Cardinals of the $P_\kappa(\lambda)$-filter games

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 @article{BenhamouGitman:supercompactGames, title={Cardinals of the $P_\kappa(\lambda)$-filter games}, author={Tom Benhamou and Victoria Gitman}, journal = {Manuscript}, year = {2025}, eprint = {2504.00119} } 

T. Benhamou and V. Gitman, “Cardinals of the $P_κ(λ)$-filter games,” Manuscript, 2025.

In an attempt to generalize reflection and compactness properties of first-order logic, Tarski [1] discovered cardinal notions whose existence require axiomatic frameworks which are strictly stronger than ${\rm ZFC}$. The fundamental discovery of Tarski provided a connection between compactness for certain strong logics and the ability to extend filters to special ultrafilters. Tarski's result initiated a fruitful line of research, isolating new large cardinal notions:

• (weak compactness) Suppose that $\kappa$ is inaccessible. Every $\kappa$-complete filter on a $\kappa$-algebra $\mathcal{A}$ ($\kappa$-complete sub-algebra of $P(\kappa)$ of size $\kappa$) can be extended to a $\kappa$-complete ultrafilter on $\mathcal{A}$ $\Longleftrightarrow$ $L_{\kappa,\kappa}$ ($L_{\kappa,\omega}$) satisfies the compactness theorem for theories of size $\kappa$.
• (strong compactness) Suppose that $\kappa$ is regular. Every $\kappa$-complete filter (on any set) can be extended to a $\kappa$-complete ultrafilter $\Longleftrightarrow$ $L_{\kappa,\kappa}$ ($L_{\kappa,\omega}$) satisfies the full compactness theorem.

Weakly compact cardinals can also be characterized by the existence of certain ultrafilters on the powerset of $\kappa$ of $\kappa$-sized models of set theory. A $\kappa$-model $M$ is an $\in$-model of size $\kappa$ satisfying a sufficiently large fragment of set theory that is closed under sequences of length less than $\kappa$. Given a $\kappa$-model $M$, an $M$-ultrafilter is a uniform ultrafilter on $P(\kappa)^M$ that is normal from the point of view of $M$, namely, it is closed under diagonal intersections of $\kappa$-length sequences that are elements of $M$. We will say that an $M$-ultrafilter is weak if it is just $\kappa$-complete. It is a folklore result that an inaccessible cardinal $\kappa$ is weakly compact if and only if every $\kappa$-model $M$ has a weak $M$-ultrafilter if and only if it has an $M$-ultrafilter. This characterization is quite different from the extension-like properties considered above, as it only mentioned the existence of certain ultrafilters. A natural question is: can the two be combined? For instance, given a $\kappa$-model $N$ extending a $\kappa$-model $M$ with a weak $M$-ultrafilter $U$, can we find a weak $N$-ultrafilter $W$ extending $U$? Keisler and Tarski [3] (see also [4] (Proposition 2.9)) showed that this filter extension property again characterizes weak compactness. However, if we remove the 'weakness' condition from the ultrafilters, then surprisingly the property becomes inconsistent, as shown by the second author [4] (Proposition 2.13). This suggests a more delicate approach, a strategic extension of an $M$-ultrafilter to an $N$-ultrafilter. Game variations of this filter extension property were considered first by Holy and Schlicht [4]. (Here we use game names that correspond to our later notation as opposed to the names originally used by Holy and Schlicht.)

Definition: Suppose that $\kappa$ is an inaccessible cardinal and $\delta\leq\kappa^+$ is an ordinal.

1. Let $wG_\delta(\kappa)$ be the following two-player game of perfect information played by the challenger and the judge. The challenger starts the game and plays a $\kappa$-model $M_0$ and the judge responds by playing an $M_0$-ultrafilter $U_0$. At stage $\gamma>0$, the challenger plays a $\kappa$-model $M_\gamma$, with $\{\langle M_\xi,U_\xi\rangle\mid \xi<\gamma\}\in M_\gamma$, elementarily extending his previous moves and the judge plays an $M_\gamma$-ultrafilter $U_\gamma$ extending $\bigcup_{\xi<\gamma}U_\xi$. The judge wins if she can continue playing for $\delta$-many steps.
2. Let $G_\delta(\kappa)$ be an analogously defined game where we additionally require that the ultrapower of $M=\bigcup_{\xi<\delta}M_\xi$ by $U=\bigcup_{\xi<\delta}U_\xi$ is well-founded.

