Some updates

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

ii/lst/05_set_theory.tex

@@ -50,7 +50,7 @@ \subsection{Axioms of \texorpdfstring{\(\symsfup{ZF}\)}{ZF}}
     where \( z \subseteq x \) means \( (\forall t)(t \in z \Rightarrow t \in x) \).
     We write \( \mathcal P(x) \) for the power set of \( x \).
     We can form the Cartesian product \( x \times y \) as a suitable subset of \( \mathcal P(\mathcal P(x \cup y)) \), as if \( z \in x, t \in y \), we have \( (z, t) = \qty{\qty{z}, \qty{z, t}} \in \mathcal P(\mathcal P(x \cup y)) \).
-    The set of all functions \( x \to y \) can be defined as a subset of \( \mathbb P(x \times y) \).
+    The set of all functions \( x \to y \) can be defined as a subset of \( \mathcal P(x \times y) \).
     \item \emph{Axiom of infinity}.
     Using our currently defined axioms, any model \( V \) must be infinite.
     For example, writing \( x^+ \) for the \emph{successor} of \( x \) defined as \( x \cup \qty{x} \), the sets \( \varnothing, \varnothing^+, \varnothing^{++}, \dots \) are distinct.

iii/forcing/01_set_theory.tex

@@ -18,7 +18,7 @@ \subsection{Introduction to independence results}
 \end{theorem}
 The continuum hypothesis is that there are no sets of cardinality strictly between \( \abs{\mathbb N} \) and \( \abs{\mathcal P(N)} = \abs{\mathbb R} \).
 \begin{definition}
-    The \emph{continuum hypothesis} \( \mathsf{CH} \) states that if \( X \subseteq \mathbb P(\mathbb N) \) is infinite, then either \( \abs{X} = \abs{\mathbb N} \) or \( \abs{X} = \abs{\mathcal P(\mathbb N)} \), or equivalently,
+    The \emph{continuum hypothesis} \( \mathsf{CH} \) states that if \( X \subseteq \mathcal P(\mathbb N) \) is infinite, then either \( \abs{X} = \abs{\mathbb N} \) or \( \abs{X} = \abs{\mathcal P(\mathbb N)} \), or equivalently,
     \[ 2^{\aleph_0} = \aleph_1 \]
 \end{definition}
 Progress was made on the continuum hypothesis in the 19th and 20th centuries.
@@ -42,7 +42,7 @@ \subsection{Systems of set theory}
 We write \( \mathrm{FV}(\varphi) \) for the set of free variables of \( \varphi \).
 We will write \( \varphi(u_1, \dots, u_n) \) to emphasise the dependence of \( \varphi \) on its free variables \( u_1, \dots, u_n \).
 By doing so, we will allow ourselves to freely change the names of the free variables, and assume that substituted variables are free.
-The syntax \( \varphi(u_0, \dots, u_n) \) does not imply that \( u_i \) occurs freely, or even at all.
+The syntax \( \varphi(u_1, \dots, u_n) \) does not imply that \( u_i \) occurs freely, or even at all.
 
 Some of the most common axioms of set theory are as follows.
 \begin{enumerate}
@@ -96,7 +96,7 @@ \subsection{Systems of set theory}
 We can then use formulas of the form
 \[ \exists C.\, (C \text{ is a class} \wedge \forall x \in C.\, \varphi) \]
 by defining it to mean that there is a formula \( \theta \) giving a class \( C \) satisfying \( \forall x \in C.\, \varphi \).
-For example, the universe class \( \mathrm{V} = \qty{x \mid x = x} \), the Russell class \( R = \qty{x \mid x \notin x} \), and the class of ordinals are all definable.
+For example, the universe class \( \mathrm{V} = \qty{x \mid x = x} \), the Russell class \( R = \qty{x \mid x \notin x} \), and the class of ordinals \( \mathrm{Ord} \) are all definable.
 Any set is a definable class.
 Classes are heavily dependent on the underlying model: if \( M = 2 \) then \( \mathrm{Ord} = 2 = M \), and if \( M = 3 \cup \qty{1} \) then \( \mathrm{Ord} = 3 \neq M \).
 
