From 9a3359dfd0b853574eff207710187fdad310800b Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Mon, 27 Nov 2023 16:07:23 +0100 Subject: [PATCH] lecture 12 --- inputs/lecture_12.tex | 229 ++++++++++++++++++++++++++++++++++++++++++ logic.sty | 2 + logic2.tex | 1 + 3 files changed, 232 insertions(+) create mode 100644 inputs/lecture_12.tex diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex new file mode 100644 index 0000000..7a44b32 --- /dev/null +++ b/inputs/lecture_12.tex @@ -0,0 +1,229 @@ +\lecture{12}{2023-11-27}{} + +\subsection{Ordinal arithmetic} + +We define $+$, $\cdot $ and exponentiation +for ordinals as follows: + +Fix an ordinal $\beta$. +We recursively define +\begin{IEEEeqnarray*}{rClClr} + \beta &+& 0 &\coloneqq & \beta\\ + \beta &+& (\alpha + 1)&\coloneqq &(\beta + \alpha),\\ + \beta &+& \lambda &\coloneqq & \sup_{\alpha < \lambda} \beta + \alpha &~ ~\text{for limit ordinals $\lambda$} +\end{IEEEeqnarray*} +(Recall that $\alpha + 1 = \alpha \cup \{\alpha\}$ +was already defined.) +\begin{IEEEeqnarray*}{rClClr} + \beta &\cdot& 0 &\coloneqq& 0,\\ + \beta &\cdot& (\alpha+1) &\coloneqq & \beta\cdot \alpha + \beta,\\ + \beta &\cdot& \lambda &\coloneqq &\sup_{\alpha < \lambda} \beta\cdot \alpha &~ ~\text{for limit ordinals $\lambda$} +\end{IEEEeqnarray*} +and +\begin{IEEEeqnarray*}{rClr} + \beta^0 &\coloneqq & 1,\\ + \beta^{\alpha+1} &\coloneqq & \beta^{\alpha} \cdot \beta,\\ + \beta^{\lambda} &\coloneqq & \sup_{\alpha < \lambda} \beta^{\alpha} &~ ~\text{for limit ordinals $\lambda$}. +\end{IEEEeqnarray*} + +\begin{example} + \leavevmode + \begin{itemize} + \item $2+ 2 =4$, + \item $1 + \omega = \sup_{n < \omega} 1 + n = \omega \neq \omega +1$, + \item $2 \cdot \omega = \sup_{n < \omega} 2\cdot n = \omega$, + \item $\omega \cdot 2 =\omega \cdot 1 + \omega = \omega + \omega$.. + \end{itemize} +\end{example} + + +\begin{warning} + Cardinal arithmetic and ordinal arithmetic are very different! + The symbols are the same, but usually we will distinguish + between the two by the symbols used for variables + (i.e.~$\alpha,\beta, \omega, \omega_1$ are viewed primarily as ordinals + and $\kappa,\lambda, \aleph_\alpha$ as cardinals). +\end{warning} + +We will very rarely use ordinal arithmetic. + +\subsection{Cofinality} + +\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 $f(\eta) \ge \xi$. +\end{definition} +\begin{remark} + If $\beta$ is a limit ordinal, + this is equivalent to + \[ + \forall \xi < \beta .~\exists \eta \alpha.~f(\eta) > \xi. + \] +\end{remark} +\begin{example} + \begin{enumerate}[(a)] + \item Look at $\omega + \omega$. + \begin{IEEEeqnarray*}{rCl} + f\colon \omega &\longrightarrow & \omega + \omega \\ + n &\longmapsto & \omega + n + \end{IEEEeqnarray*} + is cofinal. + \item Look at $\aleph_\omega$. + Then + \begin{IEEEeqnarray*}{rCl} + f\colon \omega &\longrightarrow & \aleph_{\omega} \\ + n &\longmapsto & \aleph_n + \end{IEEEeqnarray*} + is cofinal. + \end{enumerate} +\end{example} + +\begin{definition} + Let $\beta$ be an ordinal. + The \vocab{cofinality} of $\beta$, + denoted $\cf(\beta)$, + is the least ordinal $\alpha$ + such that there exists a cofinal + $f\colon \alpha \to \beta$. +\end{definition}k +\begin{example} + \begin{itemize} + \item $\cf(\aleph_\omega) = \omega$. + In fact $\cf(\aleph_{\lambda}) \le \lambda$ + for limit ordinals $\lambda$ + (consider $\alpha \mapsto \aleph_\alpha$). + \item $\cf(\aleph_{\omega + \omega}) = \omega$. + \end{itemize} +\end{example} +\begin{lemma} + For any ordinal $\beta$, + $\cf(\beta)$ is a cardinal. +\end{lemma} +\begin{proof} + Let $f\colon \alpha \to \beta$ be cofinal. + Then $\tilde{f}\colon |\alpha| \to \beta$, + the composition with $\alpha \leftrightarrow|\alpha|$ + is cofinal as well and $|\alpha| \le \alpha$. +\end{proof} + +\begin{question} + How does one imagine ordinals with + cofinality $> \omega$? +\end{question} +No idea. + +\begin{definition} + An ordinal $\beta$ is \vocab{regular} + iff $\cf(\beta) = \beta$. + Otherwise $\beta$ is called \vocab{singular}. +\end{definition} +In particular, a regular ordinal is always a cardinal. +\begin{lemma} + Let $\beta$ be an ordinal + Then $\cf(\beta)$ is a regular cardinal, + i.e. + \[\cf(\cf(\beta)) = \cf(\beta).