From 2c516f0d576cffef5bce91b76af589ed47be6bd3 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?fn=20=E2=8C=83=20=E2=8C=A5?=
 <70830482+FnControlOption@users.noreply.github.com>
Date: Thu, 20 Nov 2025 12:03:32 -0800
Subject: [PATCH 1/4] Fix typos in Isomorphic Structures

---
 content/model-theory/basics/isomorphism.tex | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/content/model-theory/basics/isomorphism.tex b/content/model-theory/basics/isomorphism.tex
index 1ba7a901..0d9c01df 100644
--- a/content/model-theory/basics/isomorphism.tex
+++ b/content/model-theory/basics/isomorphism.tex
@@ -10,7 +10,7 @@
 \olsection{Isomorphic Structures}
 
 First-order !!{structure}s can be alike in one of two ways. One way in
-which the can be alike is that they make the same !!{sentence}s
+which they can be alike is that they make the same !!{sentence}s
 true. We call such !!{structure}s \emph{elementarily equivalent}. But
 structures can be very different and still make the same !!{sentence}s
 true---for instance, one can be !!{enumerable} and the other not.
@@ -117,8 +117,8 @@
 \end{prob}
 
 \begin{defn}
-An \emph{automorphism} of a structure $\mathfrak{M}$ is an isomorphism
-of $\mathfrak{M}$ onto itself.
+An \emph{automorphism} of a structure $\Struct{M}$ is an isomorphism
+of $\Struct{M}$ onto itself.
 \end{defn}
 
 \begin{prob}

From e0ad2dc28dc7b49dfd07e143ac032b57c68be4ac Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?fn=20=E2=8C=83=20=E2=8C=A5?=
 <70830482+FnControlOption@users.noreply.github.com>
Date: Thu, 20 Nov 2025 12:08:01 -0800
Subject: [PATCH 2/4] Fix typos in Partial Isomorphisms

---
 content/model-theory/basics/partial-iso.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/content/model-theory/basics/partial-iso.tex b/content/model-theory/basics/partial-iso.tex
index 91a174ae..ea0e756e 100644
--- a/content/model-theory/basics/partial-iso.tex
+++ b/content/model-theory/basics/partial-iso.tex
@@ -152,7 +152,7 @@ \section{Partial Isomorphisms}
      $x_1$, \dots,~$x_n$, then $\Sat{M}{!A}[s_1]$ if and
      only if~$\Sat{N}{!A}[s_2]$.
    \item $I_{n+1} (\mathbf{a},\mathbf{b})$ if and only if for every
-     $a\in A$ there is a $b\in B$ such that $I_n
+     $a\in \Domain M$ there is a $b\in \Domain N$ such that $I_n
      (\mathbf{a}a,\mathbf{b}b)$, and vice-versa.
    \end{enumerate}
 \end{defn}
@@ -198,7 +198,7 @@ \section{Partial Isomorphisms}
 
   Given $a \in \Domain M$, let $!T^a_n$ be set of !!{formula}s
   $!B(x,\mathbf{y})$ of rank no greater than $n$ satisfied by
-  $\mathbf{a}a$ in $\Struct{M}$; $\tau^a_n$ is finite, so we can
+  $\mathbf{a}a$ in $\Struct{M}$; $!T^a_n$ is finite, so we can
   assume it is a single first-order !!{formula}. It follows that
   $\mathbf{a}$ satisfies $\lexists[x][!T^a_n(x,\mathbf{y})]$, which
   has quantifier rank no greater than $n+1$. By hypothesis

From 750d059b8fe4192e3e7347cb8da1bf87b377f4ff Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?fn=20=E2=8C=83=20=E2=8C=A5?=
 <70830482+FnControlOption@users.noreply.github.com>
Date: Thu, 20 Nov 2025 15:30:48 -0800
Subject: [PATCH 3/4] Fix typo in Dense Linear Orders

---
 content/model-theory/basics/dlo.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/content/model-theory/basics/dlo.tex b/content/model-theory/basics/dlo.tex
index 339a1de7..8d26dced 100644
--- a/content/model-theory/basics/dlo.tex
+++ b/content/model-theory/basics/dlo.tex
@@ -45,7 +45,7 @@ \section{Dense Linear Orders}
   a_n$. Given $a \in \Domain{M_1}$, find $b \in \Domain{M_2}$ as
   follows:
   \begin{enumerate}
-  \item if $a <_2 a_1$ let $b \in \Domain{M_2}$ be such that $b <_2
+  \item if $a <_1 a_1$ let $b \in \Domain{M_2}$ be such that $b <_2
     b_1$;
   \item if $a_n <_1 a$ let $b \in \Domain{M_2}$ be such that $b_n <_2 b$;
  \item if $a_i <_1 a <_1 a_{i+1}$ for some $i$, then let $b \in

From 162e62afc81c0f8783b19fcfe078fb928586f7e0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?fn=20=E2=8C=83=20=E2=8C=A5?=
 <70830482+FnControlOption@users.noreply.github.com>
Date: Thu, 20 Nov 2025 15:49:59 -0800
Subject: [PATCH 4/4] Fix proof of Theorem 25.9

---
 content/model-theory/basics/isomorphism.tex | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/content/model-theory/basics/isomorphism.tex b/content/model-theory/basics/isomorphism.tex
index 0d9c01df..f90d2e2c 100644
--- a/content/model-theory/basics/isomorphism.tex
+++ b/content/model-theory/basics/isomorphism.tex
@@ -94,14 +94,14 @@
   \begin{align}
     h(\Value{t}{M}[s]) 
     & = h(\Assign{f}{M}(\Value{t_1}{M}[s], \dots, \Value{t_n}{M}[s]) \notag\\
-    & = h(\Assign{f}{M}(\Value{t_1}{M'}[h \circ s], \dots,
-    \Value{t_n}{M'}[h \circ s]) \ollabel{iso-1}\\
+    & = \Assign{f}{M'}(h(\Value{t_1}{M}[s]), \dots,
+    h(\Value{t_n}{M}[s])) \ollabel{iso-1}\\
     & = \Assign{f}{M'}(\Value{t_1}{M'}[h \circ s], \dots,
     \Value{t_n}{M'}[h \circ s]) \ollabel{iso-2}\\
     & = \Value{t}{M'}[h\circ s] \notag
   \end{align}
-  Here, \olref{iso-1} follows by induction hypothesis and \olref{iso-2} by
-  \olref{defn:iso-func} of \olref{defn:isomorphism}.
+  Here, \olref{iso-1} follows by \olref{defn:iso-func} of
+  \olref{defn:isomorphism} and \olref{iso-2} by induction hypothesis.
 \end{enumerate}
 Part (b) is left as an exercise.