Hellsten [15] gave a characterization of ${\mathrm{\Pi }}_n^1$-indescribable subsets of a ${\mathrm{\Pi }}_n^1$-indescribable cardinal in terms of a natural filter base: when κ is a ${\mathrm{\Pi }}_n^1$-indescribable cardinal, a set $S\subseteq \kappa$ is ${\mathrm{\Pi }}_n^1$-indescribable if and only if $S\cap C\ne \varnothing$ for every n-club $C\subseteq \kappa$. We generalize Hellsten's characterization to ${\mathrm{\Pi }}_n^1$-indescribable subsets of $P_{\kappