集合論のメモ

$ \newcommand{\exi}{\exists\,} \newcommand{\all}{\forall} \newcommand{\equ}{\!=\!} \newcommand{\nequ}{\!\neq\!} \newcommand{\amp}{\;\&\;} \renewcommand{\Set}[2]{\left\{\;#1\mathrel{}\middle|\mathrel{}#2\;\right\}} \newcommand{\parenth}[1]{\left(\;#1\;\right)} \newcommand{\braces}[1]{\left\{\;#1\;\right\}} \newcommand{\bracket}[1]{\left[\;#1\;\right]} \newcommand{\godel}[1]{\left\ulcorner #1 \right\urcorner} $

$\text{cof}(\beta, \alpha) :\Leftrightarrow \exi f:\beta \to \alpha \left\{ \begin{array}{} & \all \alpha_0 \in \alpha \exi \beta_0 \in \beta \;\; \alpha_0 \lt f`\beta_0 \\ \land & \all \beta_0, \beta_1 \in \beta \;[ \beta_0 \in \beta_1 \Leftrightarrow f`\beta_0 \in f`\beta_1] \end{array}\right.$
$\text{cf}(\alpha) :\equiv \min_{\beta} \text{cof}(\beta, \alpha)$
$\aleph(\alpha) \text{は弱到達不可能基数} :\Leftrightarrow \left\{\begin{array}{} & \alpha \in \text{Lim} \\ \land & \text{cf}(\aleph(\alpha)) \equ \aleph(\alpha) \end{array}\right.$
$\aleph(\alpha) \text{は弱到達不可能基数} \Leftrightarrow \bracket{ \text{cf}(\alpha) \equ \alpha \equ \aleph(\alpha)}$
$\begin{array}{} & \aleph(\alpha) \text{は強到達不可能基数} \\ & \text{Inac}(\aleph(\alpha)) \end{array} :\Leftrightarrow \left\{\begin{array}{} & 0 \lt \alpha \\ \land & \text{cf}(\aleph(\alpha)) \equ \aleph(\alpha) \\ \land & \all \beta \lt \alpha \;\; 2^{\aleph(\beta)} \lt \aleph(\alpha) \end{array}\right.$

1. $\text{Inac}(\aleph(\alpha)) \Rightarrow \all u \in V_{\aleph(\alpha)} \; \all f : u \to V_{\aleph(\alpha)} \;\; \text{Ran}(f) \in V_{\aleph(\alpha)}$


2. $\all \alpha \;\; \text{cf}(\aleph(\alpha+1)) \equ \aleph(\alpha+1)$


3. $\all \alpha \in \text{Lim} \;\; \text{cf}(\aleph(\alpha)) \equ cf(\alpha)$


4. $\text{cf}(\alpha+\beta) \equ \text{cf}(\beta)$