Filter extension games and generic large cardinals

A cardinal $\kappa$ is called generically blah, where blah stands in for some large cardinal notion if $\kappa$ has some property characterizing the blah large cardinal in a forcing extension. So, for instance, a cardinal $\kappa$ is generically measurable if in a forcing extension there is an elementary embedding $j:V\to M$ with $M$ transitive and $\text{crit}(j)=\kappa$, but $M$ not necessarily contained in $V$. Equivalently, $\kappa$ is generically measurable if some forcing extension has a $V$-ultrafilter - a filter measuring $P(\kappa)^V$ that is $V$-$\kappa$-complete and $V$-normal, with a well-founded ultrapower. For the reader familiar with the previous post, this is precisely how we defined the notion of the external $M$-ultrafilter for a weak $\kappa$-model $M$. The reader of the previous post will also have noticed that for external ultrafilters, such as $M$-ultrafilters, the property of weak amenability turns out to be quite important. So what if we further require that the $V$-ultrafilter in the forcing extension is weakly amenable? This cannot be equivalent to generic measurability because $\omega_1$ can be generically measurable, if say $\omega_1$ carries a precipitous uniform normal ideal, but if a forcing extension has a weakly amenable $V$-ultrafilter, then $\kappa$ is at least weakly compact and more (follows from [1]). What if we ask that the weakly amenable $V$-ultrafilter have well-founded iterated ultrapowers? Of course, all these notions are equiconsistent with a measurable cardinal because generic measurability is already equiconsistent with it [2]. So let's look at some natural weakenings of generic measurability. What if we ask that in a forcing extension, for all sufficiently large regular $\theta$, there is an $H_\theta$-ultrafilter with a well-founded ultrapower? And what if we additionally ask that the $H_\theta$-ultrafilters are weakly amenable? We can also weaken generic measurability in a different direction by dropping the requirement that the $V$-ultrafilter produces a well-founded ultrapower, but maybe keeping weak amenability? Maybe even consider weak $V$-ultrafilters, where we drop the $V$-normality requirement?

We can also go off in an entirely different direction and care about which poset provides the forcing extension with the (weak) $V$-ultrafilter. Maybe we want the forcing extension to be by a poset of the form $P(\kappa)/I$ for a uniform normal ideal $I$ on $\kappa$. Or maybe we want the forcing to be $\gamma$-closed for some cardinal $\gamma$?

Assuming that a forcing extension has a $V$-ultrafilter has no large cardinal strength. If $\kappa$ is uncountable and regular, then the non-stationary ideal $NS_\kappa$ is normal, and so forcing with $P(\kappa)/NS_\kappa$ adds a $V$-ultrafilter. Conversely, if a forcing extension has a $V$-ultrafilter, then the ultrapower map $j:V\to M$, with $M$ potentially ill-founded, nontheless has critical point $\kappa$, which can be used to show that $\kappa$ is regular. But adding in weak amenability will give large cardinal strength.

It actually turns out that several of these variations on generic measurability are characterized by the existence of a winning strategy for the judge in the filter extension games discussed in the previous post. We will assume throughout that $\kappa$ is inaccessible. First, let's give formal definitions of some generic large cardinal notions. We will call a weak $V$-ultrafilter good if it produces a well-founded ultrapower.

Definition:

• $\kappa$ is weakly almost generically measurable with weak amenability (wa) if some forcing extension has a weakly amenable weak $V$-ultrafilter.
• $\kappa$ is almost generically measurable with weak amenability (wa) if some forcing extension has a weakly amenable $V$-ultrafilter.
• $\kappa$ is generically measurable for sets if for all sufficiently large regular $\theta$, some forcing extension has a good $H_\theta$-ultrafilter.
• $\kappa$ is generically measurable with weak amenability (wa) for sets if for all sufficiently large regular $\theta$, some forcing extension has a weakly amenable good $H_\theta$-ultrafilter.
• $\kappa$ is generically measurable with weak amenability (wa) if some forcing extension has a weakly amenable good $V$-ultrafilter.
• $\kappa$ is generically measurable with weak amenability (wa) and $\alpha$-iterability if some forcing extension has a weakly amenable $V$-ultrafilter with $\alpha$-many well-founded iterated ultrapowers.
• $\kappa$ is (any of the above notions) by $\mathcal P$ (a class of forcings) if the (weak) $V$-ultrafilter exists in a forcing extension by some $\mathbb P\in\mathcal P$.

