limit cardinal LimitCardinal 2003-12-01 06:32:53 2007-01-07 11:53:43 Definition cardinal number strong limit cardinal \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{case} \PMlinkescapeword{cases} \PMlinkescapeword{even} A {\em limit cardinal} is a cardinal $\kappa$ such that $\lambda^+<\kappa$ for every cardinal $\lambda<\kappa$. Here $\lambda^+$ denotes the cardinal successor of $\lambda$. If $2^\lambda<\kappa$ for every cardinal $\lambda<\kappa$, then $\kappa$ is called a {\em strong limit cardinal}. Every strong limit cardinal is a limit cardinal, because $\lambda^+\leq2^\lambda$ holds for every cardinal $\lambda$. Under GCH, every limit cardinal is a strong limit cardinal because in this case $\lambda^+=2^\lambda$ for every infinite cardinal $\lambda$. The three smallest limit cardinals are $0$, $\aleph_0$ and $\aleph_\omega$. Note that some authors do not count $0$, or sometimes even $\aleph_0$, as a limit cardinal. An infinite cardinal $\aleph_\alpha$ is a limit cardinal if and only if $\alpha$ is a limit ordinal (counting $0$ as a limit ordinal).