@@ -128,10 +128,10 @@ \subsection{Adding defined functions}
 \begin{example}
     The intersection \( a \cap b \) can be defined as the unique set \( c \) such that
     \[ \forall x.\, (x \in c \iff x \in a \wedge x \in b) \]
-    This definition makes sense only if there is a unique \( c \) satisfying this formula \( \varphi(c) \).
+    This definition makes sense only if there is a unique \( c \) satisfying this formula \( \varphi(a, b, c) \).
     If
     \[ M = \qty{a, c, d, \qty{a}, \qty{a, b}, \qty{a, b, c}, \qty{a, b, d}} \]
-    then it is easy to check that both \( \qty{a} \) and \( \qty{a, d} \) satisfy \( \varphi \), so intersection cannot be defined.
+    then it is easy to check that both \( \qty{a} \) and \( \qty{a, b} \) satisfy \( \varphi(\{ a, b, c \}, \{ a, b, d \}, -) \), so intersection cannot be defined.
 \end{example}
 
 \subsection{Absoluteness}
@@ -140,8 +140,8 @@ \subsection{Absoluteness}
 Other definitions need not, for example \( \mathcal P(\mathbb N) \), which need not be the true power set in a given transitive model.
 To quantify this behaviour, we need to define what it means for \( \varphi \) to hold in an arbitrary structure; this concept is called \emph{relativisation}.
 \begin{definition}
-    The quantifier \( \forall x \in a.\, \varphi \) is an abbreviation of \( \forall x.\, x \in a \Rightarrow \varphi \).
-    We use the analogous abbreviation for the existential quantifier.
+    The quantifier \( \forall x \in a.\, \varphi \) is an abbreviation of \( \forall x.\, (x \in a) \to \varphi \).
+    Similarly, \( \exists x \in a.\, \varphi \) is an abbreviation of \( \exists x.\, (x \in a) \wedge \varphi \).
     Let \( W \) be a class; we define by recursion the \emph{relativisation} \( \varphi^W \) of \( \varphi \) as follows.
     \begin{align*}
         (x \in y)^W &\equiv x \in y \\
@@ -168,7 +168,7 @@ \subsection{Absoluteness}
     We proceed by induction on the length of formulae.
     For example,
     \[ N \vDash (x \in y)^M \text{ iff } N \vDash x \in y \text{ and } x, y \in M \text{ iff } \theta(x), \theta(y), M \vDash x \in y \]
-    The cases for equality is similar, and disjunction and negation are simple.
+    The case for equality is similar, and disjunction and negation are simple.
     Finally,
     \[ N \vDash (\exists x.\, \varphi(x))^M \text{ iff } N \vDash \exists x.\, x \in M \wedge \varphi^M(x) \]
     which holds precisely when there is some \( x \in N \) such that \( N \vDash x \in M \) and \( N \vDash \varphi^M(x) \), but \( N \vDash x \in M \) if and only if \( \theta(x) \), giving the result as required.
@@ -198,7 +198,7 @@ \subsection{Absoluteness}
     \( \varphi \leftrightarrow \psi \) does not imply \( \varphi^M \leftrightarrow \psi^M \).
     Let \( \varphi(v) \) be the statement \( \forall x.\, (x \notin v) \); in \( \mathsf{ZF} \) this defines \( \varnothing \).
     Now, the following are two ways to express \( 0 \in z \).
-    \[ \psi(z) \equiv \exists y.\, (\varphi(y) \wedge y \in z);\quad \theta(z) \equiv \forall y.\, (\varphi(y) \Rightarrow y \in z) \]
+    \[ \psi(z) \equiv \exists y.\, (\varphi(y) \wedge y \in z);\quad \theta(z) \equiv \forall y.\, (\varphi(y) \to y \in z) \]
     Note that if there exists a unique \( y \) such that \( \varphi(y) \), then these are equivalent.
     However, this is often not the case, for example if
     \[ a = 0;\quad b = \qty{0};\quad c = \qty{\qty{\qty{0}}};\quad M = \qty{a, b, c} \]
@@ -207,7 +207,7 @@ \subsection{Absoluteness}
 The main obstacle to absoluteness for basic statements turns out to be transitivity of the model.
 \begin{definition}
     Given classes \( M \subseteq N \), we say that \( M \) is \emph{transitive} in \( N \) if
-    \[ \forall x, y \in N.\, (x \in M \wedge y \in x \Rightarrow y \in M) \]
+    \[ \forall x, y \in N.\, (x \in M \wedge y \in x \to y \in M) \]
 \end{definition}
 
