March 8
Brent Cody,
Virginia Commonwealth University
Characterizations of the two-cardinal weakly compact ideal
Sun proved that, assuming $\kappa$ is a weakly compact cardinal, a subset $W\subseteq\kappa$ is $\Pi^1_1$-indescribable (or equivalently weakly compact) if and only if $W\cap C\neq\emptyset$ for every $1$-club $C\subseteq \kappa$; here, a set $C\subseteq\kappa$ is $1$-club if and only if $C\in\textrm{NS}_\kappa^+$ and whenever $\alpha<\kappa$ is inaccessible and $C\cap \alpha\in\textrm{NS}_\alpha^+$ then $\alpha\in C$. We generalize Sun's characterization to $\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$, which were first defined by Baumgartner by using a natural two-cardinal version of the cumulative hierarchy. Using the minimal strongly normal ideal of non-strongly stationary sets on $P_\kappa\lambda$, which is distinct from $\textrm{NS}_{\kappa,\lambda}$ when $\kappa$ is inaccessible, we formulate a notion of $1$-club subset of $P_\kappa\lambda$ and prove that a set $W\subseteq P_\kappa\lambda$ is $\Pi^1_1$-indescribable if and only if $W\cap C\neq\emptyset$ for every $1$-club $C\subseteq P_\kappa\lambda$. We also show that elementary embeddings considered by Schanker witnessing near supercompactness lead to the definition of a normal ideal on $P_\kappa\lambda$, and indeed, this ideal is equal to Baumgartner's ideal of non-$\Pi^1_1$-indescribable subsets of $P_\kappa\lambda$. Additionally, we will discuss an application which answers a question of Cox-Lücke.