1 Introduction

This work is a contribution to the study of the tension between compactness and incompactness principles in set theory. We focus on the second uncountable cardinal, $\omega _2$ , and consider the strong compactness principle of Club Stationary Reflection and the strong incompactness principle known as the Special Aronszajn Tree Property (these are defined below).

The two properties have been shown to be consistent separately by Magidor [Reference Magidor31] and Laver and Shelah [Reference Laver and Shelah30], respectively. Since the properties represent strong forms of opposing phenomena (compactness and incompactness), it is natural to suspect that they are jointly inconsistent. The main result of this paper shows, on the contrary, that the conjunction of the two principles is consistent. More precisely, we prove the following theorem.

Our work also shows that a weaker version of stationary reflection holds in the original Laver–Shelah model (which uses a weakly compact).

We proceed to define the relevant terms and contextualize our result. If $\nu $ is a regular cardinal, we use $\operatorname {cof}(\nu )$ denote the class of ordinals with cofinality $\nu $ . We recall that if $\operatorname {cf}(\alpha )>\omega $ , then $S\subseteq \alpha $ is stationary if $S\cap C\neq \emptyset $ for each club $C\subseteq \alpha $ . We say that S reflects if there is some $\beta <\alpha $ with $\operatorname {cf}(\beta )>\omega $ so that $S\cap \beta $ is stationary in $\beta $ . If $\kappa $ is regular, we say that stationary reflection holds at $\kappa ^{++}$ if every stationary $S\subseteq \kappa ^{++}\cap \operatorname {cof}(\leq \kappa )$ reflects. Baumgartner originally showed [Reference Baumgartner5] that stationary reflection at $\omega _2$ is consistent from a weakly compact cardinal. Harrington and Shelah [Reference Harrington and Shelah18] later improved this, showing that the optimal assumption of a Mahlo cardinal suffices. One obtains stronger principles by requiring that multiple stationary sets reflect simultaneously. Recall that a collection $\{ S_i \mid i < \tau \}$ of $\tau < \alpha $ stationary subsets of $\alpha $ is said to reflect simultaneously if there is some $\beta <\alpha $ with $\operatorname {cf}(\beta )>\omega $ so that $S_i\cap \beta $ is stationary in $\beta $ for every $i < \tau $ . Magidor [Reference Magidor31] has shown that the consistency strength of “any two stationary subsets of $\omega _2\cap \operatorname {cof}(\omega )$ simultaneous reflect” implies the consistency of a weakly compact cardinal. One may also consider stronger diagonal versions of the above, defined in the natural way.

We are interested in the following very strong form of stationary reflection which implies all of the above.

We will concern ourselves with the case $\kappa =\omega $ , i.e., with stationary subsets of $\omega _2\cap \operatorname {cof}(\omega )$ . Most relevant for us, Magidor [Reference Magidor31] showed that $\mathsf {CSR}(\omega _2)$ is consistent from a weakly compact cardinal; by the above remarks, this is the optimal hypotheses.

Extensions of $\mathsf {CSR}$ to other cardinals have been shown to have limitations. For example, Jech and Shelah [Reference Jech and Shelah22] proved that for every $n < \omega $ , if every stationary subset of $ \omega _{n+3} \cap \operatorname {cof}(\omega _{n+1})$ reflects, then $\mathsf {CSR}(\omega _{n+2})$ fails. However, Jech and Shelah [Reference Jech and Shelah22], and later Cummings and Wylie [Reference Cummings and Wylie12], proved the consistency of certain best possible variations of club stationary reflection below $\aleph _\omega $ .

Limitations on stationary reflection emerge from incompactness principles. One of the most prominent of these is Jensen’s $\square _\kappa $ . In [Reference Jensen23], Jensen showed that $\square _{\kappa }$ holds in L for all $\kappa>\omega $ and that $\Box _\kappa $ implies the existence of many nonreflecting stationary subsets of $\kappa ^+$ .

