Typo fixes

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

iii/forcing/02_constructibility.tex

@@ -121,7 +121,7 @@ \subsection{G\"odel functions}
 \begin{remark}
     \begin{enumerate}
         \item The reversed order of the free variables is done purely for technical reasons.
-        \item \( \mathcal F_2 \) will correspond to disjunction for \( \Delta_0 \) formulas, intersection will correspond to intersection, \( \mathcal F_3 \) will give negation, and \( \mathcal F_9 \) and \( \mathcal F_{10} \) will give atomic formulas.
+        \item \( \mathcal F_2 \) will correspond to disjunction for \( \Delta_0 \) formulas, intersection will correspond to conjunction, \( \mathcal F_3 \) will give negation, and \( \mathcal F_9 \) and \( \mathcal F_{10} \) will give atomic formulas.
         \item \( \mathcal F_7 \) and \( \mathcal F_8 \) will deal with ordered \( n \)-tuples.
         For example, the triple \( \langle x_1, x_2, x_3 \rangle \) is formed using \( x_1 \) and \( \langle x_2, x_3 \rangle \), but it cannot be formed using \( \langle x_1, x_2 \rangle \) and \( x_3 \) without \( \mathcal F_7 \) or \( \mathcal F_8 \).
     \end{enumerate}
@@ -406,7 +406,7 @@ \subsection{Well-ordering the universe}
         \item if \( x \in \mathrm{L}_\alpha \) and \( y \in \mathrm{L}_\beta \setminus \mathrm{L}_\alpha \), then \( x <_\beta y \).
     \end{enumerate}
     For limit cases, we take unions:
-    \[ <_\gamma = \bigcup_{\alpha < \gamma} <_\gamma \]
+    \[ {<_\gamma} = \bigcup_{\alpha < \gamma} <_\gamma \]
     We now describe the construction of \( <_{\alpha + 1} \).
     To do this, we consider the ordering on \( \mathrm{L}_\alpha \), and append the singleton \( \qty{\mathrm{L}_\alpha} \).
     We then follow that by the elements of \( \mathcal D(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \setminus (\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \).
@@ -417,7 +417,7 @@ \subsection{Well-ordering the universe}
         \item Suppose that \( <_{\alpha + 1}^n \) is defined.
         We end-extend \( <_{\alpha + 1}^n \) to form \( <_{\alpha + 1}^{n + 1} \) as follows.
         Suppose \( x, y \notin \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \).
-        We say \( x <_{\alpha + 1}^{n+1} \) if either
+        We say \( x <_{\alpha + 1}^{n+1} y \) if either
         \begin{enumerate}
             \item the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the least \( i \leq 10 \) such that \( \exists u, v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \); or
             \item these indices \( i \) are equal, and the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( x = \mathcal F_i(u, v) \) is less than the \( <_{\alpha + 1}^n \)-least \( u \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) such that there exists \( v \in \mathcal D^n(\mathrm{L}_\alpha \cup \qty{\mathrm{L}_\alpha}) \) with \( y = \mathcal F_i(u, v) \); or
@@ -438,7 +438,7 @@ \subsection{The generalised continuum hypothesis in \texorpdfstring{\( \mathrm{L
     \begin{enumerate}
         \item For all \( n \in \omega \), we have \( \mathrm{L}_n = \mathrm{V}_n \).
         \item If \( M \) is infinite, then \( \abs{M} = \abs{\operatorname{Def}(M)} \).
-        \item If \( \alpha \) is an ordinal, then \( \abs{\mathrm{L}_\alpha} = \abs{\alpha} \).
+        \item If \( \alpha \) is an infinite ordinal, then \( \abs{\mathrm{L}_\alpha} = \abs{\alpha} \).
     \end{enumerate}
 \end{lemma}
 \begin{lemma}[G\"odel's condensation lemma]

iii/forcing/03_forcing.tex

@@ -643,7 +643,7 @@ \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}

iii/lc/03_reflection.tex

@@ -402,7 +402,7 @@ \subsection{The fundamental theorem on measurable cardinals}
         Let \( \gamma < \kappa \), and fix \( (A_\alpha)_{\alpha < \gamma} \) such that \( A_\alpha \in U \) for each \( \alpha < \gamma \).
         Then \( \kappa \in j(A_\alpha) \) for all \( \alpha < \gamma \).
         Then \( \bigcap_{\alpha < \gamma} A_\alpha \in U \) if and only if \( \kappa \in j\qty(\bigcap_{\alpha < \gamma} A_\alpha) \).
-        Note that being an element of \( \bigcap_{\alpha < \gamma} A_\gamma \) is a formula that says that \( \vb A \) is a sequence of objects \( A_\alpha \), the \( \alpha \)th element of this sequqence is \( A_\alpha \), and \( \beta \) is an element of each element of the sequence.
+        Note that being an element of \( \bigcap_{\alpha < \gamma} A_\gamma \) is a formula that says that \( \vb A \) is a sequence of objects \( A_\alpha \), the \( \alpha \)th element of this sequence is \( A_\alpha \), and \( \beta \) is an element of each element of the sequence.
         Therefore \( \beta \in j\qty(\bigcap_{\alpha < \gamma} A_\alpha) \) if and only if \( \beta \) is an element of all elements of the sequence \( j(\vb A) \).
         Clearly, \( j(\vb A) \) is a sequence of subsets of \( j(\kappa) \) of length \( j(\gamma) = \gamma \).
         Since \( A_\alpha \) is the \( \alpha \)th element of \( \vb A \), \( j(A_\alpha) \) is the \( j(\alpha) \)th element of \( j(\vb A) \), but \( j(\alpha) = \alpha \).

iii/mtncl/02_quantifier_elimination.tex

@@ -58,7 +58,7 @@ \subsection{Skolem functions}
 
 \subsection{Skolemisation theorem}
 \begin{theorem}
-    Every first-order language \( \mathcal L \) can be expanded to some \( \mathcal L^+ \supseteq \mathcal L \) that includes an \( \mathcal L^+ \)-theory \( \Sigma \) such that
+    Every first-order language \( \mathcal L \) can be expanded to some \( \mathcal L^+ \supseteq \mathcal L \) that admits an \( \mathcal L^+ \)-theory \( \Sigma \) such that
     \begin{enumerate}
         \item \( \Sigma \) is a Skolem \( \mathcal L^+ \)-theory;
         \item any \( \mathcal L \)-structure can be expanded to an \( \mathcal L^+ \)-structure that models \( \Sigma \); and