\] +\end{lemma} +\begin{proof} + Suppose not. + Let $f\colon \cf(\beta) \to \beta$ be cofinal + and $g\colon \cf(\cf(\beta)) \to \cf(\beta)$. + + Consider + \begin{IEEEeqnarray*}{rCl} + h\colon \cf(\cf(\beta)) &\longrightarrow & \beta \\ + \eta &\longmapsto & \sup \{f(\xi): \xi \le g(\eta)\} < \beta. + \end{IEEEeqnarray*} + Clearly this is cofinal. +\end{proof} +\begin{warning} + Note that in general, a composition of cofinal map + is not necessarily cofinal. +\end{warning} + +\begin{theorem} + Let $\kappa > \aleph_0$. + Then $\kappa^+$ is regular. +\end{theorem} +\begin{proof} + Suppose that $\cf(\kappa^+) < \kappa^+$. + Then $\cf(\kappa^+) \le \kappa$, + i.e.~there is a cofinal function $f\colon \kappa \to \kappa^+$. + By the axiom of choice, + there is a function $g$ with domain $\kappa$, + such that $g(\eta)\colon \kappa \twoheadrightarrow f(\eta)$ is onto. + Now define + \begin{IEEEeqnarray*}{rCl} + h\colon \kappa\times \kappa &\longrightarrow & \kappa^+ \\ + (\eta, \xi) &\longmapsto & g(\eta)(\xi). + \end{IEEEeqnarray*} + Clearly this is surjective, + but $|\kappa \times \kappa| < \kappa^+$, + by \yaref{thm:hessenberg}. +\end{proof} + +\begin{itemize} + \item $\aleph_0, \aleph_1, \aleph_2, \ldots$ are regular, + \item $\aleph_\omega$ is singular, + \item $\aleph_{\omega + 1}, \aleph_{\omega + 2}, \ldots$ are regular, + \item $\aleph_{\omega + \omega}$ is singular, + \item $\aleph_{\omega + \omega + 1}, \ldots$ are regular, + \item $\aleph_{\omega + \omega + \omega}$ is singular, + \item $\ldots$ + \item $\aleph_{\omega_1}$ is singular, + \item $\aleph_{\omega_1 + 1}, \ldots$ is regular, + \item $\aleph_{\omega_2}$ is singular. +\end{itemize} + +\begin{question}[Hausdorff] + Is there a regular limit cardinal? +\end{question} +Maybe. This is independent of $\ZFC$. + + +\begin{theorem}[Hausdorff] + \[ + \aleph_{\alpha+1}^{\aleph_\beta} = \aleph_\alpha^{\aleph_\beta} \cdot \aleph_{\alpha+1}. + \] +\end{theorem} +\begin{proof} + Recall that + \begin{IEEEeqnarray*}{rCl} + \aleph_{\alpha+1}^{\aleph_\beta} &=& |{}^{\aleph_\beta} \aleph_{\alpha+1}|. + \end{IEEEeqnarray*} + \begin{itemize} + \item First case: $\beta \ge \alpha+1$. + Then + \[ + \aleph_{a+1}^{\aleph_\beta} \le \aleph_{\beta}^{\aleph_\beta} + \le \left( 2^{\aleph_{\beta}} \right)^{\aleph_\beta} + = 2^{\aleph_{\beta} \cdot \aleph_\beta} = 2^{\aleph_\beta} \le \aleph_{\alpha+1}^{\aleph_\beta}. + \] + Also $\aleph_\alpha^{\aleph_{\beta} = 2^{\aleph_\beta}}$ + in this case, + so + \[ + \aleph_{\alpha+1}^{\aleph_{\beta}} = 2^{\aleph_{\beta}} + = \aleph_{\alpha}^{\aleph_\beta} = \aleph_{\alpha}^{\aleph_\beta} \cdot \aleph_{\alpha+1}. + \] + \item Second case: + Suppose $\beta < \alpha+1$. + By case hypothesis and because $\aleph_{\alpha+1}$ + is regular, + no $f\colon \aleph_{\beta} \to \aleph_{\alpha+1}$ + is unbounded. + So + \[ + {}^{\aleph_{\beta}}\aleph_{\alpha+1} = \bigcup_{\xi < \aleph_{\alpha+1}} {}^{\aleph_\beta} \xi + \] + for each $\xi < \aleph_{\alpha+1}$, $|\xi| \le \aleph_\alpha$, + hence + \[ + |{}^{\aleph_{\beta}}\xi| \le \aleph_\alpha^{\aleph_\beta} + \] + for each $\xi < \aleph_{\alpha+1}$. + Therefore, \[\aleph_{\alpha+1}^{\aleph_\beta} \le \aleph_{\alpha+1} \cdot \aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\alpha+1}^{\aleph_\beta}.\] + \end{itemize} +\end{proof} diff --git a/logic.sty b/logic.sty index af1feb3..21839af 100644 --- a/logic.sty +++ b/logic.sty @@ -139,5 +139,7 @@ \DeclareSimpleMathOperator{rk} \DeclareSimpleMathOperator{otp} +\DeclareSimpleMathOperator{cf} + \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} diff --git a/logic2.tex b/logic2.tex index e7a1b7e..a951d71 100644 --- a/logic2.tex +++ b/logic2.tex @@ -35,6 +35,7 @@ \input{inputs/lecture_09} \input{inputs/lecture_10} \input{inputs/lecture_11} +\input{inputs/lecture_12} \cleardoublepage