Here is what we know about the strength of these notions.

• $\kappa$ weakly almost generically measurable with wa if and only if it is weakly compact (follows from [3]).
• $\kappa$ is almost generically measurable with wa if and only if it is completely ineffable (follows from [3]).
• Generically measurable for sets cardinals can exists in $L$, but are stronger than completely ineffables (follows from [1]).
• Generically measurable for sets cardinals are equiconsistent with generically measurable with wa for sets cardinals (follows from [1]).

Notice here that it is possible to have good generic $H_\theta$-ultrafilters, for all sufficiently large $\theta$, but not a good generic $V$-ultrafilter (because this is much higher in consistency strength).

We will abbreviate the assertion that the judge has a winning strategy a filter game $G$ by $\text{Judge}_G$.

• $\kappa$ is weakly almost generically measurable (weakly compact) if and only if $\text{Judge}_{G_1(\kappa)}$.
• $\kappa$ is almost generically measurable with wa (completely ineffable) if and only if $\text{Judge}_{wG_\omega(\kappa)}$ (follows from [3]).
• If $\text{Judge}_{G_\omega(\kappa)}$, then $\kappa$ is generically measurable for sets with wa (by $\text{Coll}(\omega,H_\theta)$) (follows from [4]).
• $\kappa$ is generically measurable for sets with wa is equiconsistent with $\text{Judge}_{G_\omega(\kappa)}$ (follows from [4]).
• If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa$ is generically measurable with wa and $\omega_1$-iterability (by $\text{Coll}(\omega,H_\theta)$) (analogous to [5]).
• If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa $ is generically measurable with wa by $P(\kappa)/I$ for a precipitous normal ideal $I$ (follows from [6]).
• For uncountable regular cardinals $\delta$, $\text{Judge}_{G_\delta(\kappa)}$ if and only if $\kappa$ is generically measurable with wa by $\delta$-closed forcing (analogous to [5]).

Question: Can we construct a generic weakly amenable good $V$-ultrafilter not all of whose iterated ultrapowers are well-founded?

Let's move on now to generic supercompactness. In the setting of supercompactness, a $V$-ultrafilter is going to be a filter measuring $P(P_\kappa(\lambda))^V$ that is fine, $V$-$\kappa$-complete and $V$-normal. Without the $V$-normality assumption, we will say that the $V$-ultrafilter is weak. If the ultrapower is well-founded, we will say that it is good. A cardinal $\kappa$ is generically $\lambda$-supercompact if and only if some forcing extension has an elementary embedding $j:V\to M$ with $M$ transitive, $\text{crit}(j)=\kappa$, and $j''\lambda\in M$. Equivalently, $\kappa$ is generically $\lambda$-supercompact if some forcing extension has a good $V$-ultrafilter. A cardinal $\kappa$ is generically $\lambda$-strongly compact if and only if some forcing extension has an elementary embedding $j:V\to M$ with $M$ transitive, $\text{crit}(j)=\kappa$, and $j''\lambda\subseteq s$ with $|s|^M\lt j(\kappa)$. Equivalently, some forcing extension has a good weak $V$-ultrafilter. Now we can introduce all the same variations of partial generic supercompactness that we discussed above for generic measurability with many results generalizing, but also in this case a number of open questions.

We can define the notions of stationary, unbounded, and closed for subsets of $P_\kappa(\lambda)$ and show that the non-stationary ideal is fine and normal (see for, instance, \cite{jech:settheory}). This shows that the existence of a $V$-ultrafilter does not carry large cardinal strength.

Here are the generic large cardinal notions corresponding to the ones we considered for generic measurability.

• $\kappa$ is almost generically $\lambda$-strongly compact with weak amenability (wa) if some forcing extension has a weakly amenable weak $V$-ultrafilter.
• $\kappa$ is almost generically $\lambda$-supercompact with weak amenability (wa) if some forcing extension has a weakly amenable $V$-ultrafilter.
• $\kappa$ is generically $\lambda$-supercompact for sets if for all sufficiently large regular $\theta$, some forcing extension has an $H_\theta$-ultrafilter with a well-founded ultrapower.
• $\kappa$ is generically $\lambda$-supercompact with weak amenability (wa) for sets if for all sufficiently large regular $\theta$, some forcing extension has a weakly amenable good $H_\theta$-ultrafilter.
• $\kappa$ is generically $\lambda$-supercompact with weak amenability (wa) if some forcing extension has a weakly amenable good $V$-ultrafilter.
• $\kappa$ is generically $\lambda$-supercompact with weak amenability (wa) and $\alpha$-iterability if some forcing extension has a weakly amenable $V$-ultrafilter with $\alpha$-many well-founded iterated ultrapowers.
• $\kappa$ is (any of the above notions) by $\mathcal P$ (a class of forcings) if the (weak) $V$-ultrafilter exists in a forcing extension by some $\mathbb P\in\mathcal P$.