Further studies showed that variations of $\square _\kappa $ place limitations on the cofinality of reflection points, as well as the amount of simultaneous reflection. For instance, in [Reference Schimmerling37], Schimmerling introduced the hierarchy of square principles, $\square _{\kappa ,\lambda }$ , $1 \leq \lambda \leq \kappa ^+$ . As $\lambda $ increases, this hierarchy is strictly decreasing in strength; see Jensen [Reference Jensen24] for $\kappa $ regular and [Reference Cummings, Foreman and Magidor10] for $\kappa $ singular. For a regular cardinal $\kappa $ and $\lambda \leq \kappa $ , Schimmerling and independently Foreman and Magidor have observed that if $\kappa ^{<\lambda }= \kappa $ and $\square _{\kappa ,<\lambda }$ hold, then every stationary subset of $\kappa ^+$ has a stationary subset which does not reflect at any point of cofinality $\geq \lambda $ (see [Reference Cummings and Schimmerling11]). In particular, $\kappa ^{<\kappa } =\kappa $ and $\square _{\kappa ,<\kappa }$ imply that every stationary subset of $\kappa ^+$ has a stationary subset which does not reflect at any point in $\kappa ^+ \cap \operatorname {cof}(\kappa )$ . In [Reference Cummings, Foreman and Magidor10], Cummings, Foreman, and Magidor extended these results and developed the theory for $\kappa $ singular.

Other notable weakenings of $\square _\kappa $ were introduced and developed by Todorčević [Reference Todorčević40]. These principles, denoted $\square (\kappa ^+)$ and $ \square (\kappa ^+,\lambda )$ , place refined limitations on the extent of stationary reflection. See [Reference Fontanella and Hayut13, Reference Hayut and Lambie-Hanson20, Reference Rinot35].

The weakest nontrivial form of square studied by Jensen is the so-called Weak Square, denoted $\square _{\kappa }^*$ , Remarkably, $\square _\kappa ^*$ is equivalent to a key incompactness phenomenon, the existence of a special $\kappa ^+$ -Aronszajn tree. Let us recall the relevant definitions. A tree is a partially ordered set $(T,\leq _T)$ so that for each $x\in T$ , the set of $\leq _T$ -predecessors of x is well ordered; we refer to the height of x in T as the order type of this set. If $\alpha $ is an ordinal, we use Lev $_\alpha (T)$ to denote all $x\in T$ of height $\alpha $ . The height of T is the least ordinal $\alpha $ so that T has no elements of height $\alpha $ . A branch through T is a linearly ordered subset of T, and a cofinal branch is a branch which intersects every level below the height of T.

Let $\kappa $ be regular. A $\kappa $ -tree is a tree T of height $\kappa $ so that each level has size $<\kappa $ ; we will always assume that for each such tree, each node in the tree has incompatible extensions to all higher levels. $\kappa $ is said to have the tree property if every $\kappa $ -tree has a cofinal branch. K $\ddot {\text {o}}$ nig showed [Reference König25] that $\omega $ has the tree property, whereas Aronszajn has shown that the tree property fails at $\omega _1$ (the result was communicated in [Reference Kurepa29]). The extent of the tree property on cardinals greater than $\omega _1$ , a famous question of Magidor’s, is independent of $\mathsf {ZFC}$ . A watershed in our understanding is due to Mitchell and Silver [Reference Mitchell34], who showed that the tree property at $\omega _2$ is consistent from a weakly compact cardinal.

A tree which witnesses the failure of the tree property is said to be Aronszajn (i.e., a $\kappa $ -tree which has no cofinal branches); the existence of such a tree is an instance of incompactness. A particularly strong witness that a tree is Aronszajn is given by a specializing function: in the case that $\kappa =\lambda ^+$ , a specializing function is an $f:T\longrightarrow \lambda $ so that if $x<_Ty$ , then $f(x)\neq f(y)$ . (For an exploration of these concepts at an arbitrarily regular cardinal, see [Reference Krueger27].) Having a specializing function is a particularly strong witness to being Aronszajn since the function witnesses that T remains Aronszajn in any extension of that model in which $\kappa $ is still a cardinal. T is said to be a special Aronszajn tree if there is a specializing function for T. The property of interest to us is the following.

By a result of Specker [Reference Specker39], if $\kappa ^{<\kappa }$ holds, then there is a special Aronszajn tree on $\kappa ^+$ ; in particular, if the $\mathsf {CH}$ holds, then there is a special Aronszajn tree on $\omega _2$ . Jensen [Reference Jensen23] later showed that the principle $\Box ^*_\kappa $ holds if and only if there is a special Aronszajn tree on $\kappa ^+$ ; since $\kappa ^{<\kappa }$ implies $\Box ^*_\kappa $ , this strengthens Specker’s result.

