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
Canonical name	LimitCardinal
Date of creation	2013-03-22 14:04:40
Last modified on	2013-03-22 14:04:40
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	15
Author	yark (2760)
Entry type	Definition
Classification	msc 03E10
Related topic	SuccessorCardinal
Defines	strong limit cardinal