Create \ran macro

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

iii/forcing/01_set_theory.tex

@@ -338,7 +338,7 @@ \subsection{The L\'evy hierarchy}
     For the other direction, let \( X \subseteq a \) be a nonempty subset, and consider the pointwise image \( f''X \).
     This has a minimal element \( \alpha \), then for any \( z \in X \), if \( f(z) = \alpha \) then for all \( y \in X \), we have \( f(y) \geq \alpha \), so \( \langle y, z \rangle \notin r \).
     We then define well-foundedness with a \( \Sigma_1 \) formula as follows.
-    \[ \exists f.\, \qty(f \text{ is a function} \wedge \forall u \in \operatorname{ran} f.\, (u \in \mathrm{Ord}) \wedge \forall x y \in a.\, (\langle y, x \rangle \in r \to f(y) \in f(x))) \]
+    \[ \exists f.\, \qty(f \text{ is a function} \wedge \forall u \in \ran f.\, (u \in \mathrm{Ord}) \wedge \forall x y \in a.\, (\langle y, x \rangle \in r \to f(y) \in f(x))) \]
 \end{proof}
 \begin{proposition}
     The following are \( \Delta_0^{\mathsf{ZF}} \).
@@ -614,7 +614,7 @@ \subsection{Cardinal arithmetic}
     \[ \kappa^{\lambda + \mu} = \kappa^\lambda \cdot \kappa^\mu;\quad (\kappa^\lambda)^\mu = \kappa^{\lambda \cdot \mu} \]
 \end{lemma}
 \begin{definition}
-    A map between ordinals \( \alpha \to \beta \) is \emph{cofinal} if \( \sup \operatorname{ran} f = \beta \).
+    A map between ordinals \( \alpha \to \beta \) is \emph{cofinal} if \( \sup \ran f = \beta \).
     The \emph{cofinality} of an ordinal \( \gamma \), written \( \cf(\gamma) \), is the least ordinal that admits a cofinal map to \( \gamma \).
     A limit ordinal \( \gamma \) is \emph{singular} if \( \cf(\gamma) < \gamma \), and \emph{regular} if \( \cf(\gamma) = \gamma \).
 \end{definition}

iii/forcing/02_constructibility.tex

@@ -99,7 +99,7 @@ \subsection{G\"odel functions}
         \item \( \mathcal F_3(x, y) = x \setminus y \);
         \item \( \mathcal F_4(x, y) = x \times y \);
         \item \( \mathcal F_5(x, y) = \dom x = \qty{\pi_1(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
-        \item \( \mathcal F_6(x, y) = \operatorname{ran} x \qty{\pi_2(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
+        \item \( \mathcal F_6(x, y) = \ran x \qty{\pi_2(z) \mid z \in x \wedge z \text{ is an ordered pair}} \);
         \item \( \mathcal F_7(x, y) = \qty{\langle u, v, w \rangle \mid \langle u, v \rangle \in x, w \in y} \);
         \item \( \mathcal F_8(x, y) = \qty{\langle u, w, v \rangle \mid \langle u, v \rangle \in x, w \in y} \);
         \item \( \mathcal F_9(x, y) = \qty{\langle v, u \rangle \in y \in x \mid u = v} \);
@@ -166,7 +166,7 @@ \subsection{G\"odel functions}
         Then
         \begin{align*}
             \mathcal F_\varphi(a_1, \dots, a_{n-1}) &= \qty{\langle x_{n-1}, \dots, x_1 \rangle \in a_{n-1} \times \dots \times a_1 \mid \varphi(x_1, \dots, x_{n-1})} \\
-            &= \operatorname{ran}(\qty{\langle 0, x_{n-1}, \dots, x_1 \rangle \in \qty{0} \times a_{n-1} \times \dots \times a_1 \mid \theta(x_1, \dots, x_{n-1}, 0)}) \\
+            &= \ran(\qty{\langle 0, x_{n-1}, \dots, x_1 \rangle \in \qty{0} \times a_{n-1} \times \dots \times a_1 \mid \theta(x_1, \dots, x_{n-1}, 0)}) \\
             &= \mathcal F_6(\mathcal F_\theta(a_1, \dots, a_{n-1}), \mathcal F_1(\mathcal F_3(a_1, a_1), \mathcal F_3(a_1, a_1)), a_1)
         \end{align*}
         \item If \( \psi(x_1, \dots, x_n) \) is a termed formula and
@@ -241,7 +241,7 @@ \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) &= \operatorname{ran}\operatorname{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 \\&\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_6(\mathcal F_6(\mathcal F_\psi(a_1, \dots, a_n), a_1), a_1)
         \end{align*}
     \end{itemize}
@@ -261,7 +261,7 @@ \subsection{G\"odel functions}
     \end{align*}
     So
     \begin{align*}
-        \operatorname{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)} \\
+        \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})}
     \end{align*}
 \end{proof}
@@ -342,7 +342,7 @@ \subsection{The axiom of constructibility}
     \[ \forall W.\, \qty(M \cup \qty{M} \subseteq W \wedge \forall x, y \in W.\, \bigwedge_{i \leq 10} \mathcal F_i(x, y) \in W) \to Z \subseteq W \]
     The \( \Sigma_1 \) definition will use the inductive definition of the closure.
     \begin{align*}
-        \exists W.\, W \text{ is a function} &\wedge \dom W = \omega \wedge Z = \bigcup \operatorname{ran} W \\
+        \exists W.\, W \text{ is a function} &\wedge \dom W = \omega \wedge Z = \bigcup \ran W \\
         &\wedge W(0) = M \wedge W(n) \subseteq W(n+1) \\
         &\wedge \qty(\forall x, y \in W(n).\, \bigwedge_{i \leq 10} \mathcal F_i(x, y) \in W(n+1)) \\
         &\wedge \qty(\forall z \in W(n+1).\, \exists x, y \in W(n).\, \bigvee_{i \leq 10} z = \mathcal F_i(x, y))

