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:


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.

  • 5
    $\begingroup$ Easton's theorem implies that ZFC + "for all regular $\kappa$, $2^\kappa = \aleph_{\kappa^+}$" is equiconsistent with ZFC. Other variants, such as "the $\kappa^+$-th fixpoint" or $\aleph_{\aleph_{\kappa^+}}$ are possible, too. - But it is well known that ZFC+ non-SCH is much stronger than ZFC, in terms of consistency strength. $\endgroup$
    – Goldstern
    Oct 14 '17 at 11:48
  • $\begingroup$ Ah, so ZFC + CFCH is even stronger. $\endgroup$ Oct 14 '17 at 15:44
  • 2
    $\begingroup$ Under CFCH, $2^{\aleph_n} = \aleph_{\omega_{n+1}}$ and therefore $2^{\aleph_{\omega}} = \aleph_{\omega_\omega}^\omega = \aleph_{\omega_{\omega + 1}}$. By the theory of pcf, we get that if $a$ is the set of all cardinals between $\aleph_{\omega}$ and $\aleph_{\omega_\omega}$, the possible cofinalities of $\prod a$ is the regular cardinals in the range between $\aleph_{\omega_\omega + 1}$ and $\aleph_{\omega_{\omega + 1}}$ which has cardinality $|a|^+$. A recent paper of Gitik claims that it is consistent, but it is open if one can obtain this type of pcf structure in accessible cardinals. $\endgroup$
    – Yair Hayut
    Oct 14 '17 at 18:20
  • $\begingroup$ Just as a reminder, I would like to mention that despite its tempting appearance and intuitive background from natural numbers, the Natural Continuum Hypothesis (NCH) is inconsistent with ZFC due to Konig's lemma. However, it sounds interesting to me that you considered the cofinality version of CH independently. $\endgroup$ Oct 15 '17 at 11:12
  • $\begingroup$ I mainly mentioned NCH because it seems like this is a "variant" of NCH (where cardinal exponentiation speeds up) $\endgroup$ Oct 15 '17 at 21:38

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