With regard to constructing specializing functions by forcing, Baumgartner, Malitz, and Reinhardt showed [Reference Baumgartner, Malitz and Reinhard7] that $\mathsf {MA}+\neg \mathsf {CH}$ implies $\mathsf {SATP}(\omega _1)$ . Later, Laver and Shelah showed [Reference Laver and Shelah30] that $\mathsf {SATP}(\omega _2)$ is consistent from a weakly compact cardinal. Generalizing this further, Golshani and Hayut have recently shown [Reference Golshani and Hayut17], using posets which specialize with anticipation, that it is consistent that, simultaneously, for every regular cardinal $\kappa $ , $\mathsf {SATP}(\kappa ^+)$ holds. Krueger has generalized the result of Laver–Shelah (and also Abraham–Shelah [Reference Abraham and Shelah2]) in a different direction [Reference Krueger28], showing that it is consistent with the $\mathsf {CH}$ that any two countably closed Aronszajn trees on $\omega _2$ are club isomorphic. And finally, Asperó and Golshani [Reference Asperó and Golshani3] have announced a positive solution to the question of whether $\mathsf {SATP}(\omega _2)$ is consistent with the $\mathsf {GCH}$ . This work continues the study of the tension between different manifestations of compactness and incompactness phenomena in set theory, which together with the study of tension with other fundamental principles, such as approximation principles (e.g., [Reference Ben-Neria8, Reference Goldberg16]) and cardinal arithmetic (e.g., [Reference Foreman and Todorčević14, Reference Shelah38]), is central to our understanding of their extent and limitations.

We proceed to describe our result in general terms and highlight the challenges that appear in the process. Let $\kappa $ be a cardinal which is either ineffable or weakly compact in a ground model V of $\mathsf {GCH}$ ; we will specify later (Definition 2.8 ) exactly when $\kappa $ is ineffable or weakly compact. We obtain the model which witnesses Theorem 1.1 by first defining, in the extension by $\mathbb {P} = \operatorname {Col}(\omega _1,<\kappa )$ , a $\kappa ^+$ -length iteration $\mathbb {C}_{\kappa ^+} = \langle \mathbb {C}_\tau , \mathbb {C}(\tau ) \mid \tau < \kappa ^+\rangle $ of adding clubs which will eventually witnesses $\mathsf {CSR}(\omega _2)$ .

After forcing with $\mathbb {C}_{\kappa ^+}$ , we then force with a $\kappa ^+$ -iteration $\mathbb {S}_{\kappa ^+} = \langle \mathbb {S}_\tau ,\mathbb {S}(\tau ) \mid \tau < \kappa ^+\rangle $ specializing the desired Aronszajn trees $\dot {T}_\tau $ , i.e., $\mathbb {S}(\tau ) = \mathbb {S}(\dot {T_\tau })$ (see Section 1.2 for precise definitions of the posets).

To make this strategy work, we need, among other things, that all stationary subsets of $\omega _2\cap \operatorname {cof}(\omega )$ which appear in the final generic extension by $\mathbb {P} * \dot {\mathbb {C}}_{\kappa ^+} *\dot {\mathbb {S}}_{\kappa ^+}$ reflect as in the definition of $\mathsf {CSR}(\omega _2)$ . Consequently, the club-adding posets must anticipate names for stationary sets added by the later specializing iteration. In order to carry this through, we define the names $\dot {\mathbb {C}}_\tau $ and $\dot {\mathbb {S}}_\tau $ , for $\tau < \kappa ^+$ , simultaneously. More precisely, for each $\tau < \kappa ^+$ , given that the $\mathbb {P}$ -name $\dot {\mathbb {C}}_\tau $ and the $(\mathbb {P}\ast \dot {\mathbb {C}}_\tau )$ -name $\dot {\mathbb {S}}_\tau $ have been defined, we use a bookkeeping function to pick the $(\mathbb {P}*\dot {\mathbb {C}}_{\tau }*\dot {\mathbb {S}}_{\tau })$ -name $\dot {S_{\tau }}$ of a stationary subset of $\kappa \cap \operatorname {cof}(\omega )$ , and we set $\dot {\mathbb {C}}(\tau )$ to be the $(\mathbb {P}\ast \dot {\mathbb {C}}_\tau )$ -name for the poset to add, with $\dot {\mathbb {S}}_\tau $ -anticipation, the desired club. Then we select the $(\mathbb {P}*\dot {\mathbb {C}}_{\tau +1}*\dot {\mathbb {S}}_\tau )$ -name $\dot {T}_{\tau }$ for an Aronszajn tree on $\kappa $ .

