标签：varepsilon 大数 lim beta 数学 nolimits alpha omega
序数
定义
\[0=\emptyset
\]
\[n+1=n\cup\{n\}
\]
\[\alpha\le \beta=\alpha\subseteq \beta
\]
\[\exists\omega:0\in\omega,n\in\omega\implies n+1\in\omega
\]
\[\sup X=\bigcup\nolimits_{\alpha\in X}\alpha
\]
\[\lim\nolimits_{\varepsilon\to\alpha}\gamma_\varepsilon=\sup\{\gamma_\varepsilon|\varepsilon<\alpha\}
\]
加法
\[\alpha+0=\alpha
\]
\[\alpha+(\beta+1)=(\alpha+\beta)+1
\]
\[\alpha+\beta=\lim\nolimits_{\varepsilon\to\beta}\alpha+\varepsilon
\]
乘法
\[\alpha\cdot0=0
\]
\[\alpha\cdot(\beta+1)=\alpha\cdot\beta+\alpha
\]
\[\alpha+\beta=\lim\nolimits_{\varepsilon\to\beta}\alpha\cdot\varepsilon
\]
乘方
\[\alpha^0=1
\]
\[\alpha^{\beta+1}=\alpha^\beta\cdot\alpha
\]
\[\alpha+\beta=\lim\nolimits_{\varepsilon\to\beta}\alpha^\varepsilon
\]
其他定义
\[\varepsilon_0:=\lim\nolimits_{\varepsilon\to\omega}\alpha_\varepsilon,\alpha_0=\omega,\alpha_{n+1}=\omega^{\alpha_n}
\]
快速增长层级 (Fast Growing Hierarchy)
标签：varepsilon,
大数,
lim,
beta,
数学,
nolimits,
alpha,
omega
From： https://www.cnblogs.com/JerryTcl/p/16850650.html