Forcing a $\square(\kappa)$-like principle to hold at a weakly compact cardinal

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B. Cody, V. Gitman, and C. Lambie-Hanson, “A $\square(κ)$-like principle consistent with weak compactntess,” To appear in the Annals of Pure and Applied Logic.

In this paper, we introduce and investigate an incompactness principle we call $\square_1(\kappa)$, which is closely related to $\square(\kappa)$ but is consistent with weak compactness. Let us begin by recalling the basic facts about $\square(\kappa)$.

The principle $\square(\kappa)$ asserts that there is a $\kappa$-length coherent sequence of clubs $\vec{C}=\langle C_\alpha\mid\alpha\in\text{lim}(\kappa)\rangle$ that cannot be threaded. For an uncountable cardinal $\kappa$, a sequence $\vec{C}=\langle C_\alpha\mid\alpha\in\text{lim}(\kappa)\rangle$ of clubs $C_\alpha\subseteq\alpha$ is called coherent if whenever $\beta$ is a limit point of $C_\alpha$ we have $C_\beta=C_\alpha\cap\beta$. Given a coherent sequence $\vec{C}$, we say that $C$ is a thread through $\vec{C}$ if $C$ is a club subset of $\kappa$ and $C\cap\alpha=C_\alpha$ for every limit point $\alpha$ of $C$. A coherent sequence $\vec{C}$ is called a $\square(\kappa)$-sequence if it cannot be threaded, and $\square(\kappa)$ holds if there is a $\square(\kappa)$-sequence. It is easy to see that $\square(\kappa)$ implies that $\kappa$ is not weakly compact, and thus $\square(\kappa)$ can be viewed as asserting that $\kappa$ exhibits a certain amount of incompactness. The principle $\square(\kappa)$ was isolated by Todorčević [1], building on work of Jensen [2], who showed that, if $V = L$, then $\square(\kappa)$ holds for every regular uncountable $\kappa$ that is not weakly compact.

The natural ${\leq}\kappa$-strategically closed forcing to add a $\square(\kappa)$-sequence ([3], Lemma 35) preserves the inaccessibility as well as the Mahloness of $\kappa$, but kills the weak compactness of $\kappa$ and indeed adds a non-reflecting stationary set. However, if $\kappa$ is weakly compact, there is a forcing [4] which adds a $\square(\kappa)$-sequence and also preserves the fact that every stationary subset of $\kappa$ reflects. Thus, relative to the existence of a weakly compact cardinal, $\square(\kappa)$ is consistent with ${\rm Refl}(\kappa)$, the principle that every stationary set reflects. However, $\square(\kappa)$ implies the failure of the simultaneous stationary reflection principle ${\rm Refl}(\kappa,2)$ which states that if $S$ and $T$ are any two stationary subsets of $\kappa$, then there is some $\alpha<\kappa$ with ${\rm cf}(\alpha)>\omega$ such that $S\cap \alpha$ and $T\cap\alpha$ are both stationary in $\alpha$. In fact, $\square(\kappa)$ implies that every stationary subset of $\kappa$ can be partitioned into two stationary sets that do not simultaneously reflect ([4], Theorem 2.1).

If $\kappa$ is a weakly compact cardinal, then the collection of non-$\Pi^1_1$-indescribable subsets of $\kappa$ forms a natural normal ideal called the $\Pi^1_1$-indescribability ideal: $$\Pi^1_1(\kappa)=\{X\subseteq\kappa\mid \text{$X$ is not $\Pi^1_1$-indescribable}\}.$$ A set $S\subseteq\kappa$ is $\Pi^1_1$-indescribable if for every $A\subseteq V_\kappa$ and every $\Pi^1_1$-sentence $\varphi$, whenever $(V_\kappa,\in,A)\models\varphi$ there is an $\alpha\in S$ such that $(V_\alpha,\in,A\cap V_\alpha)\models\varphi$. More generally, a $\Pi^1_n$-indescribable cardinal $\kappa$ carries the analogously defined $\Pi^1_n$-indescribability ideal. It is natural to ask the question: which results concerning the nonstationary ideal can be generalized to the various ideals associated to large cardinals, such as the $\Pi^1_n$-indescribability ideals? The work of Sun [5] and Hellsten [6] showed that when $\kappa$ is $\Pi^1_n$-indescribable the collection of \emph{$n$-club} subsets of $\kappa$ is a filter-base for the filter $\Pi^1_n(\kappa)^*$ dual to the $\Pi^1_n$-indescribability ideal, yielding a characterization of $\Pi^1_n$-indescribable sets that resembles the definition of stationarity: when $\kappa$ is $\Pi^1_n$-indescribable, a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. Several recent results have used this characterization ([7], [8], [9] and [10]) to generalize theorems concerning the nonstationary ideal to the $\Pi^1_1$-indescribability ideal. For technical reasons, there has been less success with the $\Pi^1_n$-indescribability ideals for $n>1$. In this article we continue this line of research: by replacing "clubs" with "$1$-clubs" we obtain a $\square(\kappa)$-like principle $\square_1(\kappa)$ that is consistent with weak compactness but not with $\Pi^1_2$-indescribability.

