Does taking the power set give you the "next biggest cardinal" I know that if you take the power set of a set, it has a higher cardinality. Therefore there are an infinite number of them as $P^{n}(\mathbb{N})$(the nth power set of the naturals) Let's say $$C_{n} = P^{n}(\mathbb{N})$$ My question is: Given $C_n$ and $C_{n+1}$ are there cardinal numbers between them? Or does taking a power set always give you the "next greatest cardinal". 
NOTE: I don't know the formal notation for cardinals.

Not necessarily in ZFC; this is the Generalized Continuum Hypothesis when you are asking about infinite cardinals, which is shown to be independent of ZFC.
Note: the question "Does taking the power set give you the 'next biggest cardinal'" is the Generalized Continuum Hypothesis when restricted to infinite cardinals, but the claim that $\mathcal{P}^{n+1}(\mathbb{N})$ is the least cardinal greater than $\mathcal{P}^n(\mathbb{N})$ is strictly weaker.
However, it is important to note that it is false for finite cardinals; there are cardinals between $2$ and $2^2=4$, for example ($3$).
