It is a well-known fact that the Generalized Continuum Hypothesis is undecidable from ZFC. For similar sentences $\phi$, this is simply equivalent to ZFC having a model $M$ for which $M\models\phi$.

I wanted to see if there was any way to make all $\beth$-numbers limits. At first, this seemed dubious, but now, after creating a variation on GCH, I have realized it seems completely legitimate.

The variation I created is called the **Cofinal Continuum Hypothesis**, or CFCH. It is defined as follows:

$$\forall\kappa(2^\kappa=\aleph_{\kappa^+})$$

This immediately destroys the idea of the continuum hypothesis, as with this definition $2^{\aleph_0}=\aleph_\Omega$, which is very far off from $\aleph_1$.

The reason I chose specifically this definition is that its cofinality preserving properties when compared to GCH, namely that $\mathrm{cof}(2^\kappa)=\mathrm{cof}(\kappa^+)$. This property "piggybacks" off of GCH's undecidability in a sense; most proofs that a variation on GCH is inconsistent with ZFC (or similar theories) involve some failure of an inequality with $\mathrm{cof}(2^\kappa)$ and $\mathrm{cof}(\kappa)$. With this definition, it is clear to see that this problem no longer exists because in GCH $\mathrm{cof}(2^\kappa)=\mathrm{cof}(\kappa^+)$.

Because of this construction, one could generalize it to $2^\kappa$ is the $\kappa^+$-th aleph fixed point, or $2^\kappa=\aleph_{\aleph_\kappa}$. These, however, seem even more audacious than CFCH itself.

**However**, the reason I feel this is still dubious is because of it's interesting relation to the Singular Cardinal Hypothesis. It is clear that, with this definition, the first strong limit cardinal is a singular cardinal. This becomes a problem when introduced to the Singular Cardinal Hypothesis, as, assuming and CFCH and SCH:

$$\forall\alpha(\mathrm{cof}(\beth_{\omega\cdot\alpha})\neq\beth_{\omega\cdot\alpha}\rightarrow 2^{\beth_{\omega\cdot\alpha}}=\beth_{\omega\cdot\alpha}^+)$$ $$2^{\beth_\omega}=\beth_\omega^+$$ $$2^{\beth_\omega}=\aleph_{\beth_\omega^+}$$ $$\beth_\omega^+=\aleph_{\beth_\omega^+}$$

This is a contradiction, as it implies that a successor cardinal is an $\aleph$-fixed point. Thus, CFCH is inconsistent with SCH. Proving CFCH equiconsistent to ZFC would be a great feat, as it would prove $\neg$SCH equiconsistent to ZFC as well, thus showing that the existence of a measurable cardinal with Mitchell order $\kappa^{++}$ is equiconsistent to ZFC (as a result of Gitik), and every other large cardinal axiom implied by said axiom (which has not yet been done). This is why I doubt that CFCH will be proven equiconsistent with ZFC.

**Here are some other interesting facts implied by CFCH:**

- $\beth_{\omega\cdot\alpha}$ is the $\alpha$-th $\aleph$-fixed point (and thus all strong limit cardinals are $\aleph$-fixed points). This is yet another doubt I have for CFCH.
- $\mathrm{cof}(\beth_\alpha)=\mathrm{cof}(\aleph_\alpha)$ (and thus for regular cardinals $\aleph_\alpha$, $\mathrm{cof}(\beth_\alpha)=\aleph_\alpha$ and for limit ordinals $\alpha$, $\mathrm{cof}(\beth_\alpha)=\alpha$.)
- The Konig's Theorem corollaries about cofinality seem are implied by it

($\kappa<\mathrm{cof}(2^\kappa)=\mathrm{cof}(\kappa^+)=\kappa^+$ and $2^\kappa<(2^\kappa)^{\mathrm{cof}(2^\kappa)})=2^{\kappa\cdot\mathrm{cof}(2^\kappa)}=2^{\mathrm{cof}(2^\kappa)}>2^\kappa$) - As for other variations of the Generalized Continuum Hypothesis and Continuum Hypothesis:
- $\mathrm{CFCH}\rightarrow\neg\mathrm{GCH}$
- $\mathrm{CFCH}\rightarrow\neg\mathrm{SCH}$
- $\mathrm{CFCH}\rightarrow\neg\mathrm{CH}$
- $\mathrm{CFCH}\rightarrow\neg\mathrm{NCH}$ (The Natural Continuum Hypothesis, which is found in another MathExchange link "A New Continuum Hypothesis")

**In general, it would be really useful if the reader of this question reply in the comments with either a link to a website listing many interesting models of ZF and ZFC or several Forcing methods or they send me the names of the models themselves.** Of course, you could reply to this question with comments pertaining to the question as well.