Ultimately, several large cardinals in the interval between weakly compact cardinals and measurable cardinals have been characterized by the existence of a winning strategy for the judge in one of the filter games. The following list is a partial account of characterizations of this kind. Suppose that $\kappa$ is inaccessible:

1. $\kappa$ is weakly compact if and only if the judge has a winning strategy in the game $G^{(*)}_1(\kappa)$ (see the previous paragraph).
2. (Keisler-Tarski [3]) $\kappa$ is weakly compact if and only if the judge has a winning strategy in the game $wG^*_\omega(\kappa)$.
3. (Nielsen [6] (Theorem 3.4)) If the judge has a winning strategy in the game $G_n(\kappa)$ for some $1\leq n<\omega$, then $\kappa$ is $\Pi^1_{2n}$-describable and $\Pi^1_{2n-1}$-indescribable.
4. (follows from Theorem 3.12 in [6]) The judge has a winning strategy in the game $wG_\omega(\kappa)$ if and only if $\kappa$ is completely ineffable.
5. ([4] Observation 3.5) Suppose $2^\kappa=\kappa^+$. Then $\kappa$ is measurable if and only if the judge has a winning strategy in the game $G^{(*)}_{\kappa^+}(\kappa)$.

In a recent work, Foreman, Magidor and Zeman [5] continued this investigation and proved the following elegant result:

Theorem 1: The following are equiconsistent:

1. There is an inaccessible cardinal $\kappa$ such that the judge has a winning strategy in the game $G_{\omega+1}^*(\kappa)$.
2. There is a measurable cardinal.

Theorem 2: Assume that $\kappa$ is inaccessible, $2^\kappa=\kappa^+$, and that $\kappa$ does not carry a $\kappa$-complete $\kappa^+$-saturated ideal on $\kappa$. Let $\delta>\omega$ be a regular cardinal less than $\kappa^+$. If the judge has a winning strategy in the game $G^*_\delta(\kappa)$, then there is a uniform normal ideal $\mathcal{I}$ on $\kappa$ and a set $D \subseteq \mathcal{I}^+$ such that:

1. $(D,\subseteq_\mathcal{I})$ is a downward growing tree of height $\delta$.
2. $D$ is $\delta$-closed.
3. $D$ is dense in $\mathcal{I}^+$.

They also proved a partial converse:

Theorem 3 Let $\delta\leq\kappa$ be uncountable regular cardinals and $J$ be a $\kappa$-complete ideal on $\kappa$ which is $(\kappa^+,\infty)$-distributive and has a dense $\delta$-closed subset. Then the judge has a winning strategy in the game $G^*_\delta(\kappa)$, which is constructed in a natural way from the ideal $\mathcal{J}$.

In their paper, the authors asked whether the filter games can be generalized to the two-cardinal setting [5] (Question 5). In this paper we provide analogues of the filter games for filters on $P_\kappa(\lambda)$, $(w)G_\delta(\kappa,\lambda)$ and $(w)G^*_\delta(\kappa,\lambda)$. We also consider the strong game $sG_\delta(\kappa,\lambda)$ (introduced in the one cardinal context in [5]), where we require that the ultrafilter resulting from unioning up the judge's moves is countably complete. We add in the parameter $\theta$ when we need the challenger's moves to be elementary in some $H_\theta$, which as in the original filter games, affects only games of length $\delta$ with $\text{cof}(\delta)=\omega$. Often, the one cardinal $\kappa$-theory of ultrafilters, turns out to be a particular case of the two cardinal $(\kappa,\lambda)$-theory when considering the case $\kappa=\lambda$. In particular, uniform filters on $\kappa$ can be identified with (fine) uniform filters on $P_\kappa(\kappa)$, and the notions of normality and completeness coincide. Here we will also generalize the notion of a $\kappa$-model. In that sense, the two-cardinal games we introduce generalize the one-cardinal games of Holy and Schlicht. We then prove that major parts of the theory of the one cardinal filter games (and in particular all the results above) generalize to the two-cardinal settings.

