# What are some good lower bounds on the consistency of the failure of the PCF conjecture?

Shelah's celebrated theorem states that $$\aleph_\omega$$ is a strong limit cardinal, then $$2^{\aleph_\omega}<\aleph_{\omega_4}$$.

But the conjecture is that $$\omega_4$$ can be provably replaced by $$\omega_1$$. Namely, $$2^{\aleph_\omega}<\aleph_{\omega_1}$$ holds, assuming that $$\aleph_\omega$$ is a strong limit cardinal.

As far as I understand it, we know that from large cardinal assumptions it is consistent that $$2^{\aleph_\omega}$$ is arbitrarily large below $$\aleph_{\omega_1}$$ (and it is a strong limit, of course). But there is no current way to go beyond $$\aleph_{\omega_1}$$. Even Gitik's work on the subject does not translate to the $$\aleph_n$$'s.

Question. Suppose that the PCF Conjecture fails. Namely, $$\aleph_\omega$$ is a strong limit cardinal, but $$2^{\aleph_\omega}>\aleph_{\omega_1}$$. What kind of large cardinals can we expect to find in inner models?

(Of course large cardinals are necessary, since $$2^{\aleph_\omega}>\aleph_{\omega+1}$$ with $$\aleph_\omega$$ as a strong limit was shown by Gitik to be equiconsistent with the existence of a measurable $$\kappa$$ of Mitchell order $$\kappa^{++}$$.)

• YCor, I'm all in favor of informative titles, but I'm not sure what would be more informative? The PCF Conjecture is a fairly common term. Feb 11, 2019 at 17:14
• What are the best bounds you know already? From Gitik's work on SCH it follows that $2^{\aleph_0} > \aleph_{\omega+1}$ (with $\aleph_\omega$ a strong limit) is equiconsistent with a measurable cardinal $\kappa$ of Mitchell order $\kappa^{++}$. I'm guessing you already know this? It's probably worth mentioning in the question. It's hard to know what you would call a "good" bound without first knowing what you might consider an "everybody-already-knows-that" bound. Feb 11, 2019 at 18:01
• @Will: Yes, but this is just $2^{\aleph_\omega}=\aleph_{\omega+2}$. We are talking about significantly larger gaps here. Good bounds include Woodin cardinals, or proper class of strong cardinals, or a sequence of $\omega_1+1$ strong cardinals, or whatever. I'm understand there is some ambiguity in "good lower bound", but obviously Gitik's initial result about SCH is not that. Feb 11, 2019 at 18:02
• This is a good question. I don't think we know much about it yet. Something along the lines of the Gitik-Schindler-Shelah paper seems to be the state of the art. Feb 11, 2019 at 19:23
• Shelah has a lot of results that treat pcf assumptions themselves kind of like large cardinal statements. A typical result might show that a combinatorial statement implies a pcf statement, and from the pcf statement one can force the combinatorial statement, with the pcf statement being something currently intractable. Feb 12, 2019 at 14:19

As explained in the comments by Andres, it follows from the work of Gitik, Schindler and Shelah Pcf theory and Woodin cardinals that one can get $$PD$$ (Projective Determinacy).
• Well, this is very nice. I'm surprised that we don't have the machinery to prove something like $\mathsf{AD}^{L(\Bbb R)}$ from these hypotheses, seeing how we do get $\sf PD$. Feb 14, 2019 at 10:30