Naming sections

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

iii/forcing/01.tex → iii/forcing/01_set_theory.tex

@@ -1,4 +1,4 @@
-\subsection{???}
+\subsection{Introduction to independence results}
 Independence results are found across mathematical disciplines.
 \begin{enumerate}
     \item The \emph{parallel postulate} is independent from the other four postulates of Euclidean geometry.

iii/forcing/02_constructibility.tex

@@ -23,7 +23,7 @@ \subsection{Definable sets}
 where \( \ulcorner \varphi \urcorner \in V_\omega \) 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{???}
+\subsection{Defining the constructible universe}
 We define the \( \mathrm{L}_\alpha \) hierarchy by transfinite recursion as follows.
 \[ \mathrm{L}_0 = \varnothing;\quad \mathrm{L}_{\alpha + 1} = \operatorname{Def}(\mathrm{L}_\alpha);\quad \mathrm{L}_\lambda = \bigcup_{\alpha < \lambda} \mathrm{L}_\alpha;\quad \mathrm{L} = \bigcup_{\alpha \in \mathrm{Ord}} \mathrm{L}_\alpha \]
 \begin{lemma}
@@ -116,7 +116,7 @@ \subsection{G\"odel functions}
 \end{proposition}
 \begin{lemma}[G\"odel normal form]
     For every \( \Delta_0 \) formula \( \varphi(x_1, \dots, x_n) \) with free variables contained in \( x_1, \dots, x_n \), there is a term \( \mathcal F_\varphi \) built from the symbols \( \mathcal F_1, \dots, \mathcal F_{10} \) such that
-    \[ \mathsf{ZF} \vdash \forall a_1, \dots, a_n.\, \mathcal F_\varphi(a_1, \dots, a_n) = \qty{\langle x_n, \dots, x_1 \rangle \mid a_n \times \dots \times a_1 \mid \vaprhi(x_1, \dots, x_n)} \]
+    \[ \mathsf{ZF} \vdash \forall a_1, \dots, a_n.\, \mathcal F_\varphi(a_1, \dots, a_n) = \qty{\langle x_n, \dots, x_1 \rangle \mid a_n \times \dots \times a_1 \mid \varphi(x_1, \dots, x_n)} \]
 \end{lemma}
 \begin{remark}
     \begin{enumerate}

iii/forcing/main.tex

@@ -10,8 +10,8 @@
 
 \tableofcontentsnewpage{}
 
-\section{???}
-\input{01.tex}
+\section{Set theoretic preliminaries}
+\input{01_set_theory.tex}
 \section{Constructibility}
 \input{02_constructibility.tex}
 

iii/lc/01.tex → iii/lc/01_inaccessible_cardinals.tex

@@ -1,4 +1,4 @@
-\subsection{Non-examples}
+\subsection{Large cardinal properties}
 Modern set theory largely concerns itself with the consequences of the incompleteness phenomenon.
 Given any `reasonable' set theory \( T \), then G\"odel's first incompleteness theorem shows that there is \( \varphi \) such that \( T \nvdash \varphi \) and \( T \nvdash \neg\varphi \).
 To be `reasonable', the set of axioms must be computably enumerable, among other similar restrictions.
@@ -71,6 +71,7 @@ \subsection{Non-examples}
     Limit cardinals are often singular.
 \end{enumerate}
 
+\subsection{Weakly inaccessible and inaccessible cardinals}
 Motivated by these examples of properties of \( \omega \), we make the following definition.
 
 \begin{definition}
@@ -97,6 +98,7 @@ \subsection{Non-examples}
     Hence \( \kappa \) is singular, contradicting regularity.
 \end{proof}
 
