I am wondering if there are ways of defining “structure” on infinite sets that generate sequences of cardinals distinct from either the $ \aleph$ or $ \beth$ numbers? Let me explain:

Start by defining $ \beth_0=|\mathbb N|$ , which then is also the cardinality of any other countably infinite set. Next use the power set axiom to define a sequence of sets: P($ \mathbb N$ ), P(P($ \mathbb N$ )), P(P(P($ \mathbb N$ ))), …

So the first kind of “structure” we introduce is that of subsets.

By Cantor’s Theorem, $ |A|<|$ P($ A$ )$ |$ . So define the $ \beth$ numbers as the cardinality of these successive power sets: $ \beth_1=|$ P($ \mathbb N$ )$ |$ , $ \beth_2=|$ P(P($ \mathbb N$ ))$ |$ , …

Other kinds of structure on infinite sets would generate the same sequence of $ \beth$ s. For example, if we define F($ A$ ) to be the set of all functions $ A\rightarrow$ {0,1}. Then we will have $ |$ F($ \mathbb N$ )$ |=|$ P($ \mathbb N$ )$ |=\beth_1$ and further $ |$ F(F($ \mathbb N$ ))$ |=\beth_2$ , …

A second kind of structure uses ideas of well ordered sets to build up ordinals. Define $ \aleph_0=|\omega_0|$ and $ \aleph_{\gamma^+}=$ {$ \alpha:\alpha$ is an ordinal and $ |\alpha|<\aleph_{\gamma}$ }. This will get us the sequence of $ \aleph$ s.

Again other kinds of structure (e.g. building ordinals using sets with a maximum element instead of using wellorderd sets) would generate the same sequence of $ \aleph$ s.

The independence of the (generalized) continuum hypothesis means that after $ \aleph_0=\beth_0$ there is not much (other then some weak restrictions) we can prove about the relative size of $ \aleph$ s and $ \beth$ s in ZFC. And to me it seems like this is because once we get to uncountable sets there is just not enough “structure” avalaible to construct one-to-one functions and nail down their relative sizes.

So my question is are there other ways of defining “structure” on infinite sets to generate another sequence of cardinals (say $ \gimel_0,\gimel_1, \gimel_3$ …), which, at least within ZFC, are neither $ \aleph$ s nor $ \beth$ s?

Or alternatively, is there a proof that within ZFC that all cardinals will end up being either $ \aleph$ s or $ \beth$ s?