Restructure LC

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

iii/lc/01_inaccessible_cardinals.tex

@@ -17,8 +17,8 @@ \subsection{Large cardinal properties}
 We begin with some non-examples.
 
 \begin{enumerate}
-    \item \( \kappa \) is called an \emph{\( \aleph \) fixed point} if \( \kappa = \aleph_\kappa \).
-    Note that, for example, \( \omega \), \( \omega_1 \), and \( \aleph_\omega \) are not \( \aleph \) fixed points.
+    \item \( \kappa \) is called an \emph{aleph fixed point} if \( \kappa = \aleph_\kappa \).
+    Note that, for example, \( \omega \), \( \omega_1 \), and \( \aleph_\omega \) are not aleph fixed points.
     However, we have the following result.
     We say that \( F : \mathrm{Ord} \to \mathrm{Ord} \) is \emph{normal} if \( \alpha < \beta \) implies \( F(\alpha) < F(\beta) \), and if \( \lambda \) is a limit, \( F(\lambda) = \bigcup_{\alpha < \lambda} F(\alpha) \).
     One can show that every normal ordinal operation has arbitrarily large fixed points, and in particular that these fixed points may be enumerated by the ordinals.
@@ -63,7 +63,7 @@ \subsection{Large cardinal properties}
     Note that if \( \kappa \) is regular, then if \( \lambda < \kappa \), and for each \( \alpha < \lambda \) we have a set \( X_\alpha \subseteq \kappa \) of size \( \abs{X_\alpha} < \kappa \), then \( \bigcup X_\alpha \neq \kappa \).
     It is easy to show that this property is equivalent to regularity.
 
-    Then \( \omega \) is a regular cardinal.
+    We have therefore shown that \( \omega \) is a regular cardinal.
     Note that \( \aleph_1 \) is also regular, since countable unions of countable sets are countable.
     This argument generalises to all succcessor cardinals, so all successor cardinals \( \aleph_{\alpha + 1} \) are regular.
     The cardinal \( \aleph_\omega \) is not regular, as it is the union of \( \qty{\aleph_n \mid n \in \mathbb N} \), which is a subset of \( \aleph_\omega \) of cardinality \( \aleph_0 \), giving \( \cf(\aleph_\omega) = \aleph_0 \).
@@ -85,7 +85,7 @@ \subsection{Weakly inaccessible and inaccessible cardinals}
 
 Many things in the relationship of \( \mathsf{WI} \) and \( \mathsf{I} \) are unclear: \( 2^{\aleph_0} \) is clearly not inaccessible as it is not a strong limit, but it is not clear that this is not a limit.
 The \emph{generalised continuum hypothesis} \( \mathsf{GCH} \) is that for all cardinals \( \alpha \), we have \( 2^{\aleph_\alpha} = \aleph_{\alpha + 1} \), and so \( \aleph_\alpha = \beth_\alpha \).
-Thus, the notions of limit and strong limit coincide, and so the notions of inaccessible cardinals and weakly inaccessible cardinals coincide.
+Under this assumption, the notions of limit and strong limit coincide, and so the notions of inaccessible cardinals and weakly inaccessible cardinals coincide.
 
 \begin{proposition}
     Weakly inaccessible cardinals are aleph fixed points.
@@ -139,15 +139,15 @@ \subsection{Second order replacement}
 This class function has cofinal range in \( \omega_1 \), and so \( \mathrm{V}_{\omega_1} \) does not satisfy replacement.
 
 We will prove that \( \mathsf{I}(\kappa) \) implies that \( \mathrm{V}_\kappa \) models replacement.
-A set \( M \) is said to satisfy \emph{second-order replacement} \( \mathsf{SOR} \) if for every \( F : M \to M \) and every \( x \in M \), the set \( \qty{F(y) \mid y \in x} \) is in \( M \).
+A set \( M \) is said to satisfy \emph{second-order replacement} \( \mathsf{SOR} \) if for every function \( F : M \to M \) and every \( x \in M \), the set \( \qty{F(y) \mid y \in x} \) is in \( M \).
 Any model of \( \mathrm{V}_\alpha \) that satisfies second-order replacement is a model of \( \mathsf{ZFC} \), as the counterexamples to replacement are special cases of violations of second-order replacement.
 
 \begin{theorem}[Zermelo]
     If \( \kappa \) is inaccessible, then \( \mathrm{V}_\kappa \) satisfies second-order replacement.
 \end{theorem}
 We first prove the following lemmas.
 \begin{lemma}
-    If \( \kappa \) is inaccessible and \( \lambda < \kappa \), then \( \abs{\mathrm{V}_\kappa} < \kappa \).
+    If \( \kappa \) is inaccessible and \( \lambda < \kappa \), then \( \abs{\mathrm{V}_\lambda} < \kappa \).
 \end{lemma}
 \begin{proof}
     This follows by induction.
@@ -199,7 +199,7 @@ \subsection{Countable transitive models of set theory}
 Suppose \( \kappa \) is inaccessible, so \( \mathrm{V}_\kappa \vDash \mathsf{ZFC} \).
 A standard model-theoretic argument shows there is a countable elementary substructure \( (N, \in) \preceq (\mathrm{V}_\kappa, \in) \).
 In particular, \( (N, \in) \vDash \mathsf{ZFC} \).
-The proof of the downwards L\"owenheim--Skolem theorem is a Skolem hull construction, given by
+The proof of the downwards L\"owenheim--Skolem theorem that we will use is a Skolem hull construction, given by
 \[ N_0 = \varnothing;\quad N_{k+1} = N_k \cup W(N_k);\quad N = \bigcup_{k \in \mathbb N} N_k \]
 where \( W(N_k) \) is a set of witnesses for all formulas of the form \( \exists x.\, \varphi \) with parameters in \( N_k \).
 The fact that this is an elementary substructure follows from the Tarski--Vaught test.