The strength of ideal hypotheses

Last time we delved into the world of ideals and their associated properties, precipitousness and saturation. We noted that these properties could be viewed as a measure of “how close” a cardinal is to being measurable, and furthermore that all the properties are equiconsistent; i.e. that the existence of a precipitous ideal on some \kappa is equiconsistent with the existence of a measurable cardinal. But we can do better.

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                                                                    A different kind of saturation.

Recall that starting from a measurable \kappa we get that \omega_1 carries a precipitous ideal after forcing with \text{Col}(\omega,{<}\kappa). Jech et al (’80) in fact showed that if we further force with a certain poset of clubs we get that, in the final extension, even the nonstationary ideal \text{NS}_{\omega_1} on \omega_1 is precipitous.

Fact 1. \text{NS}_{\omega_1} precipitous is equiconsistent with a measurable.

A lot of questions then arise at this point. What about \text{NS}_\kappa for cardinals \kappa>\omega_1? What about stronger properties such as saturation of the ideals? Before we get too excited however, Gitik & Shelah (’97) throw a spanner in the works with their seminal \textsf{ZFC} result saying that \text{NS}_\kappa can never be saturated when \kappa>\omega_1. This still leaves us with the question of how strong a hypothesis the precipitousness of \text{NS}_\kappa is for various \kappa>\omega_1, and also how strong a hypothesis the saturation of \text{NS}_{\omega_1} is. Recall that \text{NS}_{\lambda^+} is never \lambda^+-saturated, so in terms of saturation properties we can’t assume anything stronger about \text{NS}_{\omega_1}.

As for the first question, this was (essentially) resolved prior to the Gitik-Shelah result mentioned above: Jech (’84) showed that precipitousness of \text{NS}_\kappa implies that \kappa is measurable of high Mitchell order in the core model K, and Gitik (’95) showed “the converse” that if we start with a measurable of high Mitchell order then there’s a forcing extension in which \text{NS}_\kappa is precipitous. Here I added the scare quotes as it’s not really an exact equiconsistency, as there are some technical differences between the lower and upper bounds. But we get close. Sufficiently close.

Fact 2. For regular \kappa>\omega_1, precipitousness of \text{NS}_\kappa is consistency-wise in the realm of a measurable with high Mitchell order.

As for the second question of the strength of the saturation, or even the presaturation, of \text{NS}_{\omega_1}, this was nearly resolved in the 90’s. Firstly, as I mentioned last time, Shelah (’98) showed that, assuming the existence of a Woodin cardinal, there’s a forcing extension in which \text{NS}_{\omega_1} is saturated. In the other direction Steel (’96) used the machinery of core models to prove that, assuming the existence of a measurable cardinal, the presaturation of \text{NS}_{\omega_1} implies that there’s an inner model with a Woodin cardinal. All that prevented us from having an equiconsistency at this point was then just that measurable cardinal, which at the time was used to construct the core model K.

It wasn’t until years later that Jensen & Steel (’13) managed to get rid of this assumption of a measurable, establishing the equiconsistency. Claverie & Schindler (’12) improved this even result further, by showing that the strongness of \text{NS}_{\omega_1} is equiconsistent with a Woodin, where an ideal is strong if j(\omega_1^V)=\omega_2^V with j:V\to M being the generic embedding — every presaturated ideal is clearly strong.

Fact 3. The saturation, presaturation and strongness of \text{NS}_{\omega_1} are all equiconsistent with a Woodin.

Is this then the end of the story? Almost. There is a notion which is stronger than saturation but which \text{NS}_{\omega_1} can still consistently satisfy. Say an ideal I on \kappa is dense if \mathcal P(\kappa)/I has a dense subset of size \kappa. Then every dense ideal is saturated, and Woodin (’10) has shown, via an elaborate modification of his \mathbb P_{\text{max}} forcing, that density of \text{NS}_{\omega_1} is equiconsistent with \textsf{AD}, which he had previously shown is equiconsistent with a limit of Woodins. We arrive at our last fact.

Fact 4. Density of \text{NS}_{\omega_1} is equiconsistent with a limit of Woodins.

This then concludes our journey. Aside from being an analysis of these ideals, these results also seem to give the impression that the steps from measurables to Woodins to a limit of Woodins are somehow “natural” in the large cardinal hierarchy, corresponding to natural properties holding of the canonical normal ideal on the smallest uncountable cardinal.