+\subsection{Second order replacement}
 We will now show that \( \mathsf{ZFC} \) does not prove \( \mathsf{IC} \), and we omit the result for weakly inaccessible cardinals.
 We could do this via model-theoretic means: we assume \( M \vDash \mathsf{ZFC} \), and construct a model \( N \vDash \mathsf{ZFC} + \neg \mathsf{IC} \).
 However, there is another approach we will take here.
@@ -188,7 +190,9 @@ \subsection{Non-examples}
     Since \( \lambda < \kappa \), we must have \( \mathcal P(\lambda) \in \mathrm{V}_{\lambda + 2} \subseteq \mathrm{V}_\kappa \).
     Then the image of \( \mathcal P(\lambda) \) under \( F \) is \( \kappa \notin \mathrm{V}_\kappa \) as required.
 \end{proof}
-However, it is not generally the case that if \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} \) then \( \kappa \) is inaccessible.
+
+\subsection{Countable transitive models of set theory}
+It is not generally the case that if \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} \) then \( \kappa \) is inaccessible.
 Moreover, the existence of an inaccessible cardinal is strictly stronger than the consistency of \( \mathsf{ZFC} \).
 We will show this second statement first.
 
@@ -244,6 +248,7 @@ \subsection{Non-examples}
     So, by the second incompleteness theorem, \( \mathsf{ZFC}^\star \nvdash \mathsf{IC} \).
 \end{proof}
 
+\subsection{Worldly cardinals}
 We now show that if \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} \), it is not necessarily the case that \( \kappa \) is inaccessible.
 
 Observe that \( M \neq \mathrm{V}_\alpha \) for any \( \alpha \).
@@ -255,7 +260,7 @@ \subsection{Non-examples}
 \[ N_0 = \varnothing;\quad N_{k+1} = W(N_k);\quad N = \bigcup_{k \in \mathbb N} N_k \]
 We wish to create a similar structure that is of the form \( \mathrm{V}_\alpha \) for some \( \alpha \).
 We define
-\[ \alpha_0 = 0;\quad \alpha_{k+1} = \sup{\rank(x) \mid x \in W(\mathrm{V}_{\alpha_k})};\quad \alpha = \sup{\alpha_n \mid n \in \mathbb N} \]
+\[ \alpha_0 = 0;\quad \alpha_{k+1} = \sup\qty{\rank(x) \mid x \in W(\mathrm{V}_{\alpha_k})};\quad \alpha = \sup\qty{\alpha_n \mid n \in \mathbb N} \]
 Note that \( N \subseteq \mathrm{V}_{\alpha_1} \).
 \begin{theorem}
     \( \mathrm{V}_\alpha \preceq \mathrm{V}_\kappa \) and \( \alpha < \kappa \).
@@ -297,29 +302,29 @@ \subsection{Non-examples}
 
