) preserves cardinals at least \( \kappa \).
+        \item If \( \mathbb P \) preserves cofinalities above \( \kappa \), then \( \mathbb P \) preserves cardinals above \( \kappa \).
     \end{enumerate}
 \end{lemma}
 \begin{lemma}
-    Let \( A, B, \mathbb P \in M \), let \( (\kappa \text{ is regular})^M \), let \( (\mathbb P \text{ has the } \kappa \text{-chain condition})^M \), and let \( G \) be a \( \mathbb P \)-generic filter over \( M \).
+    Let \( A, B, \mathbb P \in M \), let \( (\kappa \text{ is regular})^M \), let \( (\mathbb P \) has the \( \kappa \)-chain condition\( )^M \), and let \( G \) be a \( \mathbb P \)-generic filter over \( M \).
     Then for any \( f : A \to B \) in \( M[G] \), there is \( F : A \to \mathcal P(B) \) in \( M \) such that for all \( a \in A \), we have \( f(a) \in F(a) \) and \( (\abs{F(a)} \leq \kappa)^M \).
 \end{lemma}
 \begin{theorem}
-    Let \( \mathbb P \in M \) be a forcing poset, let \( (\kappa \text{ is regular})^M \), let \( (\mathbb P \text{ has the } \kappa \text{-chain condition})^M \).
+    Let \( \mathbb P \in M \) be a forcing poset, let \( (\kappa \text{ is regular})^M \), let \( (\mathbb P \) has the \( \kappa \)-chain condition\( )^M \).
     Then \( \mathbb P \) preserves cofinalities above \( \kappa \), and hence cardinals at least \( \kappa \).
 \end{theorem}
 On the example sheet, we show that for any infinite cardinal \( \kappa \), \( \Fn_\kappa(I, J) \) has the \( (\abs{J}^{<\kappa})^+ \)-chain condition.

iii/lc/02_measurable_cardinals.tex

@@ -318,7 +318,7 @@ \subsection{Strongly compact cardinals}
     Hence, by strong compactness of \( \kappa \), the theory \( \Phi^\star \) is satisfiable.
     Let \( M \) be a model of \( \Phi^\star \).
     Define
-    \[ U = \qty{A \mid A \vDash c \in c_A} \]
+    \[ U = \qty{A \mid M \vDash c \in c_A} \]
     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 \( c \in c_{\kappa \setminus A} \).

iii/mtncl/02_quantifier_elimination.tex

@@ -14,7 +14,7 @@ \subsection{Skolem functions}
 \begin{proposition}
     Let \( \mathcal T \) be an \( \mathcal L \)-theory, and let \( \mathcal F \) be a collection of \( \mathcal L \)-formulae including all atomic formulae and closed under Boolean operations.
     Suppose that for every formula \( \psi(\vb x, y) \in \mathcal F \), there exists \( \varphi(\vb x) \in \mathcal F \) with
-    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \psi(\vb x, y) \Leftrightarrow \varphi(\vb x)) \]
+    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \psi(\vb x, y) \leftrightarrow \varphi(\vb x)) \]
     Then, every \( \mathcal L \)-formula is equivalent to one in \( \mathcal F \) with the same free variables modulo \( \mathcal T \) (that is, \( \mathcal T \) proves they are equivalent).
 \end{proposition}
 \begin{proof}
@@ -23,7 +23,7 @@ \subsection{Skolem functions}
     Consider the formula \( \exists y,\, \psi(\vb x, y) \).
     By the inductive hypothesis, \( \psi(\vb x, y) \) is \( \mathcal T \)-equivalent to \( \psi'(\vb x, y) \in \mathcal F \).
     Then, there is some \( \varphi(\vb x) \in \mathcal F \) such that
-    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \psi'(\vb x, y) \Leftrightarrow \varphi(\vb x)) \]
+    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \psi'(\vb x, y) \leftrightarrow \varphi(\vb x)) \]
     Thus the formula \( \exists y,\, \psi(\vb x, y) \) in question is \( \mathcal T \)-equivalent to \( \varphi(\vb x) \in \mathcal F \).
 \end{proof}
 \begin{proposition}
@@ -41,7 +41,7 @@ \subsection{Skolem functions}
     \emph{Part (i).}
     Clearly, \( \varphi(\vb x, t(\vb x)) \to \exists y.\, \varphi(\vb x, y) \) in any model.
     So having Skolem functions means that
-    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \varphi(\vb x, y) \Leftrightarrow \varphi(\vb x, t(\vb x))) \]
+    \[ \mathcal T \vdash \forall \vb x.\, (\exists y.\, \varphi(\vb x, y) \leftrightarrow \varphi(\vb x, t(\vb x))) \]
     completing the proof by the previous proposition.
 
