>$\Sigma^{*}$, the set of all strings over an alphabet $\Sigma=\{0,1\}$ is countable
>the set of all infinite binary strings isn't
schizo nonsense. you can easily map one to the other. for every string s in $\Sigma^*$, just take a string of infinite 0s and replace the bottommost 0s with s and then put a 1 before the topmost symbol from s. mapping the other way is as simple as undoing this process, i.e. just look for the topmost one and cut it off and everything above it off.

cantor was an idiot, set theory and CS are flawed at their most fundamental levels