The link you provided is merely the continuum hypothesis which boils down to “there’s no size of infinity between the natural and real sets”.
So somehow adding another 2-dimensional element/variable to the already infinite set to make its cardinality bigger is equated to a new dimensional axis?
Cantor’s diagonal argument is basically how countable infinity can encompass infinite sets of countable infinity(set of odds, evens, etc), since by going a diagonal once. It would guarantee to hit every number(one to one correspondence). Although if another infinite set which composed of two variables(say BX, so it’d be XXBXX.., BXBXBX…, and XBXBX…), even if one would use the diagonal argument. There will still be an infinite set of it that didn’t got contained by the countable infinite set by reversing the components once.
Increasing, through a powerset, the cardinality of a countable infinite 2-dimensional set into an uncountable one doesn’t magically give them depth, if anything they are just innumerable greater in quantity than the former set. Yeah this is derailing, feel free to comment this on my wall if you want to continue.