Cardinal numbers with countable cofinality

Question

What does this assumption mean:

Let $k$ be any cardinal number with uncountable cofinality

Which cardinals have countable cofinality?

I know the definition of cofinality, but I'd like to see some examples of both countable and uncountable cofinality.

Answer 1

Let $\{ \kappa_n : n < \omega \}$ be any increasing sequence of cardinals, then $\sup_n \kappa_n$ is a cardinal of countable cofinality.

For example, $\aleph_0 = \sup_n n$ is an infinite cardinal of countable cofinality, and $\aleph_{\omega} = \sup_n \aleph_n$ is another, but $\aleph_1$ is not (proving this is a standard exercise).

Answer 2

$\aleph_\omega=\sup\{\aleph_n\mid n<\omega\}$ has countable cofinality.

More generally, if $\delta$ is a limit ordinal, then $\aleph_\delta$ has the same cofinality as $\delta$ (any cofinal sequence in $\delta$ can be translated to a cofinal sequence below $\aleph_\delta$). So any limit ordinal $\delta$ with countable cofinality would be mapped to a cardinal with a countable cofinality, $\aleph_\delta$.