As expected, the tension between compactness and incompactness gives rise to tension between the different parts of the forcing construction. We list three notable manifestations:

(1) Working with intermediate generic extensions. A central property of the Laver–Shelah forcing [Reference Laver and Shelah30] is the existence of intermediate forcing extensions in which regular cardinals $\alpha < \kappa $ become $\omega _2$ and the relevant portion $\dot {T}_\tau \cap (\alpha \times \omega _1)$ of the trees are Aronszajn trees on $\alpha $ . Accompanying this is machinery for projecting conditions of $\mathbb {P} * \dot {\mathbb {S}}_\tau $ to those intermediate extensions. In [Reference Laver and Shelah30], the existence of such intermediate extensions is secured by the weak compactness of $\kappa $ , and the fact that $\mathbb {P} * \dot {\mathbb {S}}_\tau $ is $\kappa $ -c.c. However, in our case, the presence of the poset $\dot {\mathbb {C}}_\tau $ prevents the initial segment $\mathbb {P} *\dot {\mathbb {C}}_\tau $ from being $\kappa $ -c.c. To overcome this difficulty, we use the fact that the full collapse poset $\mathbb {P}$ absorbs many restricted subforcings of $\mathbb {P} * \dot {\mathbb {C}}_\tau $ , which allows us to place upper bounds on various generic filters of the restricted poset. We then couple this in Section 6 with a generalization of a result of Abraham’s [Reference Abraham1] that (stated in current language) if $\dot {\mathbb {Q}}$ is an $\operatorname {Add}(\omega ,\omega _1)$ -name for an $\omega _1$ -closed poset, then $\operatorname {Add}(\omega ,\omega _1)\ast \dot {\mathbb {Q}}$ is strongly proper. This secures the existence of sufficiently many strongly generic conditions (and, in turn, the existence of intermediate extensions).

(2) Preservation of stationary sets by quotients. The ability to add a closed unbounded set through the reflection points $\alpha < \kappa $ of a stationary set $\dot {S_\tau } \subseteq \kappa \cap \operatorname {cof}(\omega )$ hinges upon the fact that many such points exist. The ineffability (in fact, just weak compactness) of $\kappa $ guarantees that for many $\alpha < \kappa $ , $\dot {S_\tau } \cap \alpha $ is a stationary subset of $\alpha $ in the restricted generic extension where $\alpha = \omega _2$ . The forcing construction of [Reference Magidor31] uses the fact that the related quotient of $\mathbb {P} * \dot {\mathbb {C}}_\tau $ by its initial segment is $\sigma $ -closed, and an argument of Baumgartner’s [Reference Baumgartner5] shows that $\sigma $ -closed posets preserve the stationarity of stationary sets of countable cofinality ordinals. By contrast, for us, the stationary sets $\dot {S}_\tau $ further rely on the specializing poset $\dot {\mathbb {S}}_\tau $ , and although the poset $\mathbb {P} * \dot {\mathbb {C}}_\tau *\dot {\mathbb {S}}_\tau $ is $\sigma $ -closed, it does not in general admit $\sigma $ -closed quotients by its natural restrictions to heights $\alpha < \kappa $ . Nevertheless, in Section 4, we analyze the Laver–Shelah iteration $\dot {\mathbb {S}}_\tau $ to prove that the relevant quotients preserve the stationary of $\dot {S_\tau } \cap \alpha $ for many suitable $\alpha < \kappa $ .

(3) Preservation of Aronszajn trees. The organization of the posets $\dot {\mathbb {C}}_\tau $ and $\dot {\mathbb {S}}_{\tau }$ , described above, guarantees that for each $\tau < \kappa ^+$ , $\dot {T}_\tau $ is a $(\mathbb {P} * \dot {\mathbb {C}}_{\tau +1}*\dot {\mathbb {S}}_\tau )$ -name of an Aronszajn tree on $\kappa $ , which is specialized by $\mathbb {P} * \dot {\mathbb {C}}_{\tau +1} * \dot {\mathbb {S}}_{\tau +1}$ .

However, in the final forcing construction, $\mathbb {S}_{\tau +1}$ follows the extended iteration $\mathbb {P} * \dot {\mathbb {C}}_{\kappa ^+}$ , and on its face, $\mathbb {P} * \dot {\mathbb {C}}_{\kappa ^+} * \dot {\mathbb {S}}_{\tau }$ might introduce a cofinal branch to $T_\tau $ , causing the specializing poset $\mathbb {S}(\tau )$ to collapse $\kappa $ . To guarantee that this cannot occur, an Aronszajn preservation theorem is required for the quotient of $\mathbb {P} * \dot {\mathbb {C}}_{\kappa ^+} * \dot {\mathbb {S}}_\tau $ by $\mathbb {P} * \dot {\mathbb {C}}_{\tau +1} * \dot {\mathbb {S}}_\tau $ . The fact that no new reals are added during the iteration, and that $\dot {\mathbb {C}}_{\kappa ^+}$ is not $\kappa $ -closed, prevents us from using known preservation arguments (for instance, those of [Reference Unger41]). Therefore, in Section 6, we develop an alternative preservation argument which fits the properties of the poset $\mathbb {P} * \dot {\mathbb {C}}_{\kappa ^+}$ , and we apply them in Section 7 to show that the tree $T_\tau $ remains Aronszajn.

