Victoria Gitman

Class choice and the surprising weakness of Kelley-Morse set theory

2025-10-06T00:00:00-04:00

V. Gitman, J. D. Hamkins, and T. Johnstone, “Class choice and the surprising weakness of Kelley-Morse set theory.”

Suppose that for every natural number $n$ there is a class $X$ for which some property $\varphi(n,X)$ holds-assume we are working in a suitable set theory with classes, including the axiom of choice and even the global choice principle. Should we expect to find a uniformizing class $Z\subseteq\omega\times V$ for which $\varphi(n,Z_n)$ holds for all the various sections? Similarly, if every ordinal $\alpha$ has a class $X$ for which $\varphi(\alpha,X)$ holds, then should we we expect to find a class $Z\subseteq{\rm Ord}\times V$ for which $\varphi(\alpha,Z_\alpha)$ holds on every section? Or if every set $x$ admits a class $X$ with $\varphi(x,X)$, then will we find a class $Z\subseteq V\times V$ which unifies these witnesses $\varphi(x,Z_x)$?

These would all be instances of the class choice scheme, a central concern of this article. The class $Z$ in each case uniformizes the witnesses, in effect choosing for each index $x$ a class witness $Z_x$ on that section. The class choice scheme is therefore a choice principle enabling a choice of classes, rather than sets.

It is relatively easy to see that the class choice scheme is not provable in Gödel-Bernays ${\rm GBC}$ set theory, even with the axiom of global choice being available in that theory. A counterexample is provided by any transitive model of ${\rm ZFC}+V={\rm HOD}$, equipped with all and only its definable classes. To see this, observe that for each $n$ there is a definable $\Sigma_n$-truth predicate (a partial satisfaction class), but by Tarski's theorem on the non-definability of truth, these cannot be unified into a single definable class. Indeed, in this case we are not really 'choosing' the classes at all, since in fact for each $n$ there is a unique $\Sigma_n$-truth predicate. Rather, what is going on is that ${\rm GBC}$ is not strong enough (although ${\rm KM}$ is) to collect these various unique witnesses together into a single uniform class. The situation resembles more a class-wise failure of collection or even replacement than choice.

Perhaps it will be a little more surprising, however, to learn that the class choice scheme is also not provable in the considerably stronger Kelley-Morse ${\rm KM}$ set theory, against which the simple truth-predicate counterexample above is not successful, as ${\rm KM}$ does prove the existence of uniform first-order truth predicate satisfaction classes relative to any class parameter. Nevertheless, we show that ${\rm KM}$ does not prove the class choice scheme, and indeed the scheme can fail even in very low-complexity first-order instances and again with merely a set of indices. Kelley-Morse set theory ${\rm KM}$ simply does not prove the class choice scheme, even in easy-seeming cases.

Furthermore, the proof methods reveal several further surprising weaknesses of ${\rm KM}$, which may call into question the suitability of this theory as a foundation of set theory. Namely, ${\rm KM}$ does not prove the Łoś theorem scheme for second-order internal ultrapowers, even in the case of large-cardinal ultrapowers, such as those arising from a normal measure on a measurable cardinal-the familiar large cardinal embeddings can fail to be elementary in the language of ${\rm KM}$ set theory. In the case of ultrapowers by an ultrafilter on $\omega$, we show, the internal ultrapower of the ${\rm KM}$ universe is not itself always even a model of ${\rm KM}$. Finally, we will show that ${\rm KM}$ does not prove that second-order logical complexity is preserved by first-order quantifiers, even bounded quantifiers. In light of all these weaknesses, the theory ${\rm KM}$ appears not quite as robust a foundational theory as might have been thought.

Meanwhile, by augmenting the theory ${\rm KM}$ with the class choice scheme, thereby forming the theory we denote ${\rm KM}^+$, we addresses all these weaknesses, and in this sense ${\rm KM}^+$ succeeds as a robust foundational treatment of second-order set theory.

Background on sets and classes

The objects of first-order set theory are sets, and set theorists commonly take the classes of a model of first-order set theory to be simply the definable collections of sets (allowing parameters), which means that the study of their properties naturally takes place in the meta-theory. A formal framework for studying sets and classes, in contrast, is provided by second-order set theory. The most natural interpretation of second-order set theory is in a two-sorted logic with separate sorts (separate variables and quantifiers) for sets and classes. The language of second-order set theory has two membership relations, one deciding set membership in sets and the other deciding set membership in classes. Because foundational axioms of second-order set theory assert that classes are extensional collections of sets, the models of second-order set theory relevant for us have the form $\langle V,\in,\mathcal S\rangle$, where $\langle V,\in\rangle$ is a model of the language of first-order set theory and $\mathcal S$ is the family of classes, that is, collections of sets. Following the standard convention, we use lowercase letters for the sets and uppercase letters for the classes. In the meta-theory of second-order set theory, we can also study hyperclasses, the definable (with parameters) collections of classes.

Any reasonable axiomatic foundation for second-order set theory should assert that the collection of all sets satisfies ${\rm ZFC}$ and that, more generally, it continues to satisfy ${\rm ZFC}$ in the extended first-order language with predicates for any finitely many classes. The more general requirement translates into the class replacement axiom, which states that the image of a set under a class function is itself a set. A reasonable foundation should also include some class existence principles and should in particular generalize the notion of classes from first-order set theory by asserting that every first-order definable collection is a class. The two most commonly considered axiomatic foundations for second-order set theory are the Gödel-Bernays axioms (${\rm GBC}$) and the Kelley-Morse axioms (${\rm KM}$). They are distinguished by the strength of their comprehension axiom schemes that decide which (second-order) definable collections are classes. The theory ${\rm GBC}$ includes only the minimal first-order comprehension and it is equiconsistent with ${\rm ZFC}$. The theory ${\rm KM}$ asserts full second-order comprehension and strength-wise it lies between the existence of a transitive model of ${\rm ZFC}$ and the existence of an inaccessible cardinal.

Class choice schemes

Axiomatizations of second-order set theory often include choice principles for classes because these imply many properties desirable in this context. A commonly used choice principle is the class choice scheme, which asserts that every $V$-indexed definable family of hyperclasses has a choice function $\mathcal F:V\to \mathcal S$. Historically, this principle can be traced to Marek's PhD thesis in the 1970s [1] and was rediscovered several times, most recently in the work of Hrbáček on nonstandard second-order set theories with infinitesimals [2] and Antos and Friedman on definable hyperclass forcing over ${\rm KM}$ models (missing reference). To give the precise statement of the class choice scheme, we use the standard notation that if $Z$ is a class, then $$Z_x=\{y\mid(x,y)\in Z\}$$ denotes the class coded on section $x$ of $Z$. We think of such a class $Z$ as naturally encoding the function $x\mapsto Z_x$, enabling us in effect to refer to functions $\mathcal F:V\to\mathcal S$ as above. In general, if a collection of classes can be coded like this by a single class $Z$, we will say that it constitutes a codable hyperclass.

Definition: The class choice scheme is the following scheme of assertions, for any formula $\varphi$ in the language of second-order set theory and allowing any class parameter $A$: $$\forall x\,\exists X\,\varphi(x,X,A)\rightarrow\exists Z\,\forall x\,\varphi(x,Z_x,A).$$

As we hinted in the introduction, the class choice scheme can also be viewed as a collection principle rather than a choice principle, because it is equivalent over ${\rm GBC}$ to the following slightly weaker-seeming principle, the class-collection scheme: $$\forall x\,\exists X\,\varphi(x,X,A)\rightarrow \exists Z\,\forall x\,\exists y\,\varphi(x,Z_y,A).$$ In this formulation, the class witnesses $X$ are merely collected as sections $Z_y$, but not necessarily on the corresponding section $Z_x$. Note first that either principle over ${\rm GBC}$ implies ${\rm KM}$, and then the further point is that with the class collection scheme one can choose for each $x$ a suitable $y$ by global choice and thereby replace the sections on the correct index, making the two principles equivalent.

The class choice scheme admits several natural weakenings, by stratifying on the logical complexity of $\varphi$ or restricting to sets of indices.

Definition:

The $\Sigma^1_n$ (or $\Pi^1_n$)-class choice scheme is the restriction of the class choice scheme to instances $\varphi$ of complexity at most $\Sigma^1_n$ (or $\Pi^1_n$).
The set-indexed class choice scheme is the restriction of the class choice scheme to \emph{set}-indexed families. That is, for any set $a$, class parameter $A$, and any formula $\varphi$ in the language of second-order set theory, we include $$\forall x\in a\,\exists X\,\varphi(x,X,A)\rightarrow\exists Z\,\forall x\in a\,\varphi(x,Z_x,A).$$
The parameter-free class choice scheme is the restriction of the class choice scheme to parameter-free formulas $\varphi(x,X)$.

