It is well known that the power set of a countably infinite set is uncountable. Therefore, if a power set is countable, then the cardinality of the set must below "countable". Formally, if $2^A$ is $\aleph_0$, then $A$ must be below $\aleph_0$.
However, I think $\aleph_0$ is the lowest possible cardinality? Are there any possible cardinalities between a finite set and a countable set?
From existing information we can only conclude that if $2^A$ is countable, then $A$ is “below countable”. How to prove that if $2^A$ is countable, then $A$ must be finite?