Filter extension games on mini supercompactness filters

As the name suggests, a measurable cardinal $\kappa$ is characterized by the existence of a measure - a non-trivial $\kappa$-complete ultrafilter on $\kappa$. Smaller large cardinals $\kappa$, such as weakly compact, ineffable, and Ramsey cardinals, can also be characterized by the existence of ultrafilters, obviously not on the entire powerset of $\kappa$, but on $\kappa$-sized chunks of the powerset. These chunks arise as powersets of $\kappa$ of $\kappa$-sized $\in$-models of set theory. Let us say that a set $M$ is a weak $\kappa$-model if it is a transitive $\in$-model of size $\kappa$ with $\kappa\in M$ satisfying the theory ${\rm ZFC}^-$ (roughly ${\rm ZFC}$ without powerset, see [1] for details). We remove the adjective 'weak' if $M$ is closed under sequence of length less than $\kappa$, $M^{\lt\kappa}\subseteq M$. In most cases we will end up dealing with, $V_\kappa$ will be a subset of $M$, but $P(\kappa)$ won't exist in $M$. Given a weak $\kappa$-model $M$, we will say that $U\subseteq P(\kappa)^M$ is an $M$-ultrafilter if it is a filter measuring every set in $P(\kappa)^M$ that is $M$-$\kappa$-complete - complete for sequences of length less than $\kappa$ that are elements of $M$, and $M$-normal - closed under diagonal intersections of sequences that are elements of $M$. The filter $U$ is, in the cases we consider, always external to $M$. We will say that an $M$-ultrafilter $U$ has the countable intersection property (this is a weaking of countable completeness) if whenever $\{A_n\mid n\lt\omega\}$ is a sequence of sets in $U$, but possibly not an element of $M$, then $\bigcap_{n\lt\omega}A_n\neq\emptyset$. It is easy to see that if an $M$-ultarfilter $U$ has the countable intersection property, then the ultrapower of $M$ by $U$ is well-founded. If $M$ is closed under sequences of length $\omega$, then an $M$-ultrafilter $U$ automatically has the countable intersection property, but otherwise we cannot argue that the ultrapower of $M$ by $U$ is well-founded, and indeed this need not be the case in general for weak $\kappa$-models. We also cannot always iterate the ultrapower construction by an $M$-ultrafilter because since the $M$-ultrafilter $U$ is external $M$, we cannot apply the ultrapower map to it to get the next ultrafilter in the iteration. However, it turns out the ultrafilter need not be an element of the model in order to be able to iterate the ultrapower construction. It suffices if the model contains sufficiently large pieces of the ultrafilter. Thus, an $M$-ultrafilter $U$ is called weakly amenable if for every sequence $\vec A=\{A_\xi\subseteq\kappa\mid\xi\lt\kappa\}$ in $M$, $\vec A\cap U\in M$. With a weakly amenable $M$-ultrafilter, we can iterate the ultrapower construction, but unless the ultrafilter has the countable intersection property [2], we are not guaranteed to have well-founded iterated ultrapowers. Indeed, a weakly amenable $M$-ultrafilter can have any number $\alpha$ of well-founded iterated ultrapowers for $0\leq\alpha\lt\omega_1$ [3], and if it has $\omega_1$-many well-founded iterated ultrapowers, then, as is in the case with a measure on $\kappa$, all the iterated ultrapowers will be well-founded.

Given a weak $\kappa$-model $M$ and an $M$-ultrafilter $U$, we can enumerate $U=\{A_\xi\mid\xi\lt\kappa\}$, consider the diagonal intersection $\Delta_{\xi\lt\kappa} A_\xi$, and ask whether it is stationary. It is not difficult to see that the stationarity is independent of the enumeration and is only a property of $U$.

For everything that follows we will be supposing that $\kappa$ is inaccessible. Let's see some characterizations of classical smaller large cardinals in terms of the existence of $M$-ultrafilters.

The strength of Ramsey embeddings comes from both the weak amenability and the countable intersection property. Even if we remove the countable intersection property, we get a large cardinal notion stronger than ineffability. Incidentally, if we require that every subset of $\kappa$ is in a $\kappa$-model for which there is a weakly amenable $M$-ultrafilter, then we get a large cardinal notion a little bit stronger than a Ramsey cardinal and requiring that every $\kappa$-model has a weakly amenable $M$-ultrafilter is outright inconsistent. [6]

Now let's ask the following question. Suppose $\kappa$ is weakly compact. Given a $\kappa$-model $M_0$, an $M_0$-ultrafilter $U_0$ and a $\kappa$-model $M_1$ extending $M_0$, can we always find an $M_1$-ultrafilter $U_1$ extending $U_0?$ Let's say that a weak $M$-ultrafilter is an $M$-ultrafilter without $M$-normality. Then if we replace $M$-ultrafilters with weak $M$-ultrafilters, the answer to our question is positive (use the tree property), but, otherwise, it is negative [7]. Alright, so what if we don't just take any $U_0$, but choose it carefully to ensure that for any $M_1$, we can extend? This strategic approach works, but it has a cost in strength that goes beyond weakly compacts. If we assume stronger large cardinals exist, we can play this game much longer. Here is a formal definition of this 'filter extension' game, and its variants, due to Holy and Schlicht [7].