We will see that the principle $\square_1(\kappa)$ holds trivially at weakly compact cardinals $\kappa$ below which stationary reflection fails. (This is analogous to the fact that $\square(\kappa)$ holds trivially for every $\kappa$ of cofinality $\omega_1$.) Thus, the task at hand is not just to show that $\square_1(\kappa)$ is consistent with the weak compactness of $\kappa$, but to show that it is consistent with the weak compactness of $\kappa$ even when stationary reflection holds at many cardinals below $\kappa$, so that nontrivial coherence of the sequence is obtained. Recall that when $\kappa$ is $\kappa^+$-weakly compact, the set of weakly compact cardinals below $\kappa$ is weakly compact and much more, so, in particular, the set of inaccessible $\alpha<\kappa$ at which stationary reflection holds is weakly compact. Brickhill and Welch showed that, assuming $V=L$, if $\kappa$ is $\kappa^+$-weakly compact and $\kappa$ is not $\Pi^1_2$-indescribable then $\square_1(\kappa)$ holds [11]. We show that the same can be forced.

Theorem: If $\kappa$ is $\kappa^+$-weakly compact and the ${\rm GCH}$ holds, then there is a cofinality-preserving forcing extension in which

We will also investigate the relationship between $\square_1(\kappa)$ and weakly compact reflection principles. The weakly compact reflection principle ${\rm Refl}_1(\kappa)$ states that $\kappa$ is weakly compact and for every weakly compact $S\subseteq \kappa$ there is an $\alpha<\kappa$ such that $S\cap\alpha$ is weakly compact. It is straightforward to see that if $\kappa$ is $\Pi^1_2$-indescribable, then ${\rm Refl}_1(\kappa)$ holds, and if ${\rm Refl}_1(\kappa)$ holds, then $\kappa$ is $\omega$-weakly compact (see [9]). However, the following results show that neither of these implications can be reversed. The first author [9] showed that if ${\rm Refl}_1(\kappa)$ holds then there is a forcing which adds a non-reflecting weakly compact subset of $\kappa$ and preserves the $\omega$-weak compactness of $\kappa$, hence the $\omega$-weak compactness of $\kappa$ does not imply ${\rm Refl}_1(\kappa)$. The first author and Hiroshi Sakai [10] showed that ${\rm Refl}_1(\kappa)$ can hold at the least $\omega$-weakly compact cardinal, and hence ${\rm Refl}_1(\kappa)$ does not imply the $\Pi^1_2$-indescribability of $\kappa$. Just as $\square(\kappa)$ and ${\rm Refl}(\kappa)$ can hold simultaneously relative to a weakly compact cardinal, we will prove that $\square_1(\kappa)$ and ${\rm Refl}_1(\kappa)$ can hold simultaneously relative to a $\Pi^1_2$-indescribable cardinal.

Theorem: Suppose that $\kappa$ is $\Pi^1_2$-indescribable and the ${\rm GCH}$ holds. Then there is a cofinality-preserving forcing extension in which

Using $n$-club subsets of $\kappa$, we formulate a generalization of $\square_1(\kappa)$ to higher degrees of indescribability. It is easily seen that $\square_n(\kappa)$ implies that $\kappa$ is not $\Pi^1_{n+1}$-indescribable. However, for technical reasons our methods do not seem to show that $\square_n(\kappa)$ can hold nontrivially when $\kappa$ is $\Pi^1_n$-indescribable. Our methods do allow for a generalization of Hellsten's $1$-club shooting forcing to $n$-club shooting, and we also show that, if $S$ is a $\Pi^1_n$-indescribable set, a $1$-club can be shot through $S$ while preserving the $\Pi^1_n$-indescribability of all $\Pi^1_n$-indescribable subsets of $S$.

Finally, we consider the influence of $\square_n(\kappa)$ on \emph{simultaneous} reflection of $\Pi^1_n$-indescribable sets. We let ${\rm Refl}_n(\kappa,\mu)$ denote the following simultaneous reflection principle: $\kappa$ is $\Pi^1_n$-indescribable and whenever $\{S_\alpha\mid\alpha<\mu\}$ is a collection of $\Pi^1_n$-indescribable sets, there is a $\beta<\kappa$ such that $S_\alpha\cap\beta$ is $\Pi^1_n$-indescribable for all $\alpha < \mu$. We show that for $n\geq 1$, if $\square_n(\kappa)$ holds at a $\Pi^1_n$-indescribable cardinal, then the simultaneous reflection principle ${\rm Refl}_n(\kappa,2)$ fails. As a consequence, we show that relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that ${\rm Refl}_1(\kappa)$ holds and ${\rm Refl}_1(\kappa,2)$ fails.


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