     \emph{Part (ii).}
@@ -124,7 +124,7 @@ \subsection{Elimination sets}
 \begin{definition}
     Let \( \mathcal T \) be an \( \mathcal L \)-theory.
     A set \( F \) of \( \mathcal L \)-formulae is an \emph{elimination set} for \( \mathcal T \) if, for every \( \mathcal L \)-formula \( \varphi \), there is a Boolean combination \( \varphi^\star \) of formulae in \( F \) such that
-    \[ \mathcal T \vdash \varphi \Leftrightarrow \varphi^\star \]
+    \[ \mathcal T \vdash \varphi \leftrightarrow \varphi^\star \]
     A theory \( \mathcal T \) has \emph{quantifier elimination} if the family of quantifier-free formulae forms an elimination set for \( \mathcal T \).
 \end{definition}
 Note that a theory having quantifier elimination depends on its underlying language.
@@ -155,7 +155,7 @@ \subsection{Elimination sets}
     Suppose that, for every \( \mathcal L \)-formula of the form
     \[ \theta(\vb x) = \exists y.\, \bigwedge_{i < n} \varphi_i(\vb x, y);\quad \varphi_i \in F \cup \neg F \]
     there exists a Boolean combination \( \theta^\star(\vb x) \) of formulae in \( F \) such that
-    \[ \mathcal T \vdash \forall \vb x.\, (\theta(\vb x) \Leftrightarrow \theta^\star(\vb x)) \]
+    \[ \mathcal T \vdash \forall \vb x.\, (\theta(\vb x) \leftrightarrow \theta^\star(\vb x)) \]
     Then \( F \) is an elimination set for \( \mathcal T \).
 \end{proposition}
 The proof is similar to a previous proposition.
@@ -264,13 +264,14 @@ \subsection{Amalgamation}
     Let \( \varphi(\vb x) \) be an existential formula.
     We will call a pair \( (\mathcal M, \vb m) \) a \emph{witnessing pair} if \( \mathcal M \) is existentially closed in \( \mathbb K \) and \( \mathcal M \vDash \varphi(\vb m) \).
     For each such pair, let
-    \[ \theta_{(\mathcal M, \vb m)} = \bigwedge \qty{\psi \text{ a literal} \mid \mathcal M \vDash \psi(\vb m)} \]
+    \[ \theta_{(\mathcal M, \vb m)}(\vb x) = \bigwedge \qty{\psi(\vb x) \text{ a literal} \mid \mathcal M \vDash \psi(\vb m)} \]
     where the \emph{literals} are the atomic formulae and their negations.
     Let
     \[ \chi(\vb x) = \bigvee_{(\mathcal M, \vb m)} \theta_{(\mathcal M, \vb m)}(\vb x) \]
-    Note that the \( \theta \) and \( \chi \) are not first-order formulae, but any conjunction or disjunction over an infinite set is logically equivalent to a finite conjunction or disjunction over a finite subset; this can be seen by applying the compactness theorem.
+    % Note that the \( \theta \) and \( \chi \) are not first-order formulae, but any conjunction or disjunction over an infinite set is logically equivalent to a finite conjunction or disjunction over a finite subset; this can be seen by applying the compactness theorem.
     It suffices to show that if \( \mathcal N \) is existentially closed in \( \mathbb K \) then
     \[ (\mathcal N \vDash \varphi(\vb n)) \iff (\mathcal N \vDash \chi(\vb n)) \]
+    Then we can use the compactness theorem twice to reduce \( \chi \) to a first-order finitary formula as required.
     If \( \vb n \in \mathcal N \) is such that \( \mathcal N \vDash \varphi(\vb n) \), then \( (\mathcal N, \vb n) \) is a witnessing pair, and thus \( \mathcal N \vDash \chi(\vb n) \) by construction.
     For the converse, if \( \mathcal N \vDash \chi(\vb n) \), there is a witnessing pair \( (\mathcal M, \vb m) \) such that \( \mathcal N \vDash \theta_{(\mathcal M, \vb m)}(\vb n) \).
     Hence, for each literal \( \psi(\vb x) \),
@@ -391,7 +392,8 @@ \subsection{Characterisations of quantifier elimination}
     We can iteratively convert existential quantifiers to universal quantifiers, noting that (iv) allows us to convert a sequence of existentials to a sequence of universals simultaneously.
 
     \emph{(v) implies (i).}
-    We use induction on the structure of \( \mathcal L \)-formulae, noting that universal formulae are preserved under extensions, and that every formula and its negation can be represented as a universal formula.
+    Note that universal formulae are preserved under extensions, and every formula and its negation can be represented as a universal formula.
+    This directly gives the result.
 \end{proof}
 Let \( \mathcal M, \mathcal N \) be \( \mathcal L \)-structures.
 If \( \mathcal M, \mathcal N \) satisfy the same quantifier-free sentences, we write \( \mathcal M \equiv_0 \mathcal N \).
@@ -410,7 +412,7 @@ \subsection{Characterisations of quantifier elimination}
     (i) implies (ii) is clear.
 
