limit cardinal

A limit cardinal is a cardinal $\kappa$ such that for every cardinal . Here ${\lambda }^{+}$ denotes the cardinal successor of $\lambda$. If for every cardinal , then $\kappa$ is called a strong limit cardinal.

Every strong limit cardinal is a limit cardinal, because ${\lambda }^{+}\le 2^{\lambda }$ 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 $0$, ${\mathrm{\aleph }}_0$ and ${\mathrm{\aleph }}_{\omega }$.Note that some authors do not count $0$, or sometimes even ${\mathrm{\aleph }}_0$, as a limit cardinal.An infinite cardinal ${\mathrm{\aleph }}_{\alpha }$ is a limit cardinalif and only if $\alpha$ is either $0$ or a limit ordinal.