1.2 Forcing

In this subsection, we review our conventions about forcing and provide explicit definitions of posets which we will use throughout the paper.

To begin, in order to anticipate working with iterations later, we will work with preorderings (i.e., relations which are transitive and reflexive) rather than partial orders. Moreover, we will use the Jerusalem convention for forcing. Thus, we view a forcing poset as a triple $(\mathbb {Q},\leq _{\mathbb {Q}},0_{\mathbb {Q}})$ , where $\leq _{\mathbb {Q}}$ is a preordering and where $0_{\mathbb {Q}}$ is a smallest element; for conditions $p,q\in \mathbb {Q}$ , we will write $p\geq _{\mathbb {Q}} q$ to indicate that p is an extension of q. When context is clear, we will drop explicit mention of $\mathbb {Q}$ in $0_{\mathbb {Q}}$ and $\leq _{\mathbb {Q}}$ . Given that we are only working with preorderings rather than partial orders, we will often have conditions $p,q\in \mathbb {Q}$ so that $p\leq _{\mathbb {Q}} q$ and $q\leq _{\mathbb {Q}} p$ , but q and p are not literally equal as sets. In this case, we will write $p=^*_{\mathbb {Q}} q$ , or simply $p=^*q$ if $\mathbb {Q}$ is clear from context.

If $\mathbb {Q}$ is a poset, we say that $\mathbb {Q}$ is $\omega _1$ -closed with sups if for any increasing sequence $\langle q_n:n\in \omega \rangle $ of conditions in $\mathbb {Q}$ , there exists a $\leq _{\mathbb {Q}}$ -least upper bound q of the sequence. Any such q is referred to as a sup of the sequence. Note that this does not say that any two compatible conditions in $\mathbb {Q}$ have a sup. Moreover, it also does not require that a sup of an increasing $\omega $ -sequence is unique. However, if $q_1$ and $q_2$ are two sups of such a sequence, then $q_1=^*_{\mathbb {Q}} q_2$ . These observations will be important later when we deal with iterations of posets with this property.

If $\mathbb {Q}$ is a poset and $q\in \mathbb {Q}$ , then we use $\mathbb {Q}/q$ to denote all conditions in $\mathbb {Q}$ which extend q.

Let $M \prec H(\theta )$ be an elementary substructure, and let $\mathbb {U} \in M$ be a poset. A condition u in $\mathbb {U}$ is $(M,\mathbb {U})$ -completely generic, if the set $\left \{ \bar {u}\in \mathbb {U}\cap M:u\geq \bar {u}\right \}$ of weaker conditions in M meets all dense subsets $D \subseteq \mathbb {U}$ which belong to M, and thus forms an $(M,\mathbb {U})$ -generic filter.

For the remainder of the paper, we fix a cardinal $\kappa $ which will be either weakly compact or ineffable (we specify in Definition 2.8 exactly when $\kappa $ is ineffable or weakly compact). In the next subsection, we will review facts about ineffability and weak compactness.

Throughout the paper, we will use $\mathbb {P}$ to denote the Levy collapse $\operatorname {Col}(\omega _1,<\kappa )$ . If $\alpha <\kappa $ is inaccessible, we use $\mathbb {P}\upharpoonright \alpha $ to denote the collapse $\operatorname {Col}(\omega _1,<\alpha )$ . We view conditions in $\operatorname {Col}(\omega _1,<\kappa )$ as countable functions p so that $\operatorname {dom}(p)\subseteq \kappa $ and so that for each $\nu \in \operatorname {dom}(p)$ , $p(\nu )$ is a countable, partial function from $\omega _1$ to $\nu $ . If G is a V-generic filter over $\mathbb {P}$ , then we use $G\upharpoonright \alpha $ to denote the V-generic filter $\left \{ p\upharpoonright \alpha :p\in G\right \}$ over $\mathbb {P}\upharpoonright \alpha $ .

For adding clubs, we generalize the club-adding poset of Magidor [Reference Magidor31] by incorporating anticipation; we only state the definition in the generality needed for our paper. Recall that if S is stationary in $\alpha $ , then the trace of S, denoted $\operatorname {tr}(S)$ , consists of all $\beta <\alpha $ so that S reflects at $\beta $ .