 \subsection{The consistency strength hierarchy}
 Let \( B \) be a base theory; we will often use \( \mathsf{ZFC} \).
-If \( T, S \) are extensions of \( B \), we say that \( T \) has lower \emph{consistency strength} than \( S \), written \( T \leq_{\mathrm{Cons}} S \), if \( B \vdash \Con(S) \to \Con(T) \).
-We say that \( T \) and \( S \) is \emph{equiconsistent}, written \( T \equiv_{\mathrm{Cons}} S \), if \( T \leq_{\mathrm{Cons}} S \) and \( S \leq_{\mathrm{Cons}} T \), and write \( T <_{\mathrm{Cons}} S \) if \( T \leq_{\mathrm{Cons}} S \) but \( S \nleq_{\mathrm{Cons}} T \).
+If \( T, S \) are extensions of \( B \), we say that \( T \) has lower \emph{consistency strength} than \( S \), written \( T \leq_{\Con} S \), if \( B \vdash \Con(S) \to \Con(T) \).
+We say that \( T \) and \( S \) is \emph{equiconsistent}, written \( T \equiv_{\Con} S \), if \( T \leq_{\Con} S \) and \( S \leq_{\Con} T \), and write \( T <_{\Con} S \) if \( T \leq_{\Con} S \) but \( S \nleq_{\Con} T \).
 \begin{remark}
     \begin{enumerate}
-        \item If \( I \) is inconsistent, then \( T \leq_{\mathrm{Cons}} I \) for all \( T \).
+        \item If \( I \) is inconsistent, then \( T \leq_{\Con} I \) for all \( T \).
         All inconsistent theories are equiconsistent.
-        In particular, \( T \) is consistent if and only if \( T <_{\mathrm{Cons}} I \).
+        In particular, \( T \) is consistent if and only if \( T <_{\Con} I \).
         We typically write \( \bot \) for an inconsistent theory.
-        \item \( <_{\mathrm{Cons}} \) is more than just `proving more theorems'.
-        If \( \varphi \) is such that \( \mathsf{ZFC} \nvdash \varphi \) and \( \mathsf{ZFC} \nvdash \neg\varphi \), it is not necessarily the case that \( \mathsf{ZFC} <_{\mathrm{Cons}} \mathsf{ZFC} + \varphi \) or \( \mathsf{ZFC} <_{\mathrm{Cons}} \mathsf{ZFC} + \neg\varphi \).
+        \item \( <_{\Con} \) is more than just `proving more theorems'.
+        If \( \varphi \) is such that \( \mathsf{ZFC} \nvdash \varphi \) and \( \mathsf{ZFC} \nvdash \neg\varphi \), it is not necessarily the case that \( \mathsf{ZFC} <_{\Con} \mathsf{ZFC} + \varphi \) or \( \mathsf{ZFC} <_{\Con} \mathsf{ZFC} + \neg\varphi \).
         For example, \( \mathsf{ZFC} + \mathsf{CH} \), \( \mathsf{ZFC} + \neg\mathsf{CH} \), and \( \mathsf{ZFC} \) are all equiconsistent.
-        \item The second incompleteness theorem shows, for suitably nice theories \( T \), that if \( T \neq \bot \) then \( T <_{\mathrm{Cons}} T + \Con(T) \).
+        \item The second incompleteness theorem shows, for suitably nice theories \( T \), that if \( T \neq \bot \) then \( T <_{\Con} T + \Con(T) \).
         Note that it is possible that \( T \) is consistent but \( T + \Con(T) \) is inconsistent, so the incompleteness theorem does not necessarily give an infinite chain of strict consistency strength inequalities.
         For example, consider
         \[ \mathsf{ZFC}^\dagger = \mathsf{ZFC} + \neg\Con(\mathsf{ZFC}) \]
         Since \( \mathsf{ZFC}^\dagger \supseteq \mathsf{ZFC} \), we must have \( \Con(\mathsf{ZFC}^\dagger) \to \Con(\mathsf{ZFC}) \), but \( \mathsf{ZFC}^\dagger \to \neg\Con(\mathsf{ZFC}) \), so \( \mathsf{ZFC}^\dagger + \Con(\mathsf{ZFC}^\dagger) \) is inconsistent.
     \end{enumerate}
 \end{remark}
 In conclusion,
-\[ \mathsf{ZFC} <_{\mathrm{Cons}} \mathsf{ZFC} + \Con(\mathsf{ZFC}) <_{\mathrm{Cons}} \mathsf{ZFC} + \mathsf{WorC} <_{\mathrm{Cons}} \mathsf{ZFC} + \mathsf{IC} \]
+\[ \mathsf{ZFC} <_{\Con} \mathsf{ZFC} + \Con(\mathsf{ZFC}) <_{\Con} \mathsf{ZFC} + \mathsf{WorC} <_{\Con} \mathsf{ZFC} + \mathsf{IC} \]
 where the second inequality uses the same argument as \( \mathsf{IC} \to \Con(\mathsf{IC} + \Con(\mathsf{IC})) \).
 
-We will see that \( \mathsf{ZFC} \equiv_{\mathrm{Cons}} \mathsf{ZFC} + \neg\mathsf{IC} \).
+We will see that \( \mathsf{ZFC} \equiv_{\Con} \mathsf{ZFC} + \neg\mathsf{IC} \).
 Many large cardinal axioms have this property that their negations are weak.
 
