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The question is in the title. Jech states in his book on page 696 that the consistency strength is "in the region of Woodin cardinals," which is frustratingly imprecise. I tried to find a reference for such, but couldn't find anything.

I gave my attempt at an inner-model-theoretic approach to a lower bound, but got stuck in the middle of it, and needed some help. As a novice in inner model theory, it would be helpful to see this written out if it's not too crazy. I'd have no idea if it's simply above my pay grade.

For an upper bound, you can certainly overshoot and use Foreman's result that the consistency of a huge cardinal implies that every regular cardinal can carry a saturated ideal. But I'm wanting something more precise. Foreman's argument doesn't look modifiable to bring this down to the Woodin realm.

Where would I find a reference which discusses this?

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  • $\begingroup$ While I can't quite answer, and surely someone will be here in a moment to provide you with a better response. This is the kind of thing you can get into the "region of a Woodin" by deriving some significant enough failure of covering of the core model below a Woodin cardinal (e.g., the fact that there is an elementary embedding in a generic extension whose critical point is $\kappa^+$). Presumably the upper bound would be something along the lines of a Woodin+measurable on top and then using the stationary tower to make the measurable into $\kappa^+$, or so. $\endgroup$
    – Asaf Karagila ♦
    Commented Jul 10, 2023 at 12:05
  • $\begingroup$ Hi, Asaf. The former is along the lines of arguments I've seen before, and this is how my attempt started. Suppose there's no inner model with a Woodin cardinal and $\kappa^+$ is the successor of a singular at which $I_{\text{NS}}$ is precipitous, I tried to reason about $K$ and $K^{\text{Ult}_G(V)}$. For example, at least they're different, because $K$ computes $\kappa^+$ correctly. From here I wasn't sure how to get a contradiction. I figured this wasn't enough of a good "attempt" to include in the question. $\endgroup$
    – Connor W
    Commented Jul 10, 2023 at 17:02
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    $\begingroup$ I suppose this is precisely a Woodin cardinal. If there is no reference with this exact result, it should be workable from the following (I haven't looked): MR1694779 Yoshinobu, Yasuo, "On strength of precipitousness of some ideals and towers", J. Math. Soc. Japan 51 (1999), no. 3, 535–541. $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 11, 2023 at 17:16
  • $\begingroup$ (Yoshinobu proves in ZFC+OR is measurable that a Woodin cardinal is equiconsistent with the non-stationary ideal on the successor of a singular cardinal being precipitous.) $\endgroup$
    – Andrés E. Caicedo
    Commented Jul 11, 2023 at 17:18
  • $\begingroup$ I am unsure it is relevant, but let me comment that Goldring proved that the non-stationary ideal over $[\mu]^{<\gamma}$ can be precipitous for regular $\gamma\le\mu$ if we collapse a Woodin cardinal larger than $\mu$ to $\mu$. (There are two different notion of stationarity for $[\mu]^{<\gamma}$ for $\mu\ge\omega_2$, but Goldring proved that both ideals are precipitous.) Goldring provided a proof only for collapsing a supercompact to $\mu$, but she stated that almost the same argument works. $\endgroup$
    – Hanul Jeon
    Commented Feb 26 at 20:16

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