PlanetMath: limit cardinal

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limit cardinal

(Definition)

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 $\lambda^+\leq2^\lambda$ 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).

"limit cardinal" is owned by yark.

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