The main results

We aim to show that ${\rm KM}$ fails to prove even the weakest instances of the class choice scheme, concerning the existence of choice functions for $\omega$-indexed parameter-free first-order definable families. This provides an interesting counterpoint to the situation in second-order arithmetic, where ${\rm Z}_2$, usually considered a tight arithmetic analogue of ${\rm KM}$, proves all $\Sigma^1_2$ instances of the class choice scheme. We also show that ${\rm KM}$ together with the set-indexed class choice scheme does not imply the class choice scheme even for parameter-free first-order formulas and that ${\rm KM}$ together with the parameter-free class choice scheme does not imply the class choice scheme. We shall freely make use of mild large cardinal assumptions when proving the main results, since these assumption do not detract from our main point that ${\rm KM}$ does not prove the desired features for a foundational second-order theory of sets and classes, as set theorists want to use the foundational theory with those and far stronger large cardinals.

Theorem: (${\rm ZFC}\,+\,$there is a Mahlo cardinal) There is a model of ${\rm KM}$ in which an instance $$\forall n{\in}\omega\,\exists X\,\varphi(n,X)\rightarrow \exists Z\,\forall n{\in}\omega\,\varphi(n,Z_n)$$ of the class choice scheme fails for a first-order formula $\varphi(x,X)$.

Theorem: (${\rm ZFC}\,+\,$there is a Mahlo cardinal) There is a model of ${\rm KM}$ in which the set-indexed class choice scheme holds, but the class choice scheme fails for a parameter-free first-order formula.

Theorem: (${\rm ZFC}\,+\,$there is an inaccessible cardinal) There is a model of ${\rm KM}$ in which the parameter-free class choice scheme holds but the class choice scheme fails for a $\Pi^1_1$-formula.

The above theorem and its proof are motivated by the work in [3].

These failures of ${\rm KM}$ to prove even the weakest instances of the class choice scheme indicates a fundamental weakness in the theory and might lead us to reconsider its prominent role in second-order set theory. To highlight the weakness, we show that the Łoś theorem scheme can fail for the internal second-order ultrapower of the ${\rm KM}$ universe.

Theorem: (${\rm ZFC}\,+\,$there is an inaccessible cardinal) There is a model of ${\rm KM}$ whose internal second-order ultrapower by an ultrafilter on $\omega$ is not itself a ${\rm KM}$-model.

Similarly, we show that ${\rm KM}$ does not prove that large cardinal embeddings, such as the ultrapower of the universe by a normal measure on a measurable cardinal, are elementary in the language of ${\rm KM}$.

Theorem: (${\rm ZFC}\,+\,$measurable cardinal with an inaccessible above) There is a model of ${\rm KM}$ with a measurable cardinal $\delta$, such that the internal ultrapower of the universe by a normal measure on $\delta$ is not elementary in the language of ${\rm KM}$ set theory.

We also show that ${\rm KM}$ does not prove that $\Sigma^1_1$ formulas are (up to equivalence) closed under first-order quantifiers, even bounded first-order quantifiers.

Theorem: (${\rm ZFC}\,+\,$measurable cardinal with an inaccessible above) The theory ${\rm KM}$ fails to establish that $\forall \eta\lt\delta\,\psi(\eta,X)$ is provably equivalent to some $\Sigma^1_1$ formula whenever $\psi$ is.

In every case, these various deficiencies of ${\rm KM}$ are remedied by replacing it with the theory which we call ${\rm KM}^+$, which augments ${\rm KM}$ with the class choice scheme. A result of Marek shows that ${\rm KM}^+$ is bi-interpretable with the first-order theory ${\rm ZFC}^-$, that is, ${\rm ZFC}$ without the powerset axiom, together with the assertion that there is a largest cardinal that is furthermore an inaccessible cardinal [1].

References

W. Marek, “On the metamathematics of impredicative set theory,” Dissertationes Math. (Rozprawy Mat.), vol. 98, p. 40, 1973.
K. Khrbachek, “Remarks on nonstandard class theory,” Fundam. Prikl. Mat., vol. 11, no. 5, pp. 233–255, 2005. Available at: http://dx.doi.org/10.1007/s10958-007-0374-0
W. Guzicki, “On weaker forms of choice in second order arithmetic,” Fund. Math., vol. 93, no. 2, pp. 131–144, 1976.

Parameter-free schemes in second-order arithmetic

2025-09-07T00:00:00-04:00

This is a talk at the MAMLS Fall Fest 2025, September 27, 2025.
Slides

The talk is similar to the talk in this post.

Reflection principles in set theory without powersets

2025-08-06T00:00:00-04:00

V. Gitman, “Reflection principles in set theory without powerset,” Manuscript, 2025.

The Reflection principle is the scheme of assertions that every formula is reflected by a transitive set, namely that for every formula $\varphi(x,a)$, with parameter $a$, there is a transitive set $S$ containing $a$ such that for all $s\in S$, $\varphi(s,a)$ holds if and only if it holds in $S$. The Reflection principle follows from ${\rm ZFC}$, with the witnessing sets being elements of the $V_\alpha$-hierarchy. Given a formula $\varphi$, the reflecting $V_\alpha$ is constructed as the union of a sequence $V_{\alpha_0}\subseteq V_{\alpha_1}\subseteq \cdots V_{\alpha_n}\subseteq\cdots$ of length $\omega$, where each $V_{\alpha_{n+1}}$ is a closure of $V_{\alpha_n}$ under witnesses for all existential sub-formulas of $\varphi$, and so reflects $\varphi$ by the Tarski-Vaught test. Since the argument uses only the existence of the $V_\alpha$-hierarchy together with Replacement to verify that the desired $\alpha$ exists, it goes through in ${\rm ZF}$ as well.

The definition of the $V_\alpha$-hierarchy requires the existence of powersets, so it is natural to ask whether the Reflection principle continues to hold in set theories without powersets. Since some naturally equivalent versions of the ${\rm ZFC}$ axioms stop being equivalent once the Powerset axiom is removed, we end up with several versions of set theory without powersets. Without the Powerset axiom, the Replacement and Collection schemes are no longer equivalent and neither are the versions of the Axiom of Choice which we use interchangeably [1]. For instance, ${\rm ZFC}$ without the Powerset axiom, with the Collection scheme instead of Replacement, although it has ${\rm AC}$, does not imply that every set can be well-ordered. Let ${\rm ZFC}-$ be the theory consisting of the axioms of ${\rm ZFC}$ (with Replacement) and the assertion that every set can be well-ordered. The theory ${\rm ZFC}-$ exhibits many undesirable behaviors. It can have models in which $\omega_1$ is a countable union of countably many sets, in which $\omega_1$ exists, but every set of reals is countable, or where the Łoś theorem can fail for ultrapowers [2]. All these issues can be eliminated by instead taking the theory ${\rm ZFC}^-$, where we replace the Replacement scheme with Collection, suggesting that this is the more natural version of set theory without powersets. The most common set-theoretic structures encountered by set theorists in which the Powerset axiom fails are $H_{\kappa^+}$, the collection of all sets whose transitive closure has size at most $\kappa,$ and these indeed satisfy ${\rm ZFC}^-$.

The Reflection principle clearly implies Collection over the other axioms of ${\rm ZFC}^-$. In ${\rm ZFC}^-$, every set can be closed under witnesses for existential sub-formulas of a given formula, but after that it appears that some class version of dependent choice is required to iterate this construction $\omega$-many times. Recall that the ${\rm DC}_\omega$-scheme is the class version of dependent choice which asserts that we can make $\omega$-many dependent choices along any definable relation without terminal nodes. More formally, the ${\rm DC}_\omega$-scheme is a scheme of assertions for every formula $\varphi(x,y,a)$, with parameter $a$, that if for every $x$, there is $y$ such that $\varphi(x,y,a)$ holds, then there is a sequence $\{b_n\mid n\lt\omega\}$ such that for every $n\lt\omega$, $\varphi(b_n,b_{n+1},a)$ holds. (In a personal communication, Freund has observed that over the axioms ${\rm ZFC}^-$ without Collection, the ${\rm DC}_\omega$-scheme is equivalent to the principle of induction along definable well-founded relations.) It is easy to see that ${\rm ZFC}^-$ together with the ${\rm DC}_\omega$-scheme imply the Reflection principle and indeed ${\rm ZFC}^-$ together with the Reflection principle imply the ${\rm DC}_\omega$-scheme because we can reflect the relation and the assertion that it has no terminal nodes to a transitive set and then use ${\rm AC}$ to construct the sequence of dependent choices. It is shown in [3] that the theory ${\rm ZFC}^-$ does not prove the ${\rm DC}_\omega$-scheme, and so it does not prove the Reflection principle.

In this article, we consider a family of partial Reflection principles and examine their status in models of ${\rm ZFC}^-$. Given a set or class $A$, let the partial Reflection Principle for $A$ be the scheme of assertions for every formula $\varphi(a)$, with parameter $a\in A$, that if $\varphi(a)$ holds, then it holds in some transitive set. We will abbreviate the partial Reflection principle for $V$, where any set can be used as a parameter, as just the partial Reflection principle and at the other extreme, we will also call the partial Reflection principle for $\emptyset$, where no parameters are allowed, the parameter-free Reflection principle. Note that if we remove the requirement of transitivity from the parameter-free Reflection principle, then it is provable in ${\rm ZFC}^-$ by standard proof-theoretic arguments. Freund considered some partial Reflection principles in [4]. He showed that, over the axioms of ${\rm ZFC}^-$ without Collection, the parameter-free Reflection principle is equivalent to the principle of induction along $\Delta_1$-definable well-founded relations and that the partial Reflection principle for $\mathbb R$ is equivalent to the principle of induction along $\Delta_1$-definable with real parameters well-founded relations [4]. Partial reflection in the context of Zermelo set theory was studied by Lévy and Vaught, who showed that Zermelo set theory together with the partial Reflection principle do not imply Replacement [5]. More recently, Bokai Yao studied the partial Reflection principle in set theories with urelements [6].

