Does ZFC permit or acknowledge the existence of infinite sets that are uncountable? Does ZFC permit or acknowledge the existence of infinite sets that are uncountable? By saying infinite sets that are uncountable, I mean that the cardinality of the sets is uncountable (that is $\ge\aleph_1$) Thanks.

Cantor's theorem tells us that if $X$ is a countably infinite set then $P(X)$ is not countable.

In ZFC we assert that if $x$ is a set then $P(x)$ is a set as well, so if you assume the axiom of infinity which asserts that indeed an infinite set exists then there exists an uncountable set as well.

There is more to that as well: How do we know an $ \aleph_1 $ exists at all?