That still feels intuitive. Like for any open interval there are uncountably many reals and a finite number of integers. It seems like nothing changes as you expand the interval.
The integers HAVE fewer subsets than the reals!
Cantor's theorem tells us that the power set of any set has more elements than said set. It's also well known that the reals have the same cardinality as the power set of the integers.
Therefore, |P(R)| > |R| = |P(N)| > |N|, so the power set of the reals has more elements than the power set of integers.
The continuous hypothesis is the belief that no cardinal exists between |N| and |P(N)|. My previous argument fails if we pick two sets A and B, such that |A| < |B| < |P(A)|.