     \emph{(ii) implies (iii).}
-    It suffices to show that \( (\mathcal A, \vb a) \to_1 (\mathcal B, e(\vb a)) \) by the existential amalgamation theorem.
+    It suffices to show that \( (\mathcal A, S) \to_1 (\mathcal B, e(S)) \) by the existential amalgamation theorem.
     Since a sentence in \( \mathcal L_S \) is finite, it can only mention finitely many of the new constants in \( S \), so it is enough to check that \( (\mathcal A, \vb a) \to_1 (\mathcal B, e(\vb a)) \) for all tuples \( \vb a \) obtainable from \( S \).
     Now, if \( \vb a \) is such a tuple and \( e : \langle S \rangle_{\mathcal A} \rightarrowtail \mathcal B \) is an embedding, then \( (\mathcal A, \vb a) \equiv_0 (\mathcal B, e(\vb a)) \), giving the required result by (ii).
 

iii/mtncl/03_ultraproducts.tex

@@ -61,6 +61,7 @@ \subsection{Filters}
         \item \( \mathcal F \) is \emph{downward directed}: if \( x, y \in \mathcal F \), then \( x \wedge y \in \mathcal F \).
     \end{enumerate}
 \end{definition}
+A filter on \( X \) may be thought of as a collection of `large' subsets of \( X \): subsets that are so large that the intersection of any two large subsets is also large.
 For property (ii), we might also say that \( \mathcal F \) is a \emph{terminal segment} of \( X \).
 % large things get "stuck in the filter"; filters measure largeness
 \begin{example}

iii/mtncl/04_types.tex

@@ -14,7 +14,7 @@ \subsection{Definitions}
 \begin{definition}
     Let \( \mathcal T \) be a theory and \( n \in \mathbb N \).
     We obtain an equivalence relation \( \sim \) on the set \( \mathcal L(\vb x) \) of \( \mathcal L \)-formulae with free variables \( \vb x \), where \( \vb x \) is a tuple of length \( n \), by setting
-    \[ \varphi(\vb x) \sim \psi(\vb x) \iff \mathcal T \vdash \forall \vb x.\, (\varphi(\vb x) \Leftrightarrow \psi(\vb x)) \]
+    \[ \varphi(\vb x) \sim \psi(\vb x) \iff \mathcal T \vdash \forall \vb x.\, (\varphi(\vb x) \leftrightarrow \psi(\vb x)) \]
     The quotient \( \mathcal B_n(\mathcal T) = \faktor{\mathcal L(\vb x)}{\sim} \) becomes a Boolean algebra by setting \( [\varphi] \bowtie [\psi] = [\varphi \bowtie \psi] \) for any logical connective \( \bowtie \), called the \emph{Lindenbaum--Tarski algebra} of \( \mathcal T \) on variables \( \vb x \).
 \end{definition}
 \begin{definition}

iii/mtncl/05_indiscernibles.tex

@@ -4,7 +4,7 @@ \subsection{???}
 \begin{definition}
 	Let \( \mathcal M \) be an \( \mathcal L \)-structure, let \( \Phi \) be a set of \( \mathcal L \)-formulae, and let \( \eta \) be a strict chain of elements of \( \mathcal M \).
 	We say that \( \eta \) is \emph{\( \Phi \)-indiscernible} in \( \mathcal M \) if
-	\[ \mathcal M \vDash \phi(\vb a) \Leftrightarrow \phi(\vb b) \]
+	\[ \mathcal M \vDash \phi(\vb a) \leftrightarrow \phi(\vb b) \]
 	for all \( \vb a, \vb b \in [\eta]^k \) of the correct length and \( \varphi \in \Phi \).
 	We simply say that \( \eta \) is a sequence of indiscernibles if the above holds where \( \Phi \) is the set of every \( \mathcal L \)-formula.
 \end{definition}
@@ -91,7 +91,7 @@ \subsection{???}
 	We want to build a theory extending \( \operatorname{Th}(\mathcal M, \eta) \), whose models include an indiscernible copy of \( \omega \).
 	First, expand \( \mathcal L \) to add \( \omega \)-many constants \( C = \qty{c_i \mid i \in \omega} \), and we build an \( \mathcal L_C \)-theory \( \mathcal T \) with the following axioms:
 	\begin{enumerate}
-		\item \( \varphi(\vb a) \Leftrightarrow \varphi(\vb b) \), for each \( \mathcal L \)-formula \( \varphi(\vb x) \) and ordered tuples \( \vb a, \vb b \in [C]^{\abs{\vb x}} \);
+		\item \( \varphi(\vb a) \leftrightarrow \varphi(\vb b) \), for each \( \mathcal L \)-formula \( \varphi(\vb x) \) and ordered tuples \( \vb a, \vb b \in [C]^{\abs{\vb x}} \);
 		\item \( \varphi(c_0, \dots, c_{k-1}) \), for each formula \( \varphi(x_0, \dots, x_{k-1}) \) in \( \operatorname{Th}(\mathcal M, \eta) \).
 	\end{enumerate}
 	We will show that this theory has a model by compactness.