Number of finite strings over a countably infinite alphabet
It's countable.  The set&nbsp;$S_\ell$ of strings of length&nbsp;$\ell$ is $\Sigma\times\dots\times\Sigma$, which is a finite product of countable sets, so is countable. Now, the set of all finite strings is $\bigcup_{\ell\geq 0}S_\ell$, which is a countable union of countable sets, which is countable.

Usually, you can only get an uncountable set from countable ones by taking the power set.