 If \( \kappa \) is the least inaccessible cardinal, then \( \mathrm{V}_\kappa \) is a model of \( \mathsf{ZFC} \), but we can show that it cannot satisfy \( \mathsf{IC} \).
@@ -362,7 +367,7 @@ \subsection{The consistency strength hierarchy}
     \item Observe that if \( S \) proves that there is a transitive model of \( T \), then \( S \vdash \Con(T^\star) \) because consistency statements are downwards absolute between transitive models.
     % Does T need to be computably enumerable?
     \item Note also that if \( S \) proves every axiom of \( T \), then \( \Con(S) \to \Con(T) \).
-    \item If \( T \) is not equiconsistent with \( \bot \), then \( \Con(T) \nrightarrow \Cons(T^\star) \).
+    \item If \( T \) is not equiconsistent with \( \bot \), then \( \Con(T) \nrightarrow \Con(T^\star) \).
 \end{enumerate}
 We can therefore show
 \[ \Con(\mathsf{ZFC} + \neg\mathsf{IC}) \nrightarrow \Con(\mathsf{ZFC} + \mathsf{IC}) \]
@@ -372,7 +377,7 @@ \subsection{The consistency strength hierarchy}
 % Why not just Con(Con(that thing))?
 Hence \( \Con(\mathsf{ZFC} + \neg\mathsf{IC}) \to \Con((\mathsf{ZFC} + \neg\mathsf{IC})^\star) \), so if the given implication were to hold, it would contradict G\"odel's second incompleteness theorem.
 Thus, if \( \mathsf{ZFC} + \neg\mathsf{IC} \) is consistent,
-\[ \mathsf{ZFC} + \neg\mathsf{IC} <_{\mathrm{Cons}} \mathsf{ZFC} + \mathsf{IC} \]
+\[ \mathsf{ZFC} + \neg\mathsf{IC} <_{\Con} \mathsf{ZFC} + \mathsf{IC} \]
 Observe that none of the proofs given in this section work for weakly inaccessible cardinals, so it is not clear that weakly inaccessible cardinals qualify as large cardinals.
 However, that under the generalised continuum hypothesis, we have \( \aleph_\alpha = \beth_\alpha \) and so the notions of weakly inaccessible cardinal and inaccessible cardinal coincide.
 In Part III Forcing and the Continuum Hypothesis, we see that if \( M \vDash \mathsf{ZFC} \), there is \( L \subseteq M \) such that \( L \) is transitive in \( M \), \( L \) contains all the ordinals of \( M \), and \( L \vDash \mathsf{ZFC} + \mathsf{GCH} \).

iii/lc/02.tex → iii/lc/02_measurable_cardinals.tex

@@ -4,7 +4,7 @@ \subsection{The measure problem}
 \begin{enumerate}
     \item \( \mu(\mathbb I) = 1 \) and \( \mu(\varnothing) = 0 \);
     \item (translation invariance) if \( X \subseteq \mathbb I \), \( r \in \mathbb R \), and \( X + r = \qty{x + r \mid x \in X} \subseteq \mathbb I \), then \( \mu(X) = \mu(X + r) \); and
-    \item (additivity) if \( (A_n)_{n \in \mathbb N} \) is a family of pairwise disjoint subsets of \( \mathbb I \), then \( \mu\qty(\bigcup_{n \in \mathbb N} A_n) = \sum_{n \in \mathbb N}\mu(A_n) \).
+    \item (countable additivity) if \( (A_n)_{n \in \mathbb N} \) is a family of pairwise disjoint subsets of \( \mathbb I \), then \( \mu\qty(\bigcup_{n \in \mathbb N} A_n) = \sum_{n \in \mathbb N}\mu(A_n) \).
 \end{enumerate}
 The \emph{measure problem} was the question as to whether such a measure function exists.
 Vitali proved that a measure cannot be defined on all of \( \mathcal P(\mathbb I) \).

iii/lc/main.tex

@@ -10,9 +10,9 @@
 
 \tableofcontentsnewpage{}
 
-\section{???}
-\input{01.tex}
-\section{???}
-\input{02.tex}
+\section{Inaccessible cardinals}
+\input{01_inaccessible_cardinals.tex}
+\section{Measurable cardinals}
+\input{02_measurable_cardinals.tex}
 
 \end{document}