 \subsection{The L\'evy hierarchy}
@@ -324,7 +324,7 @@ \subsection{The L\'evy hierarchy}
     \[ \forall x \in a.\, \langle x, x \rangle \notin r \]
 \end{proof}
 \begin{corollary}
-    The statement that \( x \) is a transitive set totally ordered by \( \in \) is \( \Delta_0 \), and thus ordinals are in fact \( \Delta_0 \).
+    The statement that \( x \) is a transitive set totally ordered by \( \in \) is \( \Delta_0 \), and thus being an ordinal is \( \Delta_0 \).
 \end{corollary}
 \begin{lemma}
     (\( \mathsf{ZF} \))
@@ -522,7 +522,7 @@ \subsection{The reflection theorem}
         \eta & \text{where } \eta \text{ is the least ordinal such that } \exists x \in W_\eta.\, \varphi_j^W(x, \vb y)
     \end{cases} \]
     We set
-    \[ G_i(\delta) = \sup\qty{F_i(\vb y) \mid y \in W_\delta^{k_i}} \]
+    \[ G_i(\delta) = \sup\qty{F_i(\vb y) \mid \vb y \in W_\delta^{k_i}} \]
     If \( \varphi_i \) is not of this form, we set \( G_i(\delta) = 0 \) for all \( \delta \).
     Finally, we let
     \[ K(\delta) = \max\qty{\delta + 1, G_1(\delta), \dots, G_n(\delta)} \]
@@ -572,7 +572,7 @@ \subsection{The reflection theorem}
     Then \( T \) is not finitely axiomatisable.
     That is, for any finite set of sentences \( \Gamma \) in \( \mathcal L_\in \) such that \( T \vdash \Gamma \), there exists a sentence \( \varphi \) such that \( T \vdash \varphi \) but \( \Gamma \nvdash \varphi \).
 \end{corollary}
-This only holds for first-order theories; for example, G\"odel--Bernays set theory is finitely axiomatisable.
+This only holds for first-order theories without classes; for example, G\"odel--Bernays set theory is finitely axiomatisable.
 \begin{proof}
     Let \( \varphi_1, \dots, \varphi_n \) be a set of sentences such that \( T \vdash \bigwedge_{i=1}^n \varphi_i \).
     Suppose that \( \bigwedge_{i=1}^n \varphi_i \) proves every axiom of \( T \).
@@ -599,7 +599,7 @@ \subsection{Cardinal arithmetic}
     The cardinal arithmetic operations are defined as follows.
     Let \( \kappa, \lambda \) be cardinals.
     \begin{enumerate}
-        \item \( \kappa + \lambda = \abs{0 \times \kappa \cup 1 \times \lambda} \);
+        \item \( \kappa + \lambda = \abs{\qty{0} \times \kappa \cup \qty{1} \times \lambda} \);
         \item \( \kappa \cdot \lambda = \abs{\kappa \times \lambda} \);
         \item \( \kappa^\lambda = \abs{\kappa^\lambda} \), the cardinality of the set of functions \( \lambda \to \kappa \);
         \item \( \kappa^{<\lambda} = \sup\qty{\kappa^\alpha \mid \alpha < \lambda, \alpha \text{ a cardinal}} \).

iii/forcing/02_constructibility.tex

@@ -18,9 +18,9 @@ \subsection{Definable sets}
 \end{remark}
 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 satisfaction relation to define a formula \( \mathsf{Sat} \), and can then prove
+We can then use Tarski's \emph{satisfaction relation} to define a formula \( \mathsf{Sat} \), and can then prove
 \[ \mathsf{Sat}(M, E, \ulcorner \varphi \urcorner, x_1, \dots, x_n) \leftrightarrow (M, \in) \vDash \varphi(x_1, \dots, x_n) \]
-where \( \ulcorner \varphi \urcorner \in V_\omega \) is the G\"odel code for \( \varphi \).
+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.
 