When passing from ultrafilters on $\kappa$ to ultrafilters on $P_\kappa(\lambda)$, a distinction appears between the existence of $\kappa$-complete ultrafilters and normal ultrafilters. Thus, even for games of length 1, it is expected that there will be a difference between the assumption that there exists a winning strategy for the judge in the game $G^*_1(\kappa,\lambda)$ and the game $G_1(\kappa,\lambda)$. We start with a simple observation that $\lambda$-supercompact/strongly compact cardinals play the role of measurable cardinals.

Theorem: Assume $2^\lambda=\lambda^+$ and $\lambda^{<\kappa}=\lambda$.

1. The judge has a winning strategy in the game $G^*_{\lambda^+}(\kappa,\lambda)$ if and only if $\kappa$ is $\lambda$-strongly compact.
2. The judge has a winning strategy in the game $G_{\lambda^+}(\kappa,\lambda)$ if and only if $\kappa$ is $\lambda$-supercompact.

Finite levels of the game

The role of weakly compact cardinals is filled by nearly $\lambda$-supercompact cardinals and nearly $\lambda$-strongly compact cardinals of Shankar and White respectively [7] [8]:

Theorem: Assume $\lambda^{<\kappa}=\lambda$.

1. The judge has a winning strategy in the game $G^*_{1}(\kappa,\lambda)$ if and only if $\kappa$ is nearly $\lambda$-strongly compact.
2. The judge has a winning strategy in the game $G_{1}(\kappa,\lambda)$ if and only if $\kappa$ is nearly $\lambda$-supercompact.

By results of Hayut and Magidor [9], these cardinals are tightly connected to $\lambda$-$\Pi^1_1$-subcompact cardinals.

Moving to longer games, more differences arise between the games $G_\delta(\kappa,\lambda)$ and $G^*_\delta(\kappa,\lambda)$ (as they do correspondingly in the one-cardinal games). For the games with weak $M$-ultrafilters, we can strengthen (1) of the above theorem to:

Theorem: The judge has a winning strategy in the game $wG^*_{\omega}(\kappa,\lambda)$ if and only if $\kappa$ is nearly $\lambda$-strongly compact

In contrast, the existence of a winning strategy for the judge in the games $G_\delta(\kappa,\lambda)$ for $1<\delta<\omega$ gives a proper consistency strength hierarchy. We generalize Theorem 3.4 of [6] using Baumgartner's $\lambda$-$\Pi^1_n$-indescribable cardinals:

Theorem: Assume $\lambda^{<\kappa}=\lambda$.

1. A winning strategy for the judge in the game $G_n(\kappa,\lambda)$ is expressible by a $\Pi^1_{2n}$-formula.
2. If the judge has a winning strategy in the game $G_n(\kappa,\lambda)$, then $\kappa$ is $\lambda$-$\Pi^1_{2n-1}$-indescribable.

For the game $wG_\omega(\kappa,\lambda)$ we have a simple equivalence:

Theorem: The judge has a winning strategy in the game $wG_\omega(\kappa,\lambda)$ if and only if $\kappa$ is completely $\lambda$-ineffable.

Generic supercompactness

Generalizing Theorem 2.1.2 of [10] on completely ineffable cardinals, we show that completely $\lambda$-ineffable cardinals can be characterized by a form of generic supercompactness, and thus, by the previous theorem, so does the existence of a winning strategy for the judge in the game $wG_\omega(\kappa,\lambda)$. A strong relation between a winning strategy for the judge and various forms of generic supercompactness persists for the stronger games $G_\omega(\kappa,\lambda,\theta)$ and $sG_{\omega}(\kappa,\lambda,\theta)$, where the union ultrafilter is required to produce a well-founded ultrapower. The results below are inspired by analogous results of Nielsen and Welch [6].