Definition: Suppose that $\delta\leq\kappa^+$ 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 $\{\langle M_\xi,U_\xi\rangle \mid \xi\lt\gamma\}\in M_\gamma$, elementarily extending his previous moves and the judge plays an $M_\gamma$-ultrafilter $U_\gamma$ extending $\bigcup_{\xi\lt\gamma}U_\xi$. 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 $M=\bigcup_{\xi\lt\delta}M_\xi$ by $U=\bigcup_{\xi\lt\delta}U_\xi$ is well-founded.
  3. Let $sG_\delta(\kappa)$ be the analogous game, but where to win the judge must satisfy the additional requirement that $U=\bigcup_{\gamma\lt\delta}U_\gamma$ has the countable intersection property. (This game was not explicitly considered in [7].)

All three games are easily seen to be equivalent whenever $\text{cof}(\delta)\neq\omega$. In the original definition of the games, there is an additional parameter, a regular cardinal $\theta$, which restricts the challenger to play (a non-transitive version of) $\kappa$-models elementary in $H_\theta$. The parameter $\theta$ is important only for $\delta$ of cofinality $\omega$ because otherwise either player has a winning strategy in a game $G_\delta(\kappa)$ if and only if the same player has a winning strategy in the game with the $\kappa$-models elementary in $H_\theta$ [7].

Obviously, $\kappa$ is weakly compact if and only if the judge has a winning strategy in the game $G_1(\kappa)$. But if the judge has a winning strategy in the game $G_2(\kappa)$, then $\kappa$ is already ineffable. More generally, Nielsen showed that if $n$ is finite and the judge has a winning strategy in $G_n(\kappa)$, then $\kappa$ is $\Pi^1_{2n-1}$-indescribable [8]. It follows from arguments in [4] that the judge has a winning strategy in the game $wG_\omega(\kappa)$ if and only if $\kappa$ is completely ineffable [8]. Nielsen and Welch showed that the judge having a winning strategy in the game $G_\omega(\kappa)$ is equiconsistent with a virtually measurable cardinal [8]. Foreman, Magidor, and Zeman showed that if the judge has a winning strategy in the game $sG_{\omega}(\kappa)$, then there is a uniform normal precipitous ideal on $\kappa$ [9]. Thus, the judge having a winning strategy in $sG_{\omega}(\kappa)$ is equiconsistent with a measurable cardinal. Finally, it is an easy observation that if $2^\kappa=\kappa^+$, then the judge has a winning strategy in the game $G_{\kappa^+}(\kappa)$ if and only if $\kappa$ is measurable.

Large cardinals characterized by the existence of a winning strategy for the judge in a filter extension game often turn out to be generic large cardinals or are equiconsistent with generic large cardinals. I will write more about this in the next post.

Now let's try to build up an analogous landscape in the so called two-cardinal setting ($\kappa$ and $\lambda$), where we replace measurable cardinals with $\lambda$-supercompact cardinals and consider mini measures on $P_\kappa(\lambda)$ [10]. We will always assume that $\lambda^{\lt\kappa}=\lambda$. To get started we need to come up with analogues of small models and their ultrafilters for the two-cardinal setting.

Definition: A transitive $\in$-model $M$ is a a weak $(\kappa,\lambda)$-model if

  1. it has size $\lambda$,
  2. $\lambda\in M$,
  3. some bijection $f:\lambda\to P_\kappa(\lambda)\in M$,
  4. $M$ satisfies ${\rm ZFC}^-$.
A weak $(\kappa,\lambda)$-model $M$ is a $(\kappa,\lambda)$-model if $M$ is closed under sequences of length less than $\kappa$.

Note that for every $\alpha\lt\lambda$, the set $X_\alpha=\{a\in P_\kappa(\lambda)\mid \alpha\in a\}$ is in $M$ by separation. Next, we need the notion of an $M$-ultrafilter.

Definition: Suppose that $M$ is a weak $(\kappa,\lambda)$-model. A set $U\subseteq P(P_\kappa(\lambda))^M$ is an $M$-ultrafilter if

  1. $U$ is a filter measuring every set in $P(P_\kappa(\lambda))^M$,
  2. $U$ is fine ($X_\alpha\in U$ for every $\alpha\lt\lambda$),
  3. $U$ is $M$-$\kappa$-complete,
  4. $U$ is $M$-normal.
A weak $M$-ultrafilter is an $M$-ultrafilter with $M$-normality removed.
A weak $M$-ultrafilter $U$ is weakly amenable if for every sequence $\vec A=\{A_\xi\subseteq P_\kappa(\lambda)\mid\xi\lt\lambda\}$ in $M$, $\vec A\cap U\in M$.

