I have a question, so assuming that we adopt the axiom(s) that cause real numbers to be cardinality Aleph_2 and not Aleph_1, is then there some intuitive explan

I have a question, so assuming that we adopt the axiom(s) that cause real numbers to be cardinality Aleph_2 and not Aleph_1, is then there some intuitive explanation how does the set of size Aleph_1 looks like, and what it could be used for?