limit cardinal

A 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 strong limit cardinal.

Every strong limit cardinal is a limit cardinal, because $\lambda^{+}\leq 2^{\lambda}$ 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 either $0$ or a limit ordinal.

Title limit cardinal LimitCardinal 2013-03-22 14:04:40 2013-03-22 14:04:40 yark (2760) yark (2760) 15 yark (2760) Definition msc 03E10 SuccessorCardinal strong limit cardinal