Models of ${\rm ZFC}^-$ in which the ${\rm DC}_\omega$-scheme fails, being the only candidates in which the partial Reflection principles can fail, are notoriously hard to construct. Currently, there are only a few known types of models of ${\rm ZFC}^-$ in which the ${\rm DC}_\omega$-scheme fails. Two of the types are constructed as submodels of forcing extensions by a tree iteration of Jensen's forcing.

Jensen's forcing $\mathbb J$ is a subposet of Sacks forcing that is constructed in $L$ using the $\diamondsuit$ principle. The poset $\mathbb J$ has the ccc and adds a unique generic real that is $\Pi^1_2$-definable as a singleton [7]. In $L$, we can appropriately define finite iterations $\mathbb J_n$ of $\mathbb J$ of length $n$. These also have the ccc and add a unique $n$-length generic sequence of reals that is $\Pi_2^1$-definable (see [8] and [3]). Again working in $L$, given a set or class tree $T$ of height $\omega$, a tree iteration $\mathbb P(\mathbb J,T)$ of $\mathbb J$ along $T$ is a poset whose elements are functions $p$ from a finite subtree $D_p$ of $T$ to $\bigcup_{n\lt\omega}\mathbb J_n$ such that if $t$ is a node on level $n$ of $D_p$, then $p(t)\in \mathbb J_n$ and whenever $s\leq t$ in $T$, then $p(s)=p(t)\upharpoonright \text{len}(s)$. The functions $p$ are ordered so that $q\leq p$ whenever $D_p\subseteq D_q$ and for every $t\in D_p$, $q(t)\leq p(t)$. An $L$-generic $G\subseteq \mathbb P(\mathbb J,T)$ adds a tree $T^G$ isomorphic to $T$ whose nodes on level $n$ are generic $n$-length sequences for $\mathbb J_n$ and the sequences extend according to the tree order. For certain sufficiently homogeneous trees, such as $T=\omega^{\lt\omega}$ or $T=\omega_1^{\lt\omega}$, the forcing $\mathbb P(\mathbb J,T)$ has the ccc and the uniqueness of generics property that the only $L$-generic filters for $\mathbb J_n$ in $L[G]$ are the nodes of $T^G$ on level $n$. Also, the collection of all $n$-generic sequences for $\mathbb J_n$ over all $n\lt\omega$, namely the elements of $T^G$, is $\Pi^1_2$-definable [3]. The class forcing $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ along the class tree ${\rm Ord}^{\lt\omega}$ has all the same properties as well [9]. Since the forcing $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ has the ccc, it is easily seen to be pretame. It follows that the forcing relation for $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ is definable and the forcing extension satisfies ${\rm ZFC}^-$ [10]. The extension has all the same cardinals as $L$, but since we added class many reals, powerset of $\omega$ does not exist. The first kind of model of ${\rm ZFC}^-$ in which the ${\rm DC}_\omega$-scheme fails is constructed in a forcing extension $L[g]$ by $\mathbb P(\mathbb J,\omega_1^{\lt\omega})$ as the $H_{\omega_1}$ of an appropriately chosen symmetric submodel $N\models {\rm ZF}+{\rm AC}_\omega+\neg{\rm DC}_\omega$ [3]. We will refer to $H_{\omega_1}^N$ as $M_{\rm{small}}^g$. The second type of model of ${\rm ZFC}^-$ in which the ${\rm DC}_\omega$-scheme fails is constructed as a submodel of a forcing extension $L[G]$ by $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ [9]. Given a set subtree $T$ of ${\rm Ord}^{\lt\omega}$, let $G_T$ be the restriction of $G$ to $\mathbb P(\mathbb J,T)$, which is easily seen to be $L$-generic for it. In $L[G]$, we let $N$ be the union of $L[G_T]$, where $T$ is a well-founded set subtree of ${\rm Ord}^{\lt\omega}$. It is shown in [9] that $N$ satisfies ${\rm ZFC}^-$, but the ${\rm DC}_\omega$-scheme fails. We will refer to the model $N$ as $M_{\rm{large}}^G$. Note that $M_{\rm{large}}^G$ has the same cardinals as $L[G]$, and so has a much more complicated structure than $M^g_{\rm{small}}$.

Working in $L$ and given an inaccessible cardinal $\kappa$, we can generalize Jensen's construction to define, the $\kappa$-Jensen forcing $\mathbb J(\kappa)$, a subposet of $\kappa$-Sacks forcing that is analogous to $\mathbb J$. The forcing $\mathbb J(\kappa)$ is $\lt\kappa$-closed, has the $\kappa^+$-cc and adds a unique generic subset of $\kappa$. Finite iterations $\mathbb J(\kappa)_n$ of $\mathbb J(\kappa)$ also have the $\kappa^+$-cc, and add a unique $n$-length generic sequence of subsets of $\kappa$. Finally, given a tree of height $\omega$, the tree iteration $\mathbb P(\mathbb J(\kappa),T)$ is defined analogously as consisting of functions on subtrees of $T$ of size less than $\kappa$. For trees such as $T=\kappa^{\lt\omega}$ and $T=(\kappa^+)^{\lt\omega}$, the poset $\mathbb P(\mathbb J(\kappa),T)$ has the $\kappa^+$-cc and the uniqueness of generics property that the only generic sequences for finite iterations $\mathbb J(\kappa)_n$ are those added explicitly on the nodes of the generic tree. [11] Given an $L$-generic $g\subseteq \mathbb P(\mathbb J(\kappa),(\kappa^+)^{\lt\omega})$, we can construct a symmetric submodel $N\models {\rm ZF}+{\rm AC}_\kappa+\neg {\rm DC}_\omega$ of $L[G]$, in which $\kappa$ remains inaccessible. From the construction, it follows that $H_{\kappa^+}^N$ is the union of $H_{\kappa^+}^{L[G_T]}$ over all well-founded subtrees $T\subseteq(\kappa^+)^{\lt\omega}$ of size $\kappa$. (Analogously, $M^g_{\rm{small}}$ is the union over all well-founded countable subtrees $T\subseteq \omega_1^{\lt\omega}$ of $H_{\omega_1}^{L[g_T]}$.), and satisfies a theory known as ${\rm ZFC}^-_I$, consisting of ${\rm ZFC}^-$ together with assertions that there exists a largest cardinal $\kappa$, which is inaccessible (in models without powersets, we define that a cardinal $\kappa$ is inaccessible if it is regular, $P(\alpha)$ exists for every $\alpha\lt\kappa$, and $|P(\alpha)|\lt\kappa$.) [11]. A model of ${\rm ZFC}^-_I$ satisfies that $V_\alpha$ exists for every $\alpha\leq\kappa$, with $V_\kappa$ satisfying ${\rm ZFC}$, and that every set has size at most $\kappa$. The failure of ${\rm DC}_\omega$ in $N$ translates into a failure of a ${\rm DC}_\omega$-scheme in $H_{\kappa^+}^N$. We will refer to $H_{\kappa^+}^N$ as $M^G_{\rm{Ismall}}$.

In this article, we show:

Theorem: Suppose $g\subseteq \mathbb P(\mathbb J,\omega_1^{\lt\omega})$ and $G\subseteq \mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ are $L$-generic.

The partial Reflection principle holds in $M_{\rm{small}}^g$.
The partial Reflection principle for $\{\omega_1\}$ fails in $M_{\rm{large}}^G$. Consequently, the partial Reflection principle fails in $M_{\rm {large}}^G$.
The partial Reflection principle for $\mathbb R$ holds in $M_{\rm{large}}^G$. Consequently, the parameter-free Reflection principle holds in $M_{\rm{large}}^G$.

Corollary: Over ${\rm ZFC}^-$, the Reflection principle is not equivalent to the partial Reflection principle.

Corollary: Over ${\rm ZFC}^-$, the partial Reflection principle is not equivalent to the partial Reflection principle for $\mathbb R$.

Theorem: Suppose $g\subseteq \mathbb P(\mathbb J(\kappa),(\kappa^+)^{\lt\omega})$ is $L$-generic. The partial Reflection principle holds in $M^g_{\rm{Ismall}}$. Consequently, over ${\rm ZFC}^-_I$, the partial Reflection principle is not equivalent to the Reflection principle.