Given a model $M$, we say that an ultrafilter $U$ on $P_\kappa(\lambda)^M$ is \emph{weakly amenable} if the restriction of $U$ to any set in $M$ of size at most $\lambda$ in $M$ is an element of $M$, that is, $M$ contains all sufficiently 'small' pieces of $U$.

1. If the judge has a winning strategy in the game $G_\omega(\kappa,\lambda,\theta)$, then in some set-forcing extension, there is a weakly amenable $H_{\theta}$-ultrafilter with a well-founded ultrapower. Thus, if the judge has a winning strategy in the game $G_\omega(\kappa,\lambda,\theta)$ for every regular $\theta\geq\lambda^+$, then $\kappa$ is generically $\lambda$-supercompact for sets with weak amenability.
2. If in a set-forcing extension, there is an elementary embedding $j:H_\theta\to M$ with $\text{crit}(j)=\kappa$, $j(\kappa)>\lambda$, $j''\lambda\in M$, and $M\subseteq V$, then the judge has a winning strategy in the game $G_\omega(\kappa,\lambda,\theta)$.

Although $(1)$ and $(2)$ above are almost converses of each other, it is unclear to us how to get an exact equivalence.

Theorem: If the judge has a winning strategy in the game $sG_\omega(\kappa,\lambda)$, then $\kappa$ is generically $\lambda$-supercompact with weak amenability and $\omega_1$-iterability.

Above $\omega$, we have the following generic supecompactness equivalence:

1. $\kappa$ is generically $\lambda$-supercompact with weak amenability by $\delta$-closed forcing.
2. The judge has a winning strategy in the game $G_{\delta}(\kappa,\lambda)$.

Precipitous ideal and closed dense subtrees

Finally, we prove a similar result to Foreman, Magidor and Zeman's Theorem 1:

Theorem: If the judge has a winning strategy in the game $sG^*_\omega(\kappa,\lambda)$, then there is a fine $\kappa$-complete precipitous ideal on $P_\kappa(\lambda)$. If the judge has a winning strategy in the game $sG_\omega(\kappa,\lambda)$, then we have, moreover, that the ideal is normal.

A major difference in our approach is that we do not pass through the game where we choose sets determining $M$-ultrafilters instead of $M$-ultrafilters. Instead, we construct a tree of $M$-ultrafilters and prove that this suffices to obtain a precipitous ideal. This approach can be used to slightly simplify the proof of Theorem 1. Of course, the observation of [5] that one can move to a game where the judge plays sets determining ultrafilters is highly interesting on its own merit.

In the last section, we switch back to the one cardinal setting and provide some additional information related to the results of (missing reference). To address Question 1, we prove that the assumptions regarding $\kappa$ carrying no $\kappa$-complete $\kappa^+$-saturated ideal of Theorem 1 are necessary:

Theorem: Suppose that $\mathcal{I}$ is a $\kappa$-complete $\kappa$-measuring ideal on $\kappa$ and there is a tree $D\subseteq \mathcal{I}^+$ such that:

1. $D$ is dense in $\mathcal{I}^+$.
2. $(D,\subseteq_\mathcal{I})$ is a downward growing tree of height $\delta$.
3. $D$ is $\delta$-closed.

Finally, to address Question $2$, given a tree $D$ dense in an ideal $\mathcal{I}$ satisfying $(1)$-$(3)$ as above, we construct a subtree $D^*\subseteq D$ and prove the following:

Theorem: Let $D$ be a dense subtree of $\mathcal{I}$ satisfying $(1)$-$(3)$. Then the hopeless ideal associated to the strategy $\sigma^D$, $\mathcal{I}(\sigma^D)=\mathcal{I}$ if and only if $D=D^*$.

Here is a diagram showing the implications and equiconsistencies between the various games and large cardinals.

References

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