Does taking the power set give you the "next biggest cardinal"

Question

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.

Answer 1

Not necessarily.

The standard axioms of set theory, $\sf ZFC$ cannot prove nor disprove whether or not, given an infinite set $A$ there is a cardinal between that of $|A|$ and $\mathcal P(A)$.

To get to the "next cardinal" we use a different method, which is rooted in the fact that cardinals in modern set theory are ordinals, and that we can prove that there is a least ordinal whose cardinality is larger than a given cardinal. This hierarchy, which begins with $\aleph_0=|\Bbb N|$ is called the $\aleph$ numbers.

We can talk about the cardinals of power sets, and these define a hierarchy called $\beth$ numbers. Namely, $\beth_0=|\Bbb N|$ and $\beth_{\alpha+1}=2^{\beth_\alpha}$.

(Both hierarchies include "limit stages", and in both cases we take "the smallest cardinal possible". But that's not important right now.)

Perhaps it should also be noted, that if you consider $\bigcup\mathcal P^n(\Bbb N)$ you get a set whose cardinality is strictly larger than all the cardinalities of $\mathcal P^n(\Bbb N)$. The cardinality of this set is denoted by $\beth_\omega$, and it is a limit cardinal to which I referred before. The hierarchy then continues onwards, to $\beth_{\omega+1},\beth_{\omega+2}$ and so on. To understand better this hierarchy (as well the $\aleph$ numbers) you should learn more about ordinal numbers.

Answer 2

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 ($3$).