[akomtu]

It's worth noting that R=2^N: reals is the set of all subsets of integers, simply because any real in binary form appears as a sequence of 1s and 0s that select a subset of N. And for some reason it's not possible to know if there's anything in between N and 2^N. This makes me think that infinite sets grow in discrete exponential jumps: N, 2^N, 2^R and so on. N seems to be the smallest infinite set.

[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