diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex index 35a9042..0f5a565 100644 --- a/inputs/lecture_01.tex +++ b/inputs/lecture_01.tex @@ -1,6 +1,6 @@ \lecture{01}{2023-10-16}{} - -Literature +\gist{% +\paragraph{Literature} \begin{itemize} \item Schindler, Set theory @@ -20,6 +20,7 @@ Literature \item Independence of $\CH$. \end{itemize} \end{itemize} +}{} \section{Naive set theory} diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index bb12c3c..889313b 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -6,6 +6,7 @@ Applications of induction and recursion: such that $x \in t$. \end{fact} \begin{proof} +\gist{% Take $R = \in $. We want a function $F$ with domain $\omega$ such that $F(0) = \{x\}$ @@ -27,16 +28,27 @@ Applications of induction and recursion: \begin{IEEEeqnarray*}{rCl} F(0) &=& \{x\},\\ F(n+1) &=& \bigcup\bigcup \ran(F\defon{n+1})\\ - &=& \bigcup \bigcup \{\{x\}, x, \bigcup x, \ldots, \underbrace{\bigcup^{n-1} x}_{F(n)}\} + &=& \bigcup \bigcup \{\{x\}, x, \bigcup x, \ldots, \underbrace{\bigcup\nolimits^{n-1} x}_{F(n)}\} = \bigcup F(n), \end{IEEEeqnarray*} i.e.~$F(n+1) = \bigcup F(n)$. +}{% + \begin{itemize} + \item Use recursion ($\in $) to define $F\colon \omega \to V$ + such that + \[ + F(0) = \{x\}, F(n+1) = \bigcup F(n). + \] + \item $\{x\} \cup \bigcup \ran(F)$ is as desired. + \end{itemize} +} \end{proof} - +\gist{% \begin{notation} Let $\OR$ denote the class of all ordinals and $V$ the class of all sets. \end{notation} +}{} \begin{lemma} There is a function $F\colon \OR \to V$ such that $F(\alpha) = \bigcup \{\cP(F(\beta)): \beta < \alpha\}$. @@ -65,7 +77,7 @@ Applications of induction and recursion: \end{enumerate} \end{proof} \begin{notation} - Usually, one write $V_\alpha$ for $F(\alpha)$. +Usually, one writes $V_\alpha$ for $F(\alpha)$. They are called the \vocab{rank initial segments} of $V$. \end{notation} \begin{lemma} diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex index 71ff02f..73dd365 100644 --- a/inputs/lecture_12.tex +++ b/inputs/lecture_12.tex @@ -52,15 +52,23 @@ We will very rarely use ordinal arithmetic. \begin{definition} Let $\alpha$, $\beta$ be ordinals. We say that $f\colon \alpha \to \beta$ is \vocab{cofinal} - iff for all $\xi < \beta$, there is some $\eta < \alpha$ - such that $f(\eta) \ge \xi$. + iff + \gist{% + for all $\xi < \beta$, there is some $\eta < \alpha$ + such that $f(\eta) \ge \xi$.% + }{% + \[ + \forall \xi < \beta.~\exists \eta < \alpha.~f(\eta) \ge \xi. + \] + } \end{definition} +\gist{% \begin{remark} If $\beta$ is a limit ordinal, this is equivalent to \[ - \forall \xi < \beta .~\exists \eta \alpha.~f(\eta) > \xi. - \] + \forall \xi < \beta .~\exists \eta < \alpha.