References

A. Zarach, “Unions of ${\rm ZF}^{-}$-models which are themselves ${\rm ZF}^{-}$-models,” in Logic Colloquium ’80 (Prague, 1980), vol. 108, Amsterdam: North-Holland, 1982, pp. 315–342.
V. Gitman, J. D. Hamkins, and T. A. Johnstone, “What is the theory $\mathsf {ZFC}$ without power set?,” MLQ Math. Log. Q., vol. 62, no. 4-5, pp. 391–406, 2016.
S.-D. Friedman, V. Gitman, and V. Kanovei, “A model of second-order arithmetic satisfying AC but not DC,” J. Math. Log., vol. 19, no. 1, pp. 1850013, 39, 2019.
A. Freund, “Set-theoretic reflection is equivalent to induction over well-founded classes,” Proc. Amer. Math. Soc., vol. 148, no. 10, pp. 4503–4515, 2020. Available at: https://doi.org/10.1090/proc/15103
A. Lévy and R. Vaught, “Principles of partial reflection in the set theories of Zermelo and Ackermann,” Pacific J. Math., vol. 11, pp. 1045–1062, 1961. Available at: http://projecteuclid.org/euclid.pjm/1103037137
B. Yao, “Axiomatization and Forcing in Set Theory with Urelements,” To appear in the Journal of Symbolic Logic, 2024.
R. Jensen, “Definable sets of minimal degree,” in Mathematical logic and foundations of set theory (Proc. Internat. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 122–128.
U. Abraham, “A minimal model for $\neg{\rm CH}$: iteration of Jensen’s reals,” Trans. Amer. Math. Soc., vol. 281, no. 2, pp. 657–674, 1984. Available at: http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/2000078
V. Gitman and R. Matthews, “ZFC without power set II: reflection strikes back,” Fund. Math., vol. 264, no. 2, pp. 149–178, 2024. Available at: https://doi.org/10.4064/fm206-11-2023
S. D. Friedman, Fine structure and class forcing, vol. 3. Walter de Gruyter & Co., Berlin, 2000, p. x+221.
S.-D. Friedman and V. Gitman, “Jensen forcing at an inaccessible and a model of Kelley-Morse satisfying ${\rm CC}$ but not ${\rm DC}_ω$,” Manuscript, 2023.

Filter extension games with mini supercompactness measures

2025-06-07T00:00:00-04:00

This is a talk at the Third Berkeley Conference on Inner Model Theory, University of California, Berkeley, June 28, 2025.
Slides

Abstract: Several classical smaller large cardinals $\kappa$, among them weakly compact, ineffable, and Ramsey cardinals, can be characterized by the existence of measure-like filters on $\kappa$-sized families of subsets of $\kappa$. It suffices to consider families that arise as $P(\kappa)^M$ for some $\kappa$-sized $\in$-model of a sufficiently large fragment of set theory. Isolating patterns in the properties of these filters has led to a better understanding of the classical large cardinals and the definition of new ones. Holy and Schlicht introduced the filter extension games of length $1$ up to $\kappa^+$ in which the first player plays an increasing sequence of these $\kappa$-sized models and the second player responds by choosing an increasing sequence of filters for them. They, and later Nielsen and Welch, used the existence of a winning strategy for one of the players in these games to characterize (or imply) classical large cardinals and define new ones. The existence of the strategy for the second player can be used to characterize (or imply) versions of generic measurability, including, as shown by Foreman, Magidor, and Zeman, the existence of precipitous ideals.

We generalize the filter extension games to the two-cardinal setting by considering corresponding filters on $\lambda$-sized families of subsets of $P_\kappa(\lambda)$ (arising as $P(P_\kappa(\lambda))^M$ for $\lambda$-sized $\in$-models). We show that the existence of a winning strategy for the second player in these games characterizes (or implies) several large cardinal notions in the neighborhood of a supercompact, including, nearly $\lambda$-supercompact, completely $\lambda$-ineffable and $\lambda$-$\Pi^1_n$-indescribable cardinals. We also connect the existence of winning strategies for the second player in these games with various notions of generic supercompactness, including the existence of precipitous ideals. This is joint work with Tom Benhamou.

Categoricty theorems for second-order set theory without powersets

2025-05-01T00:00:00-04:00

In 1904, Oswald Veblen defined that a theory is categorical if it has a unique model up to isomorphism. Categoricty quickly proved to be unattainable in the first-order setting. The upward Löwenheim–Skolem theorem showed that any first-order theory with a model of some infinite cardinality $\kappa$ has models of cardinality at least $\gamma$ for every $\gamma\geq\kappa$. The question becomes more meaningful if we ask for $\kappa$-categorical theories, those having a unique up to isomorphism model of cardinality $\kappa$. Morley famously showed that a theory is either (1) $\omega$-categorical, (2) categorical for all uncountable cardinals but not $\omega$-categorial, or (3) categorical for all infinite cardinals [1]. After a century of working in first-order logic, it surprises no one that it is bad at categorizing things. So what about the much more powerful second-order logic?

The massive difference between the two logics is how much of the set-theoretic universe they have access to. Even first-order logic does not exist entirely outside of the set-theoretic universe because it requires recursion for Tarsky's definition of truth, but it cannot detect most of its set-theoretic background. As we saw from Morley's theorem even though it might in certain instances be able to tell the difference between the countable and uncountable, it cannot recognize its own cardinality. Even more basically, a first-order model of arithmetic or set theory can easily miss its own ill-foundedness. Second-order logic is a very different story. The logic has additional (second-order) quantifiers ranging over all relations on the model. (For theories with coding such as arithmetic or set theory, we will think of the second-order quantifiers as ranging over the subsets of the model, which we refer to as the model's classes.) Now this is where a bit of confusion comes in for many people because there are two types of models in second-order logic, the (true) models and the Henkin models. In a model of second-order logic, we are allowed to quantify over all relations on the model that the underlying set-theoretic universe has, giving the model full access to its powerset. A Henkin model comes with a possibly restricted collection of relations on the model and the second-order quantifiers then only have access to this collection. Henkin models are camouflaged first-order structures because we can interpret their two sorts, the domain and the relations over the domain in a contorted, but first-order way. In contemporary set theory when we talk about 'second-order' set theory or set theory with classes, we are always dealing with Henkin models. People just don't look at second-order models anymore (caveat: unless they are studying abstract logics) because they are too powerful. But in this post, I will be dealing with the actual second-order models of set theory because I want to discuss the categoricity results available in this setting.

Given how much the (non-Henkin) models of second-order set theory know about their powerset, it should not be surprising that strong categoricity results can be obtained for them. Let ${\rm ZFC}_2$ denote the second-order ${\rm ZFC}$-axioms consisting of the usual axioms, but with Replacement having access to ALL functions on the model. How much do these models 'know'? Well, they have to be well-founded because if there was an infinitely descending membership chain, then the model would see it using the second-order Replacement axiom. So we can assume without loss of generality that they are $\in$-models. They have to be correct about cardinals and cofinalities because they likewise see all available functions on the ordinals. Given a set, they also have to have its full powerset. In other words, a set model of ${\rm ZFC}_2$ is a $V_\kappa$ and $\kappa$ must be inaccessible. In particular, any two models of ${\rm ZFC}_2$ with the same ordinals must be isomorphic. This is Zermelo's Categority Theorem for ${\rm ZFC}_2$. Beau Madison Mount recently asked me if there is a corresponding categoricty theorem for ${\rm ZFC}^-_2$ the second-order version of set theory without powersets, where we extend the Replacement axiom to all functions (note that the Collection scheme follows from second-order Replacement). The answer is yes and it is easy to see that models of ${\rm ZFC}^-_2$ are precisely the structures $H_\beta$ (the collection of all sets whose transitive closure has cardinality less than $\beta$). So here also it is the case that any two models with the same ordinals are going to be isomorphic. Let's fix a model $M$ of ${\rm ZFC}^-_2$ and see why. As before because the models are correct about well-foundedness, we can assume that $M$ is an $\in$-model. Also, as before $M$ is correct about cardinals and if it has a cardinal $\delta$, then $P(\delta)\subseteq M$ (not necessarily an element though because of the failure of powerset). Since every element of $H_{\delta^+}$ can be coded by a subset of $\delta$, it follows that $H_{\delta^+}\subseteq M$. First, suppose that $M$ has the largest cardinal $\delta$. Then as we already argued $H_{\delta^+}\subseteq M$. Now given a set $A\in M$, $M$ thinks that $A$ has size at most $\delta$ and it is correct about this. Thus, since $M$ is transitive, it is contained in $H_{\delta^+}$, and hence $M=H_{\delta^+}$. Otherwise, let $\beta$ be the supremum of the cardinals in $M$, and show similarly that $M=H_\beta$.

Väänänen has a more refined concept of categoricity which tries to limit the access of the models to the full powerset, a condition, as we have seen, under which categoricity results are too easily obtained. His concept of internal categoricity puts two models of, say ${\rm ZFC}$, into a Henkin model of second-order logic satisfying that only (relatively low complexity) definable relations over the model are required to exist [2]. The overarching Henkin model then views these ${\rm ZFC}$ models as models of ${\rm ZFC}_2$ from its own perspective, that is the models see only the classes that are available in the Henkin model containing them both. Can such a Henkin model recognize that two models of ${\rm ZFC}$ are isomorphic? What about ${\rm ZFC}^-$?

