Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.

What happens after exponentiation?

We have the following equation: $2^{N_n}=N_{2^{n}}$.

(Which says: For all finite cardinal $n$ we have: $2^{n~\text{th finite cardinal}}=2^{n}~\text{th finite cardinal}$).

What this means?

The *gap* between $N_n$ and $2^{N_n}$ is *rapidly increasing* in *exponential speed*.

Now look at the $\text{GCH}$. It says that the gap between an infinite cardinal $\kappa$ and $2^{\kappa}$ is just a constant number $1$ in cardinals. Even in models for total failure of $\text{GCH}$ we usually have a *finite* gap between $\kappa$ and $2^{\kappa}$. Now if we look at the infinite cardinals as generalization of natural numbers it seems we should restate continuum hypothesis with more *acceleration* for the function $\kappa \mapsto 2^{\kappa}$ in order to uniform the behavior of exponentiation function in finite and infinite cardinals. Note to the following statement:

For all cardinal $\kappa$ we have $2^{\kappa~\text{th infinite cardinal}}=2^{\kappa}~\text{th infinite cardinal}$.

This is a direct generalization of the equation $\forall n\in \omega~~~~~2^{N_n}=N_{2^n}$ to the following form:

**Natural Continuum Hypothesis (NCH):** $~~~\forall \kappa\in Card~~~~~2^{\aleph_{\kappa}}=\aleph_{2^{\kappa}}$

Unfortunately $\text{NCH}$ is contradictory by Konig's lemma because assuming $\text{NCH}$ we have: $\aleph_{\aleph_0}<cf(2^{\aleph_{\aleph_0}})=cf(\aleph_{\aleph_1})\leq \aleph_1$ a contradiction.

(Thanks to Ramiro and Emil for their advices.)

But the acceleration problem remains open. The main question here is this:

Can we have a rapidly increasing gap between carinals by exponentiation? In the other words:

**Question 1:** Assuming consistency of $\text{ZFC}$ (plus some large cardinal axiom or axiom of constructibility), is the following statement consistent with $\text{ZFC}$?

$\forall \kappa\in Card~~~\exists \lambda \in Card~;~~~~~\lambda\geq 2^{\kappa}~\wedge~2^{\aleph_{\kappa}}=\aleph_{\lambda}$

**Definition 1:** Let $\kappa$ be a (finite or infinite) cardinal. Define $\kappa$-based beth function as follows:

$\beth_{(\kappa)}:Ord\longrightarrow Card$

$\beth_{(\kappa)}(0):=\kappa$

$\forall \alpha\in Ord~~~\beth_{(\kappa)}(\alpha +1):=2^{\beth_{(\kappa)}(\alpha)}$

$\forall \alpha\in LimitOrd~~~\beth_{(\kappa)}(\alpha):=\bigcup_{\beta \in \alpha}\beth_{(\kappa)}(\beta)$

**Definition 2:** Let $F:Card\longrightarrow Card$ be an increasing function and $\delta$ an ordinal. We say that $F$ has *acceleration rank $\delta$* if $\delta=min\{\alpha\in Ord~|~\forall \kappa\in Card~~~\beth_{(\kappa)}(\alpha)\leq F(\kappa) < \beth_{(\kappa)}(\alpha+1)\}$.

For example the functions $\kappa\mapsto \kappa^{+}$, $\kappa\mapsto 2^{\kappa}$, $\kappa\mapsto 2^{2^{\kappa}}$ have acceleration ranks $0, 1, 2$ respectively.

**Question 2:** Is there any limitation for acceleration of the continuum function? Precisely is the following statement true?

"For any ordinal $\delta$ there is an increasing function $F:Card\longrightarrow Card$ with acceleration rank $\delta$ such that assuming consistency of $\text{ZFC}$ (and some large cardinal axiom) one can prove the consistency of $\text{ZFC}$ with the statement $\forall \kappa\in Card~~~2^{\aleph_{\kappa}}=\aleph_{F(\kappa)}$."

According to Emil's interesting comment I added his question here:

**Question 3:** Assuming consistency of $\text{ZFC}$ (and some large cardinal axiom or axiom of constructibility), is the following consistent with $\text{ZFC}$?

$\forall \kappa\in Card~~~~\exists \lambda\in Card~~~~~2^{\aleph_{\kappa}}=\aleph_{\lambda}$

Note that it is not trivial that one can have a "cardinal index" for $\aleph$ as value of $2^{\aleph_{\kappa}}$ everywhere. Perhaps we will need some non-cardinal ordinals as indexes to avoid inconsistency.

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