Definition 1.3 Let $\mathbb {S}$ be a cardinal-preserving poset in some model W, and let $\dot {S}$ be an $\mathbb {S}$ -name for a stationary subset of $\omega _2\cap \operatorname {cof}(\omega )$ . We let $\mathsf {CU}(\dot {S},\mathbb {S})$ denote the poset, defined in W, where conditions are closed, bounded subsets c of $\omega _2$ so that

$$ \begin{align*}\Vdash_{\mathbb{S}}\check{c}\subseteq\operatorname{tr}(\dot{S})\cup\left(\omega_2\cap\operatorname{cof}(\omega)\right). \end{align*} $$

The ordering is end-extension.

We emphasize that in order to be a condition in $\mathsf {CU}(\dot {S},\mathbb {S})$ , a given closed, bounded subset of $\omega _2$ must be outright forced by $\mathbb {S}$ to be contained, mod cofinality $\omega $ points, in $\operatorname {tr}(\dot {S})$ . Since any condition c in $\mathsf {CU}(\dot {S},\mathbb {S})$ can be extended by placing an ordinal of cofinality $\omega $ above $\max (c)$ , we see that $\mathsf {CU}(\dot {S},\mathbb {S})$ does add a club subset of $\omega _2$ of the model. Moreover, the poset is trivially $\omega _1$ -closed, so preserves $\omega _1$ . However, preservation of $\omega _2$ is a nontrivial matter.

We now review the definition of the poset which we will use to specialize Aronszajn trees on $\omega _2$ . The poset itself will decompose such a tree into a union of $\omega _1$ -many antichains, which in this case is equivalent to having a specializing function.

Definition 1.4 Suppose that T is an Aronszajn tree on $\omega _2$ . Let $\mathbb {S}(T)$ denote the poset where conditions are functions f with countable domain $\operatorname {dom}(f)\subseteq \omega _1$ , and where for each $\alpha \in \operatorname {dom}(f)$ , $f(\alpha )\subseteq T$ is a countable antichain in $<_T$ . Recalling that we are using the Jerusalem convention for forcing, we say that g extends f, written $f\leq g$ , if $\operatorname {dom}(f)\subseteq \operatorname {dom}(g)$ and if for all $\alpha \in \operatorname {dom}(f)$ , $f(\alpha )\subseteq g(\alpha )$ .

It is clear that $\mathbb {S}(T)$ is $\omega _1$ -closed. Moreover, if a tree $T'$ is not Aronszajn, then the analogously defined poset $\mathbb {S}(T')$ will collapse $\omega _2$ .

1.3 Weak compactness

In this final subsection, we review facts about the ineffability of $\kappa $ . However, we will also need various facts about the weak compactness of $\kappa $ , and so we begin with these.

Definition 1.5 $\mathcal {F}_{\operatorname {WC}}$ is the filter generated by subsets A of $\kappa $ for which there is some $U \subseteq V_\kappa $ and a $\Pi ^1_1$ -statement $\Phi $ , satisfied by $(V_\kappa ,\in ,U)$ , so that

$$\begin{align*}A = \{ \alpha <\kappa \mid \alpha \text{ is regular, and } (V_\alpha,\in,U \cap V_\alpha) \models \Phi\}.\end{align*}$$

The filter $\mathcal {F}_{\operatorname {WC}}$ is $\kappa $ -closed as well as normal. It will be helpful at later parts in our argument to phrase membership in the weakly compact filter in terms of embeddings. The idea will be that for a subset B of $\kappa $ , where B is a member of a $\kappa $ -model M, $B\in \mathcal {F}_{\operatorname {WC}}$ iff for all M-normal ultrafilters U, $\kappa \in j_U(B)$ , where $j_U$ is the ultrapower embedding. We make this precise in the following few items.

Definition 1.6 Suppose that $\alpha $ is an inaccessible cardinal. We say that a transitive set M is an $\alpha $ -model if $M\models \mathsf {ZFC}^-$ , $|M|=\alpha $ , $\alpha \in M$ , and $\,^{<\alpha }M\subseteq M$ .

Weak compactness is naturally associated with various embedding properties; here, we mention the following result from [Reference Hauser19].

Proposition 1.3 For any $\kappa $ -model M, there exist a $\kappa $ -model N and an elementary embedding $j:M\longrightarrow N$ so that $\operatorname {crit}(j)=\kappa $ and $j,M\in N$ .