Let's first give a more precise definition of internal categoricity. Our language consists of two unary predicates $M$ and $N$, which are going to split the domain into the models $M$ and $N$, and binary predicates $E^M$ and $E^N$, which are going to be the membership relations on $M$ and $N$ respectively. Our second-order theory is going to say that (1) $\Pi^1_1$-comprehension holds (the model has every relation defined by a formula with a single universal second-order quantifier followed by a first-order formula of any complexity), (2) $M$ together with $E^M$ is a model of ${\rm ZFC}_2$ (with the respect to the sets available in the Henkin model), (3) similarly $N$ together with $E^N$ is a ${\rm ZFC}_2$ model, and finally (4) there is an isomorphism $F$ between the ordinals of $M$ and the ordinals of $N$. The reason we use $\Pi^1_1$-comprehension is that it suffices for the existence of solutions to recursions on (second-order) relations along the ordinals (either of $M$ or $N$ since they are isomorphic). Via a recursion on the rank of the elements we can define an isomorphism between $M$ and $N$: given that we already have an isomorphism on all sets of rank less than some $\alpha$, we extend it in the obvious fashion to an isomorphism on all elements of rank $\alpha$. Actually, much less than $\Pi^1_1$-comprehension is supposed to suffice for building the isomorphism. The point of internal categoricity is that you don't need the entire set theoretic background to verify that two models are isomorphic, you only need a Henkin model having some very concretely definable relations.

Can we extend the same internal categoricity result to models of ${\rm ZFC}^-$? Yes, the same argument using a recursion on rank will work, but I don't know in this case whether we can weaken the comprehension assumption. In the ${\rm ZFC}$ case the elements of the recursion are actually elements of the model (they are sets in $M$ or $N$) and not second-order objects and so $\Pi^1_1$-comprehension is overkill for the existence of a solution. But since a model of ${\rm ZFC}^-$ can have class many elements of a certain rank, our recursion in this case definitely has second-order initial segments.

References

M. Morley, “Categoricity in power,” Trans. Amer. Math. Soc., vol. 114, pp. 514–538, 1965. Available at: https://doi.org/10.2307/1994188
J. Väänänen, “Tracing internal categoricity,” Theoria, vol. 87, no. 4, pp. 986–1000, 2021. Available at: https://doi.org/10.1111/theo.12237

Reflection principles in set theory without powersets

2025-04-23T00:00:00-04:00

In a ${\rm ZFC}$ universe $V$, given any formula $\varphi(x,a)$ with parameter $a$, there is a transitive set containing $a$ such that the formula $\varphi(x,a)$ is absolute between the set and the entire universe. Namely, the set reflects the universe with respect to the formula $\varphi(x,a)$. The scheme of assertions that every formula (with parameters) is reflected in this way down to some transitive set is called the Reflection principle. To prove the Reflection principle, we will crucially utilize the von Neumman $V_\alpha$-hierarchy. Choose some $V_{\alpha_0}$ containing $a$. Given $V_{\alpha_n}$, choose $V_{\alpha_{n+1}}$ such that for any subformula $\psi$ of $\varphi$ and parameters $\bar b$ from $V_{\alpha_n}$, if $V$ satisfies $\exists y\,\psi(y,\bar b)$, then $V_{\alpha_{n+1}}$ already has some such $y$. Let $\alpha$ be the supremum of the $\alpha_n$ for $n\lt\omega$. By the Tarski-Vaught test, $V_\alpha$ reflects $V$ for the formula $\varphi(x,a)$. Let's think more carefully now about which axioms of ${\rm ZFC}$ we used to carry out our argument. The existence of the von Neumann hierarchy uses the Powerset axiom together with the Replacement scheme. We also use Replacement to conclude that given a $V_{\alpha_n}$, the required $V_{\alpha_{n+1}}$ exists. Notice that the argument does not make use of the Axiom of Choice and so the Reflection principle holds in ${\rm ZF}$ as well. In fact, it is not difficult to show that Replacement, Collection, and Reflection are all eqiuvalent over Zermelo set theory (${\rm ZF}$ without Replacement). But what happens if we work in a set theory where powersets are not available?

First, we need to discuss what a set theory without powersets looks like. It is tempting to say that it is simply the axioms of ${\rm ZFC}$ with the Powerset axiom removed. But the issue is that without powersets, we need to be more careful about some of the other axiom choices we make among axioms that are equivalent in the presence of powersets. For instance, the Replacement and Collection schemes are not equivalent without the Powerset axiom around. If we take the version of ${\rm ZFC}$ with Replacement minus Powerset, then we get a very unpleasant set theory where Collection can fail, $\omega_1$ can be a countable union of countably many sets, every set of reals can be countable while $\omega_1$ exists, where Łoś's theorem can fail for ultrapowers, etc. [1] This can be fixed if we replace Replacement by Collection, but even then another issue arises with the usually equivalent versions of the Axiom of Choice. As is well known, over the other axioms of ${\rm ZFC}$, ${\rm AC}$, the assertion that every family of sets has choice function, is equivalent to Zorn's Lemma, the assertion that every partial order which has an upper bound for every chain has at least one maximal element, and is in turn equivalent to the well-ordering principle, the assertion that every set can be well-ordered. All known proofs of the equivalences make a crucial use of the Powerset axiom. The theory consisting of ${\rm ZF}$ without Powerset, Collection, and ${\rm AC}$ does not prove the well-ordering principle and it is an open question whether the same theory with ${\rm AC}$ replaced by Zorn's Lemma implies ${\rm AC}$ [2]. Taking all these subtleties into account leads us to conclude that the most robust set theory without powersets is ${\rm ZFC}^-$, the theory consisting of the axioms of ${\rm ZFC}$ without Powerset, with Collection, and with the well-ordering principle. The theory ${\rm ZFC}^-$ has many natural models. If $\kappa$ is a regular cardinal, then $H_{\kappa^+}$, the collection of all sets whose transitive closure has size at most $\kappa$ is a model of ${\rm ZFC}^-$. Class forcing extensions by pretame forcing, although they may fail to have powersets, also satisfy ${\rm ZFC}^-$. The theory ${\rm ZFC}-$, where we keep Replacement, but take the well-ordering principle as the choice axiom still has all the unpleasantness described above, but provides an interesting variation on the no powersets theme.

Now we can get back to our question in a more formal fashion. Does the theory ${\rm ZFC}^-$ imply the Reflection principle? Because we have Collection, given any set, we can build a larger set with existential witnesses for some collection of formulas with parameters from the original set. But now now it should be clear that we need some version of dependent choice for definable relations in order to be able to repeat this construction $\omega$-many times. So let the ${\rm DC}$-scheme assert that we can make $\omega$-many dependent choices along any definable (with parameters) relation without terminal nodes. More formally the ${\rm DC}$-scheme is a scheme of assertions for every formula $\varphi(x,y,a)$ with parameter $a$ that if $\forall x\,\exists y\,\varphi(x,y,a)$ holds, then there is a sequence $\{x_n\mid n\lt\omega\}$ such that $\varphi(x_n,x_{n+1},a)$ holds for all $n\lt\omega$. Indeed, it turns out that over ${\rm ZFC}^-$, the Reflection principle is equivalent to the ${\rm DC}$-scheme because if the Reflection principle holds, then we can reflect the formula $\varphi(x,y,a)$ to a transitive set and then use ${\rm AC}$ to find the sequence of dependent choices. But alas, the ${\rm DC}$-scheme is independent of ${\rm ZFC}^-$, and so it immediately follows that the Reflection principle is as well [3]. We currently have two very different models of ${\rm ZFC}^-$ in which the ${\rm DC}$-scheme fails. Both are obtained as inner models of forcing extensions by a tree iteration of Jensen's forcing.

Jensen's forcing $\mathbb J$ is a subposet of Sacks forcing that is constructed in $L$ using the $\diamondsuit$-principle. Unlike the full Sacks forcing, $\mathbb J$ has the ccc ($\diamondsuit$ is used to seal maximal antichains). It adds a unique generic real whose singleton is $\Pi^1_2$-definable. [4] A finite $n$-length iteration $\mathbb J_n$ of Jensen's forcing also has the ccc and adds a unique $n$-length generic sequence of reals that is again $\Pi^1_2$-definable [5]. A tree iteration of Jensen's forcing along a tree $\mathcal T$ of height $\omega$, $\mathbb P(\mathbb J,\mathcal T)$, adds a tree isomorphic to $\mathcal T$ whose $n$-level nodes are $n$-length generic sequences for $\mathbb J_n$ that are coherent in the sense that if $s\leq t$ are two nodes on $\mathcal T$, then the sequence on node $s$ is the restriction of the sequence on node $t$. Crucially, for certain trees $\mathcal T$ such as $\omega^{\lt\omega}$ or $\omega_1^{\lt\omega}$, the poset $\mathbb P(\mathbb J,\mathcal T)$ has the ccc and in its forcing extension, the only $n$-length generic sequences of reals for $\mathbb J_n$ are those explicitly appearing on the nodes of the generic tree, and the tree itself is $\Pi^1_2$-definable [3]. The key theme with Jensen's forcing and its iterations is uniqueness of generics.

