Set of all countably infinite strings over a finite alphabet >1
Take the alphabet $\Sigma = \{0,1\}$ and consider strings whose positions are indexed by&nbsp;$\mathbb{N}$ (also known as $\omega$-words).  There is a one-to-one correspondence between these strings and subsets of&nbsp;$\mathbb{N}$ given by $x_0x_1x_2\dots \leftrightarrow \{i\in\mathbb{N}\mid x_i=1\}$. Therefore, there are as many $\omega$-words as there are subsets of&nbsp;$\mathbb{N}$, and this is an uncountable number.

This holds for any countably infinite word length.