However, we are mostly interested in a different case, namely, when j is the elementary embeddings associated with an M-normal ultrafilter on $\kappa $ and where N is the ultrapower of M. A filter $U \subseteq \mathcal {P}(\kappa ) \cap M$ is an M-normal ultrafilter if U is an M-ultrafilter, and for every $A \in U$ and regressive function $f : A \to \kappa $ in M, there exists some $A' \subseteq A$ in U so that $f\upharpoonright A'$ is constant. We note that being M-normal implies that U is closed under intersections of $< \kappa $ -sequences in M consisting of sets in U.

It is routine to verify that each elementary embedding $j : M \to N$ as in the proposition above gives rise to an M-normal ultrafilter $U_j = \{ A \in \mathcal {P}(\kappa ) \cap M \mid \kappa \in j(A)\}$ . Conversely, we can associate to each M-normal ultrafilter U its ultrapower embedding $j_U : M \to N \cong \operatorname {\mathrm {Ult}}(M,U)$ .

Proposition 1.4 Let M be a $\kappa $ -model, and let $B \in M$ be a subset of $\kappa $ . If $B \in U$ for every M-normal ultrafilter U on $\kappa $ , then $B\in \mathcal {F}_{\operatorname {WC}}$ .

Proof Let $M,B$ be as in the statement of the proposition, and fix a subset $E_M \subseteq \kappa \times \kappa $ so that $(\kappa ,E_M)$ is isomorphic to M. To prove that $B \in \mathcal {F}_{\operatorname {WC}}$ , it suffices to show that there is a $\Pi ^1_1$ statement $\Psi $ satisfied by $(V_\kappa ,\in , E_M,B)$ so that the set $\{ \alpha <\kappa : \alpha \text { is regular and } (V_\alpha ,\in ,E_M \cap V_\alpha , B \cap \alpha ) \models \Psi \}$ is contained in B.

To begin, we observe that the assertions that “ $E_M$ is well-founded,” that “ $(\kappa ,E_M)$ is isomorphic to a transitive $\kappa $ -model,” and that “B is represented in $(\kappa ,E_M)$ by $b \in \kappa $ ” are all within the class of $\Pi ^1_1$ formulas over $(V_\kappa ,\in ,E_M,B)$ (see [Reference Krueger26, Section 2] for details). Let $\Phi _0$ denote their conjunction.

We also note that for a subset $U_M \subseteq \kappa $ , the assertion “ $U_M$ codes a subset of $M \cap \mathcal {P}(\kappa )$ which is an M-normal ultrafilter” is $\Sigma ^0_\omega $ . Therefore, the assertion $\Phi _1$ stating that

$$\begin{align*}``\forall\, U_M \subseteq \kappa \text{, if } U_M \text{ codes an } M\text{-normal ultrafilter, then }B \in U_M \text{''}\end{align*}$$

is $\Pi ^1_1$ . Let $\Phi $ be the conjunction $\Phi _0\wedge \Phi _1$ , a $\Pi ^1_1$ formula satisfied in $(V_\kappa ,\in ,E_M,B)$ .

Define $X:=\{ \alpha <\kappa : \alpha \text { is regular, and } (V_\alpha ,\in ,E_M \cap V_\alpha , B \cap \alpha ) \models \Phi \}$ , and we show that $X\subseteq B$ . Fix some $\alpha \in X$ . Then the relation $E_M \upharpoonright \alpha = E_M \cap V_\alpha \subseteq \alpha \times \alpha $ is well founded, and it codes an $\alpha $ -model $M_\alpha $ with $B \cap \alpha $ represented in $(\alpha ,E_M\upharpoonright \alpha )$ by the same element $b \in \kappa $ which represents B in $(\kappa ,E_M)$ . Let $i_\alpha : M_\alpha \to M_\kappa $ be the elementary embedding resulting from the identifications $M_\alpha \cong (\alpha ,E_M\upharpoonright \alpha ) \prec (\kappa ,E_M) \cong M$ . It is straightforward to verify that $\operatorname {\mathrm {cp}}(i_\alpha ) = \alpha $ , $i_\alpha (\alpha ) =\kappa $ , and $i_\alpha (B \cap \alpha ) = B$ .