The first model is obtained by forcing over $L$ with $\mathbb P(\mathbb J,\omega_1^{\lt\omega})$. So let $G\subseteq \mathbb P(\mathbb J,\omega^{\lt\omega})$ be $L$-generic and let $L[G]$ be the resulting forcing extension. There is a symmetric submodel $W$ of $L[G]$ (satisfying ${\rm ZF})$ which has the tree of the generic sequences, but no branch through the tree, witnessing a $\Pi^1_2$-definable failure of ${\rm DC}_{\omega}$, but at the same time ${\rm AC}_\omega$ holds in $W$ [3]. It follows that $H_{\omega_1}^W$ satisfies ${\rm ZFC}^-$ (use ${\rm AC}_\omega$ in $W$ to conclude that choice holds), but fails the ${\rm DC}$-scheme. Note that this model of ${\rm ZFC}^-$ has $\omega$ as the largest cardinal. The second model is obtained by forcing over $L$ with the class tree iteration $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$, which we can argue also has both the ccc and the uniqueness of generics property [6]. So let $G\subseteq \mathbb P(\mathbb J,\omega^1_{\lt\omega})$ be $L$-generic and let $L[G]$ be the resulting forcing extension. The extension satisfies satisfies ${\rm ZFC}^-$ because class forcing with the ccc is pretame [7], but does not have powerset of $\omega$. In this case, we consider the inner model $M$ that is the union of the $L[G_T]$, where $T$ is a set-sized well-founded subtree of ${\rm Ord}^{\lt\omega}$ and $G_T=G\upharpoonright \mathbb P(\mathbb J,T)$. It is argued in [6] that $M$ satisfies ${\rm ZFC}^-$, but the ${\rm DC}$-scheme fails. The model $M$ has does not have powerset of $\omega$, but it does have unboundedly many cardinals, indeed all the same cardinals as $L$.

Alright so the Reflection principle is independent of ${\rm ZFC}^-$. But what about the weak Reflection principle, where given a formula $\varphi(a)$, we ask that there is a transitive set containing $a$ in which $\varphi(a)$ holds?

Let's argue that the weak Reflection principle holds in the small model $H_{\omega_1}^W$. Fix a formula $\varphi(x)$ and $a\in H_{\omega_1}^W$ such that $H_{\omega_1}^W\models\varphi(a)$. We can assume without loss of generality, by replacing $a$ with its transitive closure, that $a$ is transitive. The full forcing extension $L[G]$ can construct a countable submodel of $H_{\omega_1}^W$ satisfying $\varphi(a)$. The universe $L[G]$ then satisfies that there is a real coding a well-founded model with a transitive set $a'$ isomorphic to $a$ and satisfying $\varphi(a')$. This is a $\Sigma^1_2$-assertion about $a$, which must, by Shoenfield's absoluteness principle, be absolute between $L[G]$ and $W$. Thus, $W$ also has a countable well-founded model satisfying $\varphi(b)$ for some transitive in it set $b$ isomorphic to $a$. The collapse of this model is a countable transitive model containing $a$ because, by the transitivity assumption on $b$, it must get mapped to $a$, and satisfying $\varphi(a)$. But the collapsed model must be inside $H_{\omega_1}^W$ precisely because it is countable and transitive.

Now let's argue that the weak Reflection principle fails in the large model $M$. Let $a=H_{\omega_1}^L$ and consider the formula $\varphi(a)$ asserting that every $n$-length $L$-generic sequence for $\mathbb J_n$ can be extended to an $L$-generic $n+1$-length sequence for $\mathbb J_{n+1}$. We need the parameter $a$ for the definition of $\mathbb J_n$ and to check the $L$-genericity of the sequence. The class generic tree of the generic sequences for $\mathbb J_n$ is contained in $M$ and so $M$ satisfies $\varphi(a)$. The model $M$ is the union of the models $L[G_T]$, where $T$ is a well-founded subtree of ${\rm Ord}^{\lt\omega}$ and $G_T$ is the restriction of $G$ to $T$. The crucial fact here is that each $L[G_T]$ contains exactly the generic $n$-length sequences for $\mathbb J_n$ which occur in $T$ (the proof is similar to Theorem 3.1 in [8]). Suppose some set $A$ in $M$ contains $a$ and satisfies $\varphi(a)$. Then $A\in L[G_T]$ for some well-founded subtree $T$. The set $A$ is correct about a sequence being $n$-generic for $\mathbb J_n$ because this can be checked with $a$. But then because all the generic sequences in $A$ must come from $T$ we can use $A$ to construct a branch through $T$ contradicting that $T$ is well-founded.

The Reflection principle also implies Collection over ${\rm ZFC}-$. A natural question is then whether the weak Reflection principle implies Collection over ${\rm ZFC}-$. Currently there is no easy answer because every model of ${\rm ZFC}-$ which we have fundamentally fails the weak Reflection principle. To build any one of these models, we start out with a specially chosen forcing product $\mathbb P_\alpha$ of length $\alpha$. We then let our model $W$ be the union of the models $V[G_\xi]$ for $\xi\lt\alpha$. Because of the special properties of the chosen forcing, we can show that $W\models{\rm ZFC}-$, but Collection fails because we cannot collect the generics $G_\xi$. Thus, $W$ satisfies that every generic $G_\xi$ can be extended to a generic for a longer product, but this cannot hold true in any set in $W$ because all the sets are in some $V[G_\xi].$ Of course, we can ensure that our sets are correct about genericity by including the full powerset of the forcing in the set.

Now let's weaken our Reflection principle further by considering the sentence Reflection principle, where as the name suggests, we reflect only sentences. Does ${\rm ZFC}^-$ imply the sentence Reflection principle? If we don't ask that the reflecting set is transitive, then the answer is yes. If a model of ${\rm ZFC}^-$ satisfies a sentence $\varphi$, then it also satisfies $\text{Con}(\varphi)$ (Anton Freund convinced me of this). So we just need to argue that we can carry out the Henkin construction to build a model of $\varphi$, but thinking through it carefully, we can see that Replacement suffices to build a Henkin model. So what about a transitive set? We don't know the answer.

Meahwhile, let's argue that the large model $M$ satisfies the sentence Reflection principle. Suppose that some sentence $\varphi$ holds in $M$. Since the forcing $\mathbb P(\mathbb J, {\rm Ord}^{\lt\omega})$ is pretame, it has definable forcing relations. So there is a condition $p\in \mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$ forcing that (the definition of) $M$ satisfies $\varphi$. In $L$, we can obviously build a countable $\in$-model $N$ reflecting enough properties of forcing in general and the forcing $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$, in particular, together with the assertion that there is a condition $p$ forcing that $M$ satisfies $\varphi$. Let $L_\alpha$ be the collapse of $N$. Then the model $L_\alpha$ has a forcing $\mathbb Q$, the collapse of $\mathbb P(\mathbb J,{\rm Ord}^{\lt\omega})$, and a condition $q\in \mathbb Q$ forcing that a transitive submodel of the forcing extension satisfies $\varphi$. Let $g\subseteq \mathbb Q$ be some $L_\alpha$-generic filter in $L$. Then $L_\alpha[g]$ really does have a transitive submodel $m$ satisfying $\varphi$. Thus, $L$ has a transitive set $m$ satisfying $\varphi$ and $m$ is obviously in $M$. So as long as we construct our models failing the Reflection principle as inner models of forcing extensions, we don't have much hope of violating the sentence Reflection principle.

References

V. Gitman, J. D. Hamkins, and T. A. Johnstone, “What is the theory $\mathsf {ZFC}$ without power set?,” MLQ Math. Log. Q., vol. 62, no. 4-5, pp. 391–406, 2016.
A. Zarach, “Unions of ${\rm ZF}^{-}$-models which are themselves ${\rm ZF}^{-}$-models,” in Logic Colloquium ’80 (Prague, 1980), vol. 108, Amsterdam: North-Holland, 1982, pp. 315–342.
S.-D. Friedman, V. Gitman, and V. Kanovei, “A model of second-order arithmetic satisfying AC but not DC,” J. Math. Log., vol. 19, no. 1, pp. 1850013, 39, 2019.
R. Jensen, “Definable sets of minimal degree,” in Mathematical logic and foundations of set theory (Proc. Internat. Colloq., Jerusalem, 1968), North-Holland, Amsterdam, 1970, pp. 122–128.
U. Abraham, “A minimal model for $\neg{\rm CH}$: iteration of Jensen’s reals,” Trans. Amer. Math. Soc., vol. 281, no. 2, pp. 657–674, 1984. Available at: http://dx.doi.org.ezproxy.gc.cuny.edu/10.2307/2000078
V. Gitman and R. Matthews, “ZFC without power set II: reflection strikes back,” Fund. Math., vol. 264, no. 2, pp. 149–178, 2024. Available at: https://doi.org/10.4064/fm206-11-2023
S. D. Friedman, Fine structure and class forcing, vol. 3. Walter de Gruyter & Co., Berlin, 2000, p. x+221.
V. Gitman, “Parameter-free schemes in second-order arithmetic,” To appear in the Journal of Symbolic Logic, 2024.