Schanker introduced the nearly $\lambda$-supercompact cardinal as the analogue, in the two-cardinal setting, of the weakly compact cardinal via the existence of $M$-ultrafilters characterization [11]. A cardinal $\kappa$ is nearly $\lambda$-supercompact if every $A\subseteq\lambda$ is in a weak $(\kappa,\lambda)$-model for which there is an $M$-ultrafilter with a well-founded ultrapower. As with weakly compact cardinals, equivalently, every $A\subseteq\lambda$ is in a $(\kappa,\lambda)$-model $M$ for which there is an $M$-ultrafilter, and equivalently every $(\kappa,\lambda)$-model has an $M$-ultrafilter. Nearly $\lambda$-supercompact cardinals are quite strong, they are $\theta$-supercompact for every $\theta$ with $2^{\theta^{\lt\kappa}}\leq\lambda$, but, as Schanker showed, a nearly $\lambda$-supercompact cardinal need not even be measurable [11]. If we replace $M$-ultrafilters by weak $M$-ultrafilters in the definition of nearly $\lambda$-supercompact cardinal, then we get White's notion of nearly $\lambda$-strongly compact cardinal. As in the one-cardinal setting, we get that we can extend weak $M$-ultrafilters [12], but extending $M$-ultrafilters is inconsistent [10]. So now we can bring in the games.

Definition: Suppose that $\delta\leq\lambda^+$ and $\theta\geq\lambda$ are regular cardinals.

  1. Let $wG_\delta(\kappa,\lambda)$ be the following two player game of perfect information played by the challenger and judge. The challenger starts the game and plays a $(\kappa,\lambda)$-model $M_0$ and the judge responds by playing an $M_0$-ultrafilter $U_0$. At stage $\gamma>0$, the challenger plays a $(\kappa,\lambda)$-model $M_\gamma$, with $\{\langle M_\xi,U_\xi\rangle\mid \xi\lt\gamma\}\in M_\gamma$ elementarily extending his previous moves, and the judge plays an $M_\gamma$-ultrafilter $U_\gamma$ extending $\bigcup_{\xi\lt\gamma}U_\xi$. The judge wins if she can continue playing for $\delta$-many steps.
  2. Let $G_\delta(\kappa,\lambda)$ be the analogous game, but where to win the judge must satisfy the additional requirement that the ultrapower of $M=\bigcup_{\gamma\lt\delta}M_\gamma$ by $U=\bigcup_{\gamma\lt\delta}U_\gamma$ is well-founded.
  3. Let $sG_\delta(\kappa,\lambda)$ be the analogous game, but where to win the judge must satisfy the additional requirement that $U=\bigcup_{\gamma\lt\delta}U_\gamma$ has the countable intersection property.
  4. Let $(w/s)G^{*}_\delta(\kappa,\lambda)$ be the analogously defined games, but where the judge is only required to play weak $M_\xi$-ultrafilters.
We add the parameter $\theta$ to obtain the analogous games $(w/s)G^{(*)}(\kappa,\lambda,\theta)$, where the challenger instead plays a (non-transitive version of a) $(\kappa,\lambda)$-models elementary in $H_\theta$.

The parameter $\theta$, as in the original filter games, affects only games of length $\delta$ with $\text{cof}(\delta)=\omega$. Also, if $\text{cof}(\delta)\neq\omega$, then the three games are all equivalent. Now let's characterize a bunch of partially supercompact large cardinals in terms of the existence of a winning strategy for the judge in the games. [10]

For definitions of $\lambda$-ineffable and $\lambda$-$\Pi^1_n$-indescribable cardinals, see [10].

Again, as in the one-cardinal games, the existence of a winning strategy for the judge is connected to the existence of various forms of generic supercompactness. This is explored in the next post.

References

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  2. K. Kunen, “Some applications of iterated ultrapowers in set theory,” Ann. Math. Logic, vol. 1, pp. 179–227, 1970.
  3. V. Gitman and P. D. Welch, “Ramsey-like cardinals II,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 541–560, 2011.
  4. 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.
  5. W. Mitchell, “Ramsey cardinals and constructibility,” J. Symbolic Logic, vol. 44, no. 2, pp. 260–266, 1979.
  6. V. Gitman, “Ramsey-like cardinals,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 519–540, 2011.
  7. P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals,” Fund. Math., vol. 242, no. 1, pp. 49–74, 2018.
  8. D. Saattrup Nielsen and P. Welch, “Games and Ramsey-like cardinals,” J. Symb. Log., vol. 84, no. 1, pp. 408–437, 2019.
  9. M. Foreman, M. Magidor, and M. Zeman, “Games with filters I,” Journal of Mathematical Logic, pp. to appear, 2023.
  10. T. Benhamou and V. Gitman, “Cardinals of the $P_κ(λ)$-filter games,” Manuscript, 2025.
  11. J. A. Schanker, “Partial near supercompactness,” Ann. Pure Appl. Logic, vol. 164, no. 2, pp. 67–85, 2013. Available at: https://doi.org/10.1016/j.apal.2012.08.001
  12. D. Buhagiar and M. Džamonja, “Square compactness and the filter extension property,” Fund. Math., vol. 252, no. 3, pp. 325–342, 2021. Available at: https://doi.org/10.4064/fm787-4-2020