~f(\eta) > \xi. + \] \end{remark} \begin{example} \begin{enumerate}[(a)] @@ -79,7 +87,7 @@ We will very rarely use ordinal arithmetic. is cofinal. \end{enumerate} \end{example} - +}{} \begin{definition} Let $\beta$ be an ordinal. The \vocab{cofinality} of $\beta$, @@ -87,7 +95,7 @@ We will very rarely use ordinal arithmetic. is the least ordinal $\alpha$ such that there exists a cofinal $f\colon \alpha \to \beta$. -\end{definition}k +\end{definition} \begin{example} \begin{itemize} \item $\cf(\aleph_\omega) = \omega$. @@ -139,7 +147,7 @@ In particular, a regular ordinal is always a cardinal. Clearly this is cofinal. \end{proof} \begin{warning} - Note that in general, a composition of cofinal map + Note that in general, a composition of cofinal maps is not necessarily cofinal. \end{warning} diff --git a/jrpie-gist.sty b/jrpie-gist.sty new file mode 100644 index 0000000..405fecf --- /dev/null +++ b/jrpie-gist.sty @@ -0,0 +1,14 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{jrpie-gist}[2023/01/22 - gist version for lecture notes] + +% TODO gist info +% TODO link to long version (provide link to main document) + +% TODO \phantomsection to cross link +\newcommand{\gist}[2]{% + \ifcsname EnableGist\endcsname% + #2% + \else% + #1% + \fi% +} diff --git a/logic.sty b/logic.sty index 1484949..594650f 100644 --- a/logic.sty +++ b/logic.sty @@ -9,6 +9,7 @@ \usepackage{mkessler-code} \usepackage{jrpie-math} \usepackage{jrpie-yaref} +\usepackage{jrpie-gist} \usepackage[normalem]{ulem} \usepackage{pdflscape} \usepackage{longtable} @@ -22,6 +23,7 @@ \usepackage{listings} \usepackage{multirow} \usepackage{float} +\usepackage{scalerel} %\usepackage{algorithmicx} \newcounter{subsubsubsection}[subsubsection] @@ -107,14 +109,14 @@ \DeclareSimpleMathOperator{Con} -\DeclareMathOperator{\Zermelo}{Z} -\DeclareSimpleMathOperator{ZF} -\DeclareSimpleMathOperator{ZFC} -\DeclareSimpleMathOperator{BG} -\DeclareSimpleMathOperator{BGC} -\DeclareSimpleMathOperator{HOD} -\DeclareSimpleMathOperator{OD} -\DeclareSimpleMathOperator{AC} +\DeclareMathOperator{\Zermelo}{\mathsf{Z}} +\DeclareMathOperator{\ZF}{\mathsf{ZF}} +\DeclareMathOperator{\ZFC}{\mathsf{ZFC}} +\DeclareMathOperator{\BG}{\mathsf{BG}} +\DeclareMathOperator{\BGC}{\mathsf{BGC}} +\DeclareMathOperator{\HOD}{\mathsf{HOD}} +\DeclareMathOperator{\OD}{\mathsf{OD}} +\DeclareMathOperator{\AC}{\mathsf{AC}} \newcommand{\AxC}{\yarefs{ax:c}} \newcommand{\AxExt}{\yarefs{ax:ext}} % AoE \newcommand{\AxFund}{\yarefs{ax:fund}} % AoF @@ -144,5 +146,5 @@ \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} %\newcommand\diagi{\mathop{\large \Delta}\limits} -\newcommand\diagi{\mathop{\large \Delta}} +\newcommand\diagi{\mathop{\scalerel*{\Delta}{\sum}}} diff --git a/logic2-gist.tex b/logic2-gist.tex new file mode 100644 index 0000000..ad0eca5 --- /dev/null +++ b/logic2-gist.tex @@ -0,0 +1,2 @@ +\def\EnableGist{} +\input{logic2} diff --git a/logic2.tex b/logic2.tex index 047291b..e8feadd 100644 --- a/logic2.tex +++ b/logic2.tex @@ -42,11 +42,17 @@ \input{inputs/lecture_16} \input{inputs/lecture_17} \input{inputs/lecture_18} -\input{inputs/lecture_19} -\input{inputs/lecture_20} -\input{inputs/lecture_21} -\input{inputs/lecture_22} -\input{inputs/lecture_23} +\gist{ + \input{inputs/lecture_19} + \input{inputs/lecture_20} + \input{inputs/lecture_21} + \input{inputs/lecture_22} + \input{inputs/lecture_23} +}{ + \newpage + \section{Forcing} + The rest of the course is not relevant for the exam. +} \cleardoublepage