It follows that $U_{\alpha } = \{ A \subseteq \alpha : \alpha \in i_\alpha (A)\}$ is an $M_\alpha $ -normal ultrafilter. Let $j_\alpha : M_\alpha \to N_\alpha $ be the induced ultrapower embedding, and let $k_\alpha : N_\alpha \to M$ be the factor map given by $k_\alpha ([f]_{U_\alpha }) = i_\alpha (f)(\alpha )$ . We note that ${\operatorname {\mathrm {cp}}}(k_\alpha )> \alpha $ . Since $(V_\alpha ,\in ,E_M\upharpoonright \alpha ,B \cap \alpha )$ satisfies $\Phi $ , $B \cap \alpha \in U_\alpha $ . The last implies that $\alpha \in j_\alpha (B \cap \alpha )$ , which in turn implies that $\alpha = k_\alpha (\alpha ) \in k_\alpha \circ j_\alpha (B \cap \alpha ) = i_\alpha (B \cap \alpha ) = B$ .▪

We now review the relevant facts about ineffable cardinals (see [Reference Baumgartner4] for the details). A cardinal $\lambda $ is ineffable if for any sequence $\vec {A}=\langle A_\nu :\nu <\lambda \rangle $ so that $A_\nu \subseteq \nu $ for all $\nu <\lambda $ , there is an $A\subseteq \lambda $ so that $\left \{\nu <\lambda :A_\nu =A\cap \nu \right \}$ is stationary in $\lambda $ . Such a set A is said to be coherent for $\vec {A}$ .

Notation 1.7 In the case that $\kappa $ is ineffable, we will denote the ineffability ideal on $\kappa $ by $\mathcal {I}_{in}$ throughout the paper, and we will let $\mathcal {F}_{in}$ denote the filter on $\kappa $ which is dual to $\mathcal {I}$ .

$\mathcal {I}_{in}$ consists of all $S\subseteq \kappa $ so that for some sequence $\vec {A}$ as above, no stationary subset of S is coherent for $\vec {A}$ . In the case that $\kappa $ is ineffable, $\mathcal {I}_{in}$ is a proper, normal ideal on $\kappa $ . Finally, we mention the following theorem of Baumgartner [Reference Baumgartner4, Theorem 7.2] (see [Reference Holy and Lücke21] for more information).

Theorem 1.5 (Baumgartner)

$F_{WC}$ is contained in $\mathcal {F}_{in}$ .

In fact, Baumgartner showed that $\mathcal {F}_{WC}$ is contained in the filter dual to the weak ineffability ideal on $\kappa $ .

Notation 1.8 $\mathcal {F}$ will denote either $\mathcal {F}_{WC}$ or $\mathcal {F}_{in}$ throughout this paper depending on whether $\kappa $ is weakly compact or ineffable, respectively. $\mathcal {I}$ denotes the ideal dual to $\mathcal {F}$ . We will specify in Definition 2.8 in the next section exactly when $\mathcal {F}$ is equal to $\mathcal {F}_{WC}$ or equal to $\mathcal {F}_{in}$ .

Remark 1.6

1. (1) In our paper, we only use the ineffability of $\kappa $ to prove Proposition 3.12 and in the corollaries of this proposition. All other results in our paper can be carried out assuming only that $\kappa $ is weakly compact.

2. (2) As is immediate from Theorem 1.5 and the fact that $\mathcal {F}$ is either $\mathcal {F}_{WC}$ or $\mathcal {F}_{in}$ , all $\mathcal {F}$ -positive sets are also $F_{WC}$ -positive.

3. (3) Thus, if $B\in \mathcal {F}^+$ and if M is a $\kappa $ -model with $B\in M$ , then by Proposition 1.4, there is some M-normal ultrafilter U on $\kappa $ so that $B\in U$ . We will use this fact throughout the paper.

As an illustration of the type of argument in (3) above, we prove the following fact which we will need in the proof of Proposition 4.4.

Lemma 1.7 Suppose that $B\in \mathcal {F}^+$ . Then $B\backslash \operatorname {tr}(B)\in \mathcal {I}$ .

Proof Suppose otherwise, for a contradiction. Then $B\backslash \operatorname {tr}(B)\in \mathcal {F}^+$ . Let $M^*$ be a $\kappa $ -model containing B, and hence $B\backslash \operatorname {tr}(B)$ . By Proposition 1.4, there is an $M^*$ -normal ultrafilter U so that, letting $j:M^*\longrightarrow N$ be the ultrapower embedding, $\kappa \in j(B\backslash \operatorname {tr}(B))$ . However, B is a stationary subset of $\kappa $ , since $B\in \mathcal {F}^+$ . Thus $\kappa \in j(\operatorname {tr}(B))$ . Since $\kappa \in j(B)$ also, we have $\kappa \in j(B)\cap j(\operatorname {tr}(B))=j(B\cap \operatorname {tr}(B))$ , which is a contradiction.▪