Minor tweaks

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

iii/cat/07_additive_and_abelian_categories.tex

@@ -60,7 +60,7 @@ \subsection{Additive categories}
         Then \( \mathcal C \) has a unique semi-additive structure.
     \end{enumerate}
 \end{lemma}
-We adopt the convention that morphisms into a product are denoted with column vectors, and morphisms out of a product are denoted with row vectors.
+We adopt the convention that morphisms into a product are denoted with column vectors, and morphisms out of a coproduct are denoted with row vectors.
 \begin{proof}
     \emph{Part (i).}
     The unique morphism \( 0 \to 0 \) is both the identity and a zero morphism.
@@ -443,7 +443,7 @@ \subsection{The five lemma}
     Then,
     \begin{enumerate}
         \item if \( u_1 \) is epic and \( u_2, u_4 \) are monic, then \( u_3 \) is monic;
-        \item if \( u_5 \) is monic and \( u_2, u_4 \) are spic, then \( u_3 \) is epic.
+        \item if \( u_5 \) is monic and \( u_2, u_4 \) are epic, then \( u_3 \) is epic.
     \end{enumerate}
     Thus if \( u_1, u_2, u_4, u_5 \) are isomorphisms, \( u_3 \) is an isomorphism.
 \end{lemma}

iii/forcing/01_set_theory.tex

@@ -434,7 +434,7 @@ \subsection{Transfinite recursion}
     \begin{enumerate}
         \item \( A \) and \( F \) are absolute for \( W \);
         \item \( R \) is absolute for \( W \) and \( (\text{\( R \) is set-like on \( A \)})^W \);
-        \item for all \( x \in W \), then \( \operatorname{pred}(A, x, R) \subseteq W \).
+        \item for all \( x \in W \), \( \operatorname{pred}(A, x, R) \subseteq W \).
     \end{enumerate}
     Then \( G \) is absolute for \( W \).
 \end{theorem}
@@ -592,7 +592,7 @@ \subsection{Cardinal arithmetic}
 In this subsection, we will use the axiom of choice.
 We recall the following basic definitions and results.
 \begin{definition}
-    The \emph{cardinality} of a set \( x \), written \( \abs{x} \) is the least ordinal \( \alpha \) such that there is a bijection \( x \to \alpha \).
+    The \emph{cardinality} of a set \( x \), written \( \abs{x} \), is the least ordinal \( \alpha \) such that there is a bijection \( x \to \alpha \).
 \end{definition}
 This definition only makes sense given the well-ordering principle.
 \begin{definition}

iii/forcing/02_constructibility.tex

@@ -19,7 +19,7 @@ \subsection{Definable sets}
 This definition involves a quantification over infinitely many formulas, so is not yet fully formalised.
 One method to do this is to code formulas as elements of \( \mathrm{V}_\omega \), called \emph{G\"odel codes}.
 We can then use Tarski's \emph{satisfaction relation} to define a formula \( \mathsf{Sat} \), and can then prove
-\[ \mathsf{Sat}(M, E, \ulcorner \varphi \urcorner, x_1, \dots, x_n) \leftrightarrow (M, \in) \vDash \varphi(x_1, \dots, x_n) \]
+\[ \mathsf{Sat}(M, \in, \ulcorner \varphi \urcorner, x_1, \dots, x_n) \leftrightarrow (M, \in) \vDash \varphi(x_1, \dots, x_n) \]
 where \( \ulcorner \varphi \urcorner \in \mathrm{V}_\omega \) is the G\"odel code for \( \varphi \).
 We will later use a different method to formalise it, but for now we will assume that this is well-defined.
 
@@ -166,7 +166,7 @@ \subsection{G\"odel functions}
         \begin{align*}
             \mathcal F_\varphi(a_1, \dots, a_{n-1}) &= \qty{\langle x_{n-1}, \dots, x_1 \rangle \in a_{n-1} \times \dots \times a_1 \mid \varphi(x_1, \dots, x_{n-1})} \\
             &= \ran(\qty{\langle 0, x_{n-1}, \dots, x_1 \rangle \in \qty{0} \times a_{n-1} \times \dots \times a_1 \mid \theta(x_1, \dots, x_{n-1}, 0)}) \\
-            &= \mathcal F_6(\mathcal F_\theta(a_1, \dots, a_{n-1}), \mathcal F_1(\mathcal F_3(a_1, a_1), \mathcal F_3(a_1, a_1)), a_1)
+            &= \mathcal F_6(\mathcal F_\theta(a_1, \dots, a_{n-1}, \mathcal F_1(\mathcal F_3(a_1, a_1), \mathcal F_3(a_1, a_1))), a_1)
         \end{align*}
         \item If \( \psi(x_1, \dots, x_n) \) is a termed formula and
         \[ \varphi(x_1, \dots, x_{n+1}) = \psi(x_1, \dots, x_{n-1}, x_{n+1}/x_n) \]

iii/forcing/03_forcing.tex

@@ -96,7 +96,7 @@ \subsection{Forcing posets}
     \[ \Fn(I, J) = \qty{p \mid \abs{p} < \omega \wedge p \text{ is a function} \wedge \dom p \subseteq I \wedge \ran p \subseteq J} \]
     We let \( \leq \) be the reverse inclusion on \( \Fn(I, J) \), so \( q \leq p \) if and only if \( q \supseteq p \).
     The maximal element \( \Bbbone \) is the empty set.
-    Then \( (\Fn(I, J), \leq, \varnothing) \) is a forcing poset, and moreover, the preorder is separative.
+    Then \( (\Fn(I, J), \supseteq, \varnothing) \) is a forcing poset, and moreover, the preorder is separative.
 \end{example}
 \begin{remark}
     When \( \alpha \) is an ordinal, the forcing poset \( \Fn(\alpha \times \omega, 2) \) is often written \( \operatorname{Add}(\omega, \alpha) \), denoting the idea that we are adding \( \alpha \)-many subsets of \( \omega \).