For what follows, we assume that $\lambda^{\lt\kappa}=\lambda$. Here is what we know about the strength of these notions [5]. See previous post for definitions of nearly $\lambda$-strongly and supercompact cardinals.

• $\kappa$ almost generically $\lambda$-strong with wa if and only if it is nearly $\lambda$-strongly compact.
• $\kappa$ is almost generically $\lambda$-supercompact with wa if and only if it is completely $\lambda$-ineffable.
• The least $\kappa$ for which there is some $\lambda$ such that $\kappa$ is almost generically $\lambda$-supercompact with wa is not generically $\lambda$-supercompact with wa for sets.

Question: Do nearly $\lambda$-supercompact cardinals have a generic large cardinal characterization?

Question: Are generically $\lambda$-supercompact cardinals for sets equiconsistent with generically $\lambda$-supercompact cardinals for sets with wa?

For the corresponding generic measurability notions, it was shown that generically measurable for sets cardinals are generically measurable with wa for sets in $L$ [4], but of course at this large cardinal level canonical models are not available.

Question: Can we separate the following notions via equivalence or equiconsistency:

• generically $\lambda$-supercompact for sets with wa
• generically $\lambda$-supercompact with wa
• generically $\lambda$-supercompact with wa and $\omega_1$-iterability ($\alpha$-iterability for some $\alpha>1$)
• generically $\lambda$-supercompact

As with versions of generic measurability, several of the partial generic supercompacts are characterized by the existence of a winning strategy for the judge in one of the filter extension games on mini-supercompactness measures discussed in the previous post [5].

• $\kappa$ is almost generically $\lambda$-strongly compact (nearly $\lambda$-strongly compact) if and only if:
  • $\text{Judge}_{G_1(\kappa)}$
  • $\text{Judge}_{wG^*_\omega(\kappa)}$
• $\kappa$ is almost generically $\lambda$-supercompact with wa (completely $\lambda$-ineffable) if and only if $\text{Judge}_{wG_\omega(\kappa)}$.
• If $\text{Judge}_{G_\omega(\kappa)}$, then $\kappa$ is generically $\lambda$-supercompact for sets with wa (by $\text{Coll}(\omega,H_\theta)$).
• If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa$ is generically $\lambda$-supercompact with wa and $\omega_1$-iterability (by $\text{Coll}(\omega,H_\theta)$).
• If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa $ is generically $\lambda$-supercompact by $P(\kappa)/I$ for a precipitious normal fine ideal $I$.
• For uncountable regular cardinals $\delta$, $\text{Judge}_{G_\delta(\kappa)}$ if and only if $\kappa$ is generically $\lambda$-supercompact with wa by $\delta$-closed forcing.

References

1. S. Dimopoulos, V. Gitman, and D. Saattrup Nielsen, “The virtual large cardinal hierarchy,” Fund. Math., vol. 266, no. 3, pp. 237–262, 2024. Available at: https://doi.org/10.4064/fm139-6-2024
2. T. Jech, Set theory. Berlin: Springer-Verlag, 2003, p. xiv+769.
3. F. G. Abramson, L. A. Harrington, E. M. Kleinberg, and W. S. Zwicker, “Flipping properties: a unifying thread in the theory of large cardinals,” Ann. Math. Logic, vol. 12, no. 1, pp. 25–58, 1977.
4. D. Saattrup Nielsen and P. Welch, “Games and Ramsey-like cardinals,” J. Symb. Log., vol. 84, no. 1, pp. 408–437, 2019.
5. T. Benhamou and V. Gitman, “Cardinals of the $P_κ(λ)$-filter games,” Manuscript, 2025.
6. M. Foreman, M. Magidor, and M. Zeman, “Games with filters I,” Journal of Mathematical Logic, pp. to appear, 2023.