Comment by nrds

Only countably many of those subsets can be distinguished from other ones. So "anything I could ever imagine caring about" is surely still a countable set.

I think the argument you are trying to make rests on a pretty serious fallacy generalizing "I care about some subsets of natural numbers" (and maybe " I care about subsets of natural numbers, in general") to "I care about all subsets of natural numbers, including undefinable ones".