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The reals are a superset of the integers, and a bijective mapping cannot be formed between the two, thus there are "more" elements in the set of reals than the set of integers.

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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.

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

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

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