 \subsection{Defining the constructible universe}
@@ -63,7 +63,7 @@ \subsection{Defining the constructible universe}
         Then
         \[ \bigcup a = \qty{x \in \mathrm{L}_\alpha \mid (\mathrm{L}_\alpha, \in) \vDash \exists z.\, (z \in a \wedge x \in z)} \in \operatorname{Def}(\mathrm{L}_\alpha) \]
         \item For infinity, note that
-        \[ \omega = \qty{n \in \mathrm{L}_\omega \mid (\mathrm{L}_\omega, \in) \vDash n \in \mathrm{Ord}} \in \operatorname{Def}(\mathrm{L}_\alpha) \]
+        \[ \omega = \qty{n \in \mathrm{L}_\omega \mid (\mathrm{L}_\omega, \in) \vDash n \in \mathrm{Ord}} \in \operatorname{Def}(\mathrm{L}_\omega) \]
         \item Consider separation.
         Let \( \varphi \) be a formula, and let \( a, \vb u \in \mathrm{L}_\alpha \).
         We claim that
@@ -123,8 +123,7 @@ \subsection{G\"odel functions}
         \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 \( \mathcal F_7 \) and \( \mathcal F_8 \) will deal with ordered \( n \)-tuples.
-        The triple \( \langle x_1, x_2, x_3 \rangle \), this is formed using \( x_1 \) and \( \langle x_2, x_3 \rangle \).
-        However, it cannot be formed using \( \langle x_1, x_2 \rangle \) and \( x_3 \).
+        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}
 \end{remark}
 \begin{proof}
@@ -241,7 +240,8 @@ \subsection{G\"odel functions}
         The equalities are termed formulas as above, so \( \psi \) is a termed formula.
         Then
         \begin{align*}
-            \mathcal F_\varphi(a_1, \dots, a_n) &= \ran\ran\{\langle x_{n+2}, \dots, x_1\rangle \times a_j \times a_i \times a_n \times \dots \times a_1 \\&\quad\quad\quad\mid x_i = x_{n+1} \wedge x_j = x_{n+2} \wedge x_{n+1} \in x_{n+2}\} \\
+            \mathcal F_\varphi(a_1, \dots, a_n) &=
+            \ran\ran\{\langle x_{n+2}, \dots, x_1\rangle \times a_j \times a_i \times a_n \times \dots \times a_1 \mid \\&\quad\quad\quad x_i = x_{n+1} \wedge x_j = x_{n+2} \wedge x_{n+1} \in x_{n+2}\} \\
             &= \mathcal F_6(\mathcal F_6(\mathcal F_\psi(a_1, \dots, a_n), a_1), a_1)
         \end{align*}
     \end{itemize}
@@ -257,19 +257,23 @@ \subsection{G\"odel functions}
     Now
     \begin{align*}
         \mathcal F_{\theta \wedge \psi}(a_1, \dots, a_n, \mathcal F_2(a_j, a_j)) &= \mathcal F_{\theta \wedge \psi}\qty(a_1, \dots, a_n, \bigcup a_j) \\
-        &= \qty{\langle x_{n+1}, \dots, x_1 \rangle \in \qty(\bigcup a_j) \times a_n \times \dots \times a_1 \mid x_{n+1} \in x_j \wedge \forall k \leq n.\, x_k \in a_k \wedge \psi(x_1, \dots, x_{n+1})}
+        &= \Big\{\langle x_{n+1}, \dots, x_1 \rangle \in \qty(\bigcup a_j) \times a_n \times \dots \times a_1 \mid \\
+        &\quad\quad\quad x_{n+1} \in x_j \wedge \forall k \leq n.\, x_k \in a_k \wedge \psi(x_1, \dots, x_{n+1})\Big\}
     \end{align*}
     So
     \begin{align*}
-        \ran(\mathcal F_{\theta \wedge \psi}\qty(a_1, \dots, a_n, \bigcup a_j)) &= \qty{\langle x_n, \dots, x_1 \rangle \in a_n \times \dots \times a_1 \mid \exists u.\, \langle u, x_n, \dots, x_1 \rangle \in \mathcal F_{\theta \wedge \psi}\qty(a_1, \dots, a_n, \bigcup a_j)} \\
-        &= \qty{\langle x_n, \dots, x_1 \rangle \in a_n \times \dots \times a_1 \mid \exists x_{n+1} \in x_j.\, \psi(x_1, \dots, x_{n+1})}
