An example of an infinite set with $S$ with there exists some cardinality between $S$ and $P(S)$. I just read about Continuum Hypothesis which states that there is no set $S$ with the cardinality of $S$ is strictly larger than $\mathbb{N}$ and strictly smaller than $\mathbb{R}$.
I recall that in a math class we proved that the power set $P(\mathbb{N})$ have the same cardinality with $\mathbb{R}$. So I wonder if there exists an infinite set $S$ such that there is a set $X$ with the cardinality of $X$ is strictly greater than the cardinality of $S$ and strictly smaller than the cardinality of $P(S)$. I thought a lot but couldn't find an example.
I don't know much about set theory thus I'd be glad if you couldd find an easy example.
 A: This is a great question! Unfortunately, the answer is (at first glance) a bit unsatisfying:

It is undecidable from the axioms of ZFC (= standard set theory) whether such an $S$ exists.

Specifically, the statement that no such $S$ exists is called the generalized continuum hypothesis (GCH). Godel showed that if ZFC is consistent, then ZFC+GCH is too - that is, ZFC can't disprove GCH. Later, Cohen showed the opposite: that if ZFC is consistent, then ZFC+$\neg$GCH is as well - ZFC can't prove GCH.
I said that this is unsatisfying at first glance. When we unwind the proofs, some really cool mathematics pops up! Specifically, Godel proved that

If $V$ is any model of ZFC (think: $V$ is the universe of all sets), then there's a subclass $L$ of $V$ (think: collection of some, but maybe not all, sets) which satisfies ZFC+GCH.

$L$ is called the constructible universe, and has a number of nice properties; Godel's discover of $L$ was the beginning of inner model theory.
Now it turns out that shrinking the universe can't make GCH false:

If $V=L$, then every inner model of $V$ satisfies GCH.

(Here "inner model" is a bit technical.)
So we need to go the other way! Cohen developed a technique - called forcing - which lets us expand the universe of sets! One incorrect-but-morally-correct (:P) way of phrasing his result is:

If $V$ is a model of ZFC, then there is a slightly larger model $W$ such that $W$ satisfies $\neg$ GCH.

And in fact, we can also make GCH true by going bigger!

If $V$ is a model of ZFC, then there is a slightly larger model $W$ such that $W$ satisfies GCH.

Finally, Easton showed that GCH can fail in weird places:

If $V$ is a model of ZFC+GCH, then - for each cardinal $\kappa$ in $V$ - there is a slightly larger model $W$ where GCH holds below $\kappa$ (that is, $W$ thinks that if $\vert S\vert<\kappa$ then there's nothing between $S$ and $\mathcal{P}(X)$) but fails above $\kappa$.

This kills off much hope of getting a concrete description of what failures of GCH look like.
However, it's not the end of the story; there's a ton of rich combinatorial questions around GCH, especially questions like "what can the smallest failure of GCH look like?", but I think I'll stop here.
A: It is consistent with the usual $\mathsf{ZFC}$ axioms of set theory that there be such a set, and it is also consistent with them that no such set exist. The statement that there is no such set is precisely the Generalized Continuum Hypothesis ($\mathsf{GCH}$), which is known to be consistent with $\mathsf{ZFC}$. The more specific statement that there is no set whose cardinality is strictly between those of $\Bbb N$ and the power set of $\Bbb N$ is the Continuum Hypothesis ($\mathsf{CH}$), whose negation is also known to be consistent with $\mathsf{ZFC}$.
Thus, you can consistently have it either way.
