More fixes

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

iii/lc/02_medium_large_cardinals.tex

@@ -217,7 +217,7 @@ \subsection{Weakly compact cardinals}
 
     If \( 2^\lambda \geq \kappa \), there is a family of pairwise distinct elements \( (f_\alpha)_{\alpha < \kappa} \) of \( B_\lambda \) of length \( \kappa \).
     Define a colouring \( c \) of \( \kappa \) as follows.
-    A pair \( \alpha, \beta \) is red if the truth value of \( \alpha < \beta \) is the same as the truth value of \( f_\alpha \leq_{\mathrm{lex}} f_\beta \).
+    A pair \( \alpha, \beta \) is red if the truth value of \( \alpha \leq \beta \) is the same as the truth value of \( f_\alpha \leq_{\mathrm{lex}} f_\beta \).
     A pair is blue otherwise.
     Let \( H \) be a monochromatic set for \( c \).
     If \( H \) is red, then \( f_\alpha \) forms a \( \leq_{\mathrm{lex}} \)-increasing sequence of length \( \kappa \).
@@ -336,4 +336,3 @@ \subsection{Strongly compact cardinals}
     This is a \( \kappa \)-complete filter on \( \kappa \).
     If \( U \) extends \( F \) then \( U \) must be nonprincipal, so by the Keisler--Tarski theorem, \( F \) can be extended to a \( \kappa \)-complete nonprincipal ultrafilter on \( \kappa \) as required.
 \end{proof}
-

iii/lc/04_large_large_cardinals.tex

@@ -189,7 +189,7 @@ \subsection{The upper limit}
     \begin{enumerate}
         \item The combinatorial lemma was proven using \( \mathsf{AC} \), and it is not known whether the proof works without it.
         \item To prove Kunen's lemma, we did not need that \( \lambda \) is inaccessible.
-        More explicitly, if \( j : \mathrm{V}_\alpha \to M \) is an elementary embedding with critical point \( \kappa \) such that \( \hat\alpha + 2 \leq \alpha \) (to guarantee that \( f \in \mathrm{V}_\alpha \)), then \( \mathrm{V}_{\hat\kappa + 1} \nsubseteq M \).
+        More explicitly, if \( j : \mathrm{V}_\alpha \to M \) is an elementary embedding with critical point \( \kappa \) such that \( \hat\kappa + 2 \leq \alpha \) (to guarantee that \( f \in \mathrm{V}_\alpha \)), then \( \mathrm{V}_{\hat\kappa + 1} \nsubseteq M \).
     \end{enumerate}
 \end{remark}
 \begin{corollary}

iii/lc/main.tex

@@ -64,8 +64,8 @@ \section*{Diagram of large cardinal properties}
 % SC -> strong on ES3 46--48, shows how to make strongly compact cardinals embedding cardinals
 
 The `small large cardinals' are usually considered those cardinals consistent with \( \mathrm{V} = \mathrm{L} \), and such large cardinal properties are usually downwards absolute.
-Note that \( L \) has no measurable cardinals.
-Indeed, if \( \mathrm{V} = \mathrm{L} \) and \( U \) is a \( \kappa \)-complete nonprincipal ultrafilter on \( \kappa \), then the ultrapower embedding \( j_U : L \to M \) must map to an inner model strictly smaller than \( L \), but such an inner model cannot exist.
+Note that \( \mathrm{L} \) has no measurable cardinals.
+Indeed, if \( \mathrm{V} = \mathrm{L} \) and \( U \) is a \( \kappa \)-complete nonprincipal ultrafilter on \( \kappa \), then the ultrapower embedding \( j_U : \mathrm{L} \to M \) must map to an inner model strictly smaller than \( \mathrm{L} \), but such an inner model cannot exist.
 
 There are certain large cardinals called \emph{Woodin cardinals} which sit between strong and strongly compact cardinals.
 They represent another boundary between sizes of large cardinal axioms, just like measurable cardinals; smaller large cardinals are sometimes called `medium-sized large cardinals', and the others are called `large large cardinals'.