[yiransheng]

What you described is called Continuum hypothesis[1]. Surprisingly it cannot be proven to be either true or false from(is independent of) ZFC set theory.

[1] https://en.m.wikipedia.org/wiki/Continuum_hypothesis

[IngoBlechschmid]

Indeed, and nowadays set theorists have rich experiences both in worlds where the Continuum Hypothesis holds and in worlds where it does not.

I tried to explain the resulting "multiverse philosophy" (not really related to the idea of physical alternate universes) here: https://iblech.gitlab.io/bb/multiverse.html