I have a set $S$ which is countable.
I have defined a subset $U \subset S$ that:
- cannot have a cardinality higher than $|S|$ (because it's a subset);
- is infinite (because I proved that).
Now I'd like to prove that $U$ is countable and I'd like to do it with the following statement:
If $U$ weren't countable, then I'd have a set for which the cardinality is non-finite and $|U| < \aleph_0$ at the same time, disproving the generalized continuum hypothesis. Given that the continuum hypothesis is independent from ZFC, either $U$ is independent too or it is countable.
Then, with this statement, I'd just have to show that my set is not independent from ZFC (which shouldn't be that hard) and I'm done.
Is it right?