Filter extension games and generic large cardinals

2025-04-20T00:00:00-04:00

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 $\text{Judge}_{G_\omega(\kappa,\theta)}$ for every regular $\theta>\kappa$, 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 $\text{Judge}_{G_\omega(\kappa,\theta)}$ for every regular $\theta>\kappa$ (follows from [4]).
If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa$ is generically measurable with wa and $\omega_1$-iterability (by $\text{Coll}(\omega,H_{\lambda^+})$) (analogous to [5]).
Suppose $2^\kappa=\kappa^+$. If $\text{Judge}_{sG_\omega(\kappa)}$, then $\kappa $ is generically measurable with wa by $P(\kappa)/I$ for a precipitous uniform 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, [2]). 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 for sets cardinals equiconsistent with generically $\lambda$-supercompact for sets cardinals 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:
- $\text{Judge}_{G_1(\kappa,\lambda)}$
- $\text{Judge}_{wG^*_\omega(\kappa,\lambda)}$
$\kappa$ is almost generically $\lambda$-supercompact with wa (completely $\lambda$-ineffable) if and only if $\text{Judge}_{wG_\omega(\kappa,\lambda)}$.
If $\text{Judge}_{G_\omega(\kappa,\lambda,\theta)}$ for every regular $\theta>\lambda$, then $\kappa$ is generically $\lambda$-supercompact for sets with wa (by $\text{Coll}(\omega,H_\theta)$).
If $\text{Judge}_{sG_\omega(\kappa,\lambda)}$, then $\kappa$ is generically $\lambda$-supercompact with wa and $\omega_1$-iterability (by $\text{Coll}(\omega,H_{\lambda^+})$).
Suppose $2^\lambda=\lambda^+$. If $\text{Judge}_{sG_\omega(\kappa,\lambda)}$, then $\kappa $ is generically $\lambda$-supercompact by $P(\kappa)/I$ for a precipitious normal fine ideal $I$.
For uncountable regular cardinals $\delta$, $\text{Judge}_{G_\delta(\kappa,\lambda)}$ if and only if $\kappa$ is generically $\lambda$-supercompact with wa by $\delta$-closed forcing.

References

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
T. Jech, Set theory. Berlin: Springer-Verlag, 2003, p. xiv+769.
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.
D. Saattrup Nielsen and P. Welch, “Games and Ramsey-like cardinals,” J. Symb. Log., vol. 84, no. 1, pp. 408–437, 2019.
T. Benhamou and V. Gitman, “Cardinals of the $P_κ(λ)$-filter games,” Manuscript, 2025.
M. Foreman, M. Magidor, and M. Zeman, “Games with filters I,” Journal of Mathematical Logic, pp. to appear, 2023.

Filter extension games on mini supercompactness filters

2025-04-09T00:00:00-04:00

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 set $A\in M$ with $|A|^M=\kappa$, $\vec A\cap U\in M$. 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 [4].

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.

A cardinal $\kappa$ is weakly compact if and only if: (folklore)
- Every $A\subseteq\kappa$ is in a weak $\kappa$-model $M$ for which there is an $M$-ultrafilter with a well-founded ultrapower.
- Every $A\subseteq\kappa$ is in a $\kappa$-model $M$ for which there is an $M$-ultrafilter.
- Every $\kappa$-model $M$ has an $M$-ultrafilter.
A cardinal $\kappa$ is ineffable if and only if: [5]
- Every $A\subseteq\kappa$ is in a weak $\kappa$-model $M$ for which there is an $M$-ultrafilter with a stationary diagonal intersection.
- Every $A\subseteq\kappa$ is in a $\kappa$-model $M$ for which there is an $M$-ultrafilter with a stationary diagonal intersection.
- Every $\kappa$-model $M$ has an $M$-ultrafilter with a stationary diagonal intersection.
A cardinal $\kappa$ is Ramsey if and only if every $A\subseteq\kappa$ is in a weak $\kappa$-model $M$ for which there is a weakly amenable $M$-ultrafilter with the countable intersection property. [6]

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. [7]

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 [8]. 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 [8].

Definition: Suppose that $\delta\leq\kappa^+$.

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.
Let $G_\delta(\kappa)$ be the 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.
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 [8].)

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 $\theta>\kappa$, which restricts the challenger to play (a non-transitive version of) $\kappa$-models elementary in $H_\theta$. We will call these games $G_\delta(\kappa,\theta)$. 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 $G_\delta(\kappa,\theta)$ [8].

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 [9]. It follows from arguments in [5] that the judge has a winning strategy in the game $wG_\omega(\kappa)$ if and only if $\kappa$ is completely ineffable [9]. Nielsen and Schindler showed that the judge having a winning strategy in the game $G_\omega(\kappa)$ is equiconsistent with a virtually measurable cardinal [9]. Foreman, Magidor, and Zeman showed that if $2^\kappa=\kappa^+$ and the judge has a winning strategy in the game $sG_{\omega}(\kappa)$, then there is a precipitous uniform normal ideal on $\kappa$ [10]. 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 $P_\kappa(\lambda)$ [11]. 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

it has size $\lambda$,
$\lambda\in M$,
some bijection $f:\lambda\to P_\kappa(\lambda)\in M$,
$M$ satisfies ${\rm ZFC}^-$.

A weak $(\kappa,\lambda)$-model $M$ is a $(\kappa,\lambda)$-model if $M$ is closed under sequences of length less than $\kappa$, $M^{\lt\kappa}\subseteq M$.

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

$U$ is a filter measuring every set in $P(P_\kappa(\lambda))^M$,
$U$ is fine - $X_\alpha\in U$ for every $\alpha\lt\lambda$,
$U$ is $M$-$\kappa$-complete,
$U$ 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 set $A\in M$ with $|A|^M=\lambda$, $\vec A\cap U\in M$.

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 [12]. 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 [12]. 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 [13], but extending $M$-ultrafilters is inconsistent [11]. So now we can bring in the games.

Definition: Suppose that $\delta\leq\lambda^+$.

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.
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.
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.
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 regular $\theta>\lambda$ 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. [11]

$\kappa$ is nearly $\lambda$-strongly compact if and only if:
- The judge has a winning strategy in the game $G^*_1(\kappa,\lambda)$.
- The judge has a winning strategy in the game $wG^*_\omega(\kappa,\lambda)$.
$\kappa$ is nearly $\lambda$-supercompact if and only if the judge has a winning strategy in the game $G_1(\kappa,\lambda)$.
If the judge has a winning strategy in the game $G_2(\kappa,\lambda)$, then $\kappa$ is $\lambda$-ineffable.
If the judge has a winning strategy in the game $G_n(\kappa,\lambda)$ for some finite $n$, then $\kappa$ is $\lambda$-$\Pi^1_{2n-1}$-indescribable.
$\kappa$ is completely $\lambda$-ineffable if and only if the judge has a winning strategy in the game $wG_\omega(\kappa,\lambda).$
Suppose that $2^\lambda=\lambda^+$. $\kappa$ is $\lambda$-strongly compact if and only if the judge has a winning strategy in the game $G^*_{\lambda^+}(\kappa,\lambda)$.
Suppose that $2^\lambda=\lambda^+$. $\kappa$ is $\lambda$-supercompact if and only if the judge has a winning strategy in the game $G_{\lambda^+}(\kappa,\lambda)$.
Suppose that $2^\lambda=\lambda^+$. If the judge has a winning strategy in the game $sG_\omega(\kappa,\lambda)$, then there is a precipitous normal fine ideal on $P_\kappa(\lambda)$.

For definitions of $\lambda$-ineffable and $\lambda$-$\Pi^1_n$-indescribable cardinals, see [11].

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

V. Gitman, J. D. Hamkins, and T. A. Johnstone, “What is the theory $\mathsf {ZFC}$ without power set?,” MLQ Math. Log. Q., vol. 62, no. 4-5, pp. 391–406, 2016.
K. Kunen, “Some applications of iterated ultrapowers in set theory,” Ann. Math. Logic, vol. 1, pp. 179–227, 1970.
V. Gitman and P. D. Welch, “Ramsey-like cardinals II,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 541–560, 2011.
H. Gaifman, “Elementary embeddings of models of set-theory and certain subtheories,” in Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967), Providence R.I.: Amer. Math. Soc., 1974, pp. 33–101.
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.
W. Mitchell, “Ramsey cardinals and constructibility,” J. Symbolic Logic, vol. 44, no. 2, pp. 260–266, 1979.
V. Gitman, “Ramsey-like cardinals,” The Journal of Symbolic Logic, vol. 76, no. 2, pp. 519–540, 2011.
P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals,” Fund. Math., vol. 242, no. 1, pp. 49–74, 2018.
D. Saattrup Nielsen and P. Welch, “Games and Ramsey-like cardinals,” J. Symb. Log., vol. 84, no. 1, pp. 408–437, 2019.
M. Foreman, M. Magidor, and M. Zeman, “Games with filters I,” Journal of Mathematical Logic, pp. to appear, 2023.
T. Benhamou and V. Gitman, “Cardinals of the $P_κ(λ)$-filter games,” Manuscript, 2025.
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
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

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

2025-04-04T00:00:00-04:00

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.

