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 $|\alpha|\le\aleph_{\gamma}$}. 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?

Clarification: I think I was unclear. Comments are right that in ZFC all cardinals are $\aleph$s, since any set can be well-ordered. But it is undecidable which $\aleph$s the $\beth$s correspond to, right? Even though we know there exists a well-ordered set with cardinality $\beth_1$, we cannot say which $\aleph$ that is without deciding the continuum hypothesis. So the question is are there other ways of defineing "structure" to generate cardinals that are similarly undecidable in ZFC? Or are the $\aleph$s or $\beth$s all there is?

I am wondering if there are ways of defining "structure" on infinite sets that generate sequences of cardinals that cannot be proved to have the same cardinality as 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 $|\alpha|\le\aleph_{\gamma}$}. 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, then can't be shown to have the same cardinality as either some $\aleph$s or some $\beth$s?

Or alternatively, is there a proof that within ZFC that defining cardinals using either the method of $\aleph$s or $\beth$s somehow exhausts possible ways of cardinal definition?

Clarification: I want be clear that I understood comments that in ZFC all cardinals are $\aleph$s, since any set can be well-ordered. But it is undecidable which $\aleph$s the $\beth$s correspond to, right? Even though we know there exists a well-ordered set with cardinality $\beth_1$, we cannot say which $\aleph$ that is without deciding the continuum hypothesis. So the question is are there other ways of defining "structure" to generate cardinals that are similarly undecidable in ZFC? Or are the $\aleph$s or $\beth$s all there is?

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?

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 $|\alpha|\le\aleph_{\gamma}$}. 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?

Clarification: I think I was unclear. Comments are right that in ZFC all cardinals are $\aleph$s, since any set can be well-ordered. But it is undecidable which $\aleph$s the $\beth$s correspond to, right? Even though we know there exists a well-ordered set with cardinality $\beth_1$, we cannot say which $\aleph$ that is without deciding the continuum hypothesis. So the question is are there other ways of defineing "structure" to generate cardinals that are similarly undecidable in ZFC? Or are the $\aleph$s or $\beth$s all there is?