+        \ran(\mathcal F_{\theta \wedge \psi}\qty(a_1, \dots, a_n, \bigcup a_j))
+        &= \Big\{\langle x_n, \dots, x_1 \rangle \in a_n \times \dots \times a_1 \mid \\
+        &\quad\quad\quad \exists u.\, \langle u, x_n, \dots, x_1 \rangle \in \mathcal F_{\theta \wedge \psi}\qty(a_1, \dots, a_n, \bigcup a_j)\Big\} \\
+        &= \Big\{\langle x_n, \dots, x_1 \rangle \in a_n \times \dots \times a_1 \mid \\
+        &\quad\quad\quad \exists x_{n+1} \in x_j.\, \psi(x_1, \dots, x_{n+1})\Big\}
     \end{align*}
 \end{proof}
 \begin{definition}
     A class \( C \) is \emph{closed under G\"odel functions} if whenever \( x, y \in C \), we have \( \mathcal F_i(x, y) \in C \) for \( i \in \qty{1, \dots, 10} \).
     Given a set \( b \), we let \( \mathrm{cl}(b) \) be the smallest set \( C \) containing \( b \) as a subset that is closed under G\"odel functions.
 \end{definition}
-For example, \( \mathrm{cl}(\varnothing) = \varnothing \), \( \qty{a, b} \in \mathrm{cl}(\qty{a, b}) \), and \( \mathrm{cl}(b) = \mathrm{cl}(\mathrm{cl}(b)) \).
+For example, \( \mathrm{cl}(\varnothing) = \varnothing \), \( a, b \in \mathrm{cl}(\qty{a, b}) \), and \( \mathrm{cl}(b) = \mathrm{cl}(\mathrm{cl}(b)) \).
 \begin{definition}
     Let \( b \) be a set.
     Define \( \mathcal D^n(b) \) inductively by
@@ -331,7 +335,7 @@ \subsection{The axiom of constructibility}
     The \emph{axiom of constructibility} is the statement \( \mathrm{V} = \mathrm{L} \).
     Equivalently, \( \forall x.\, \exists \alpha \in \mathrm{Ord}.\, (x \in \mathrm{L}_\alpha) \).
 \end{definition}
-We will show that if \( \mathsf{ZF} \) is consistent, then so is \( \mathsf{ZF} + (\mathrm{V} = \mathrm{L}) \), because \( \mathrm{L} \) is a model of \( \mathsf{ZF} + (\mathrm{V} = \mathrm{L}) \).
+We will show that if \( \mathsf{ZF} \) is consistent, then so is \( \mathsf{ZF} + (\mathrm{V} = \mathrm{L}) \), by demonstrating that \( \mathrm{L} \) is a model of \( \mathsf{ZF} + (\mathrm{V} = \mathrm{L}) \).
 To do this, we will show that being constructible is absolute.
 \begin{lemma}
     \( Z = \mathrm{cl}(M) \) is \( \Delta_1^{\mathsf{ZF}} \).
@@ -513,7 +517,7 @@ \subsection{The generalised continuum hypothesis in \texorpdfstring{\( \mathrm{L
     Then fix \( \varphi \) such that
     \[ \mathsf{ZFC} + \mathrm{V} = \mathrm{L} + \mathsf{GCH} \vdash \varphi \wedge \neg\varphi \]
     Then
-    \[ \mathsf{ZF} \vDash (\varphi \wedge \neg\varphi)^L \]
+    \[ \mathsf{ZF} \vdash (\varphi \wedge \neg\varphi)^L \]
     By relativisation, \( \varphi^L \wedge \neg(\varphi^L) \).
     Hence \( \mathsf{ZF} \) is inconsistent.
 \end{proof}
@@ -549,7 +553,7 @@ \subsection{Combinatorial properties}
 \end{lemma}
 \begin{proof}
     If \( (A_\alpha)_{\alpha < \omega_1} \) is a \( \diamondsuit \)-sequence, then for all \( X \subseteq \omega \), there is \( \alpha > \omega \) such that \( X = A_\alpha \).
-    Thus \( \qty{A_\alpha \mid \alpha \in \omega_1 \mid A_\alpha \subseteq \omega} = \mathcal P(\omega) \).
+    Thus \( \qty{A_\alpha \mid \alpha \in \omega_1 \wedge A_\alpha \subseteq \omega} = \mathcal P(\omega) \).
 \end{proof}
 \begin{theorem}
     If \( \mathrm{V} = \mathrm{L} \), then \( \diamondsuit \) holds.