The gap between weak compactness and strong compactness is quite large, especially considering Magidor's identity crises (missing reference) that the first strongly compact cardinal can consistently be supercompact. Recently, many intermediate filter-extension-like principles have been studied. For example, Hayut [2] analyzed level-by-level strong compactness, square principles, and subcompact cardinals which are tightly connected to the results of this paper.

Weakly compact cardinals can alternatively 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 ${\rm ZFC}^-$ that is closed under sequences of length less than $\kappa$. Canonical examples of $\kappa$-models are elementary substructures of $H_{\kappa^+}$ of size $\kappa$ that are closed under $\lt\kappa$-sequences. Given a $\kappa$-model $M$, a weak $M$-ultrafilter is an ultrafilter on $P(\kappa)^M$ that is $\kappa$-complete from the point of view of of the model, namely, it is closed for sequences that are elements of $M$. A weak $M$-ultrafilter is an $M$-ultrafilter if it is additionally normal from the point of view of the model: closed under diagonal intersections for sequnces from $M$. 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.

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.
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.

In the original definition of the games, there is an additional parameter, a regular cardinal $\theta$, which restricts the challenger to play $\kappa$-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 $H_\theta$ [4]. The weak game $wG_\delta(\kappa)$ and the game $G_\delta(\kappa)$ are easily seen to be equivalent for games of length $\delta$ with $\text{cof}(\delta)\neq\omega$. We can weaken the game $(w)G_\delta(\kappa)$ by requiring the judge to play only weak $M$-ultrafilters and call the resulting game $(w)G^*_\delta(\kappa)$. The game $wG^*_\delta(\kappa)$ was considered in [5] under the name Welch game. (Although in the Welch game, the challenger plays a $\kappa$-algebra rather than a $\kappa$-model, for all practical matters, the games are equivalent as the powerset of a $\kappa$-model is a $\kappa$-algebra and any $\kappa$-algebra can be absorbed into the powerset of a $\kappa$-model.).

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:

$\kappa$ is weakly compact if and only if the judge has a winning strategy in the game $G^{(*)}_1(\kappa)$ (see the previous paragraph).
(Keisler-Tarski [3]) $\kappa$ is weakly compact if and only if the judge has a winning strategy in the game $wG^*_\omega(\kappa)$.
(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.
(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.
([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:

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

Their result is more delicate and involves the construction of ideals containing a dense closed tree.

Theorem 2: Assume that $\kappa$ is inaccessible, $2^\kappa=\kappa^+$, and that $\kappa$ does not carry a $\kappa$-complete $\kappa^+$-saturated ideal. Let $\omega\lt\delta\lt\kappa^+$ be a regular cardinal. 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:

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

In fact, it is possible to construct such a dense set $D$ where (1) and (2) above hold with the almost containment $\subseteq^*$ in place of $\subseteq_I$.

They also proved a partial converse:

Theorem 3 Let $\delta\leq\kappa$ be uncountable regular cardinals and $\mathcal J$ 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] (Q. 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 $\lambda^{\lt\kappa}=\lambda$.

The judge has a winning strategy in the game $G^*_{\lambda^+}(\kappa,\lambda)$ if and only if $\kappa$ is $\lambda$-strongly compact.
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 Schankar and White respectively [7] [8]:

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

The judge has a winning strategy in the game $G^*_{1}(\kappa,\lambda)$ if and only if $\kappa$ is nearly $\lambda$-strongly compact.
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 $1\lt\delta\lt\omega$ gives a proper consistency strength hierarchy. We generalize Theorem 3.4 of [6] using Baumgartner's $\Pi^1_n$-$\lambda$-indescribable cardinals:

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

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

Theorem:

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.
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:

Theorem The following are equivalent for a cardinal $\kappa$ and an uncountable regular cardinal $\delta\leq\lambda$.

$\kappa$ is generically $\lambda$-supercompact with weak amenability by $\delta$-closed forcing.
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 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). Q.1 asks whether any the assumptions of Theorem 2 can be dropped. We prove that the assumption about $\kappa$ carrying no $\kappa$-complete $\kappa^+$-saturated ideal of Theorem 2 is necessary using the following theorem:

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:

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

Then there is a winning strategy $\sigma$ for the game $wG^*_\delta(\kappa)$ such that for every partial run $R$ of the game played according to $\sigma$, the associated hopeless ideal $\mathcal{I}(R,\sigma)$ is not $\kappa^+$-saturated.

Q.2 asks whether there is a correspondence between the ideal constructed in Theorem 2 and the strategy constructed in Theorem 3. We show how one can slightly refine the construction of the dense tree $D$ in the previous theorem, and build a special $D^*\subseteq D$ which gives such a correspondence:

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

A. Tarski, “Some Problems and Results Relevant to the Foundations of Set Theory,” in Logic, Methodology and Philosophy of Science, vol. 44, E. Nagel, P. Suppes, and A. Tarski, Eds. Elsevier, 1966, pp. 125–135. Available at: https://www.sciencedirect.com/science/article/pii/S0049237X09705774
Y. Hayut, “Partial Strong Compactness and Squares,” Fundamenta Mathematicae, vol. 246, pp. 193–204, 2019.
H. J. Keisler and A. Tarski, “From Accessible to Inaccessible Cardinals,” Journal of Symbolic Logic, vol. 32, no. 3, pp. 411–411, 1967.
P. Holy and P. Schlicht, “A hierarchy of Ramsey-like cardinals,” Fund. Math., vol. 242, no. 1, pp. 49–74, 2018.
M. Foreman, M. Magidor, and M. Zeman, “Games with filters I,” Journal of Mathematical Logic, pp. to appear, 2023.
D. Saattrup Nielsen and P. Welch, “Games and Ramsey-like cardinals,” J. Symb. Log., vol. 84, no. 1, pp. 408–437, 2019.
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
P. A. White, “Some Intuition behind Large Cardinal Axioms, Their Characterization, and Related Results,” Master's thesis, Virginia Commonwealth University, 2019.
Y. Hayut and M. Magidor, “Subcompact cardinals, type omission, and ladder systems,” J. Symb. Log., vol. 87, no. 3, pp. 1111–1129, 2022. Available at: https://doi.org/10.1017/jsl.2022.11
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.

Parameter-free schemes in second-order arithmetic

2025-01-09T00:00:00-05:00

This is a talk at the Online Logic Seminar, February 6, 2025.

Abstract: Second-order arithmetic has two types of objects: numbers and sets of numbers, which we think of as the reals. Which sets (reals) have to exist in a model of second-order arithmetic is determined by the various set-existence axioms. These usually come in the form of schemes, of which the most common are the comprehension scheme, the choice scheme, and the collection scheme. The comprehension scheme $\Sigma^1_n$-${\rm CA}$ asserts for a $\Sigma^1_n$-formula $\varphi(n,A)$, with a set parameter $A$, that the collection it determines is a set. The choice scheme $\Sigma^1_n$-${\rm AC}$ asserts for a $\Sigma^1_n$-formula $\varphi(n,X,A)$ that if for every number $n$ there is a set $X$ such that $\varphi(n,X,A)$ holds, then there is a single set $Y$ such that its slice $Y_n$ is a witness for $n$. The collection scheme $\Sigma^1_n$-${\rm Coll}$ asserts more generally that among the slices of $Y$, there is a witness for every $n$. The full comprehension scheme for all second-order assertions is denoted by ${\rm Z}_2$, the full choice scheme by ${\rm AC}$, and the full collection scheme by ${\rm Coll}$. Although the theories ${\rm Z}_2$+${\rm AC}$ and ${\rm Z}_2$ are equiconsistent, Feferman and Lévy showed that ${\rm AC}$ is independent of ${\rm Z}_2$. It is also not difficult to see that ${\rm Coll}$ implies ${\rm Z}_2$ over $\Sigma^1_0$-${\rm CA}$, and hence that ${\rm Coll}$ implies ${\rm AC}$ over $\Sigma^1_0$-${\rm CA}$.

In this talk, I will explore how significant the inclusion of set parameters is in the second-order set-existence schemes. Let ${\rm Z}_2^{-p}$, ${\rm AC}^{-p}$, and ${\rm Coll}^{-p}$ denote the respective parameter-free schemes. H. Friedman showed that the theories ${\rm Z}_2$ and ${\rm Z}_2^{-p}$ are equiconsistent and recently Kanovei and Lyubetsky showed that the theory ${\rm Z}_2^{-p}$ can have extremely badly behaved models in which the sets aren't even closed under complement. They also constructed a more 'nice' model of ${\rm Z}_2^{-p}$ in which $\Sigma^1_2$-${\rm CA}$ holds, but $\Sigma^1_4$-${\rm CA}$ fails. They asked whether one can construct a model of ${\rm Z}_2^{-p}$ in which $\Sigma^1_2$-${\rm CA}$ holds, but there is an optimal failure of $\Sigma^1_3$-${\rm CA}$. I will answer their question by constructing such a model. I will also construct a model of ${\rm Z}_2^{-p}+{\rm Coll}^{-p}$ in which $\Sigma^1_2$-${\rm CA}$ holds, but ${\rm AC}^{-p}$ fails, thus showing that ${\rm Coll}^{-p}$ does not imply ${\rm AC}^{-p}$ even over $\Sigma^1_2$-${\rm CA}$.