iii/forcing/03_forcing.tex

@@ -90,7 +90,7 @@ \subsection{Forcing posets}
 \end{proposition}
 \begin{example}
     For sets \( I, J \), we let \( \operatorname{Fn}(I, J) \) denote the set of all finite partial functions from \( I \) to \( J \).
-    \[ \operatorname{Fn}(I, J) = \qty{p \mid \abs{p} < \omega \wedge p \text{ is a function} \wedge \dom p \subseteq I \wedge \operatorname{ran} p \subseteq J} \]
+    \[ \operatorname{Fn}(I, J) = \qty{p \mid \abs{p} < \omega \wedge p \text{ is a function} \wedge \dom p \subseteq I \wedge \ran p \subseteq J} \]
     We let \( \leq \) be the reverse inclusion on \( \operatorname{Fn}(I, J) \), so \( q \leq p \) if and only if \( q \supseteq p \).
     The maximal element \( \Bbbone \) is the empty set.
     Then \( (\operatorname{Fn}(I, J), \leq, \varnothing) \) is a forcing poset, and moreover, the preorder is separative.
@@ -204,7 +204,7 @@ \subsection{Dense sets and genericity}
     Then for all \( i \in I \) and \( j \in J \), the following are dense.
     \begin{enumerate}
         \item \( D_i = \qty{q \in \operatorname{Fn}(I, J) \mid i \in \dom q} \);
-        \item \( R_j = \qty{q \in \operatorname{Fn}(I, J) \mid j \in \operatorname{ran} q} \).
+        \item \( R_j = \qty{q \in \operatorname{Fn}(I, J) \mid j \in \ran q} \).
     \end{enumerate}
 \end{example}
 \begin{definition}
@@ -277,7 +277,7 @@ \subsection{Names}
 \( \mathbb P \)-names are denoted with overdots, such as in \( \dot x \).
 \begin{definition}
     The \emph{range} of a \( \mathbb P \)-name \( \dot x \) is
-    \[ \operatorname{ran}(\dot x) = \qty{\dot y \mid \exists p \in \mathbb P \mid \langle p, \dot y \rangle \in \dot x} \]
+    \[ \ran(\dot x) = \qty{\dot y \mid \exists p \in \mathbb P \mid \langle p, \dot y \rangle \in \dot x} \]
 \end{definition}
 \begin{remark}
     Alternatively, by transfinite recursion on rank, we could define the class of \( \mathbb P \)-names over \( \mathrm{V} \) in the following way.

iii/lc/01_inaccessible_cardinals.tex

@@ -51,7 +51,7 @@ \subsection{Large cardinal properties}
     Cantor's theorem shows that \( \aleph_\alpha \leq \beth_\alpha \), and the continuum hypothesis is the assertion that \( \aleph_1 = \beth_1 \).
     Note that \( \kappa \) is a strong limit cardinal if and only if \( \kappa = \beth_\lambda \) for some limit ordinal \( \lambda \).
     In particular, \( \mathsf{ZFC} \vdash \Sigma \mathsf{C} \).
-    \item If \( s : n \to \omega \) for \( n < \omega \), then \( \sup(s) = \bigcup \operatorname{ran}(s) < \omega \).
+    \item If \( s : n \to \omega \) for \( n < \omega \), then \( \sup(s) = \bigcup \ran(s) < \omega \).
     This gives rise to the following definition.
     \begin{definition}
         Let \( \lambda \) be a limit ordinal.

iii/lc/main.tex

@@ -15,4 +15,26 @@ \section{Inaccessible cardinals}
 \section{Measurable cardinals}
 \input{02_measurable_cardinals.tex}
 
+\newpage
+\section*{Diagram of large cardinal properties}
+Under suitable consistency assumptions, large cardinal properties that appear in higher positions on this diagram have strictly higher consistency strength than properties appearing lower down the diagram.
+% https://q.uiver.app/#q=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
+\[\begin{tikzcd}
+	& {\mathsf{SC}(\kappa)} \\
+	{\mathsf{Ulam}(\kappa)} & {\mathsf{M}(\kappa)} & {\mathsf{RVM}(\kappa)} \\
+	{\mathsf{WC}(\kappa)} & {\mathsf{W}(\kappa)} \\
+	& {\mathsf{I}(\kappa)} & {\mathsf{WI}(\kappa)} \\
+	& {\mathsf{Wor}(\kappa)}
+	\arrow[from=1-2, to=2-2]
+	\arrow[from=2-2, to=2-3]
+	\arrow[from=2-3, to=4-3]
+	\arrow[from=2-2, to=3-2]
+	\arrow[from=3-2, to=4-2]
+	\arrow[from=4-2, to=4-3]
+	\arrow[from=4-2, to=5-2]
+	\arrow[tail reversed, from=3-1, to=3-2]
+	\arrow[shift left=2, from=2-2, to=2-1]
+	\arrow["{\text{min.}}", shift left=2, from=2-1, to=2-2]
+\end{tikzcd}\]
+
 \end{document}

util.sty

@@ -223,6 +223,7 @@
 \DeclareMathOperator{\res}{res}
 \DeclareMathOperator{\Con}{Con}
 \DeclareMathOperator{\cf}{cf}
+\DeclareMathOperator{\ran}{ran}
 % https://github.com/wspr/unicode-math/issues/457
 \AtBeginDocument{%
 \newcommand{\dashrightarrow}{\mathrel{\rightdasharrow}}