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- W2886005405 abstract "Hellsten cite{MR2026390} gave a characterization of $Pi^1_n$-indescribable subsets of a $Pi^1_n$-indescribable cardinal in terms of a natural filter base: when $kappa$ is a $Pi^1_n$-indescribable cardinal, a set $Ssubseteqkappa$ is $Pi^1_n$-indescribable if and only if $Scap Cneqemptyset$ for every $n$-club $Csubseteq kappa$. We generalize Hellsten's characterization to $Pi^1_n$-indescribable subsets of $P_kappalambda$, which were first defined by Baumgartner. After showing that under reasonable assumptions the $Pi^1_0$-indescribability ideal on $P_kappalambda$ equals the minimal emph{strongly} normal ideal $text{NSS}_{kappa,lambda}$ on $P_kappalambda$, and is not equal to $text{NS}_{kappa,lambda}$ as may be expected, we formulate a notion of $n$-club subset of $P_kappalambda$ and prove that a set $Ssubseteq P_kappalambda$ is $Pi^1_n$-indescribable if and only if $Scap Cneqemptyset$ for every $n$-club $Csubseteq P_kappalambda$. We also prove that elementary embeddings considered by Schanker cite{MR2989393} witnessing emph{near supercompactness} lead to the definition of a normal ideal on $P_kappalambda$, and indeed, this ideal is equal to Baumgartner's ideal of non--$Pi^1_1$-indescribable subsets of $P_kappalambda$. Additionally, as applications of these results we answer a question of Cox-Lucke cite{MR3620068} about $mathcal{F}$-layered posets, provide a characterization of $Pi^m_n$-indescribable subsets of $P_kappalambda$ in terms of generic elementary embeddings, prove several results involving a two-cardinal weakly compact diamond principle and observe that a result of Pereira cite{MR3640048} yeilds the consistency of the existence of a $(kappa,kappa^+)$-semimorasses $musubseteq P_kappakappa^+$ which is $Pi^1_n$-indescribable for all $n<omega$." @default.
- W2886005405 created "2018-08-22" @default.
- W2886005405 creator A5017436648 @default.
- W2886005405 date "2018-07-31" @default.
- W2886005405 modified "2023-09-27" @default.
- W2886005405 title "Characterizations of the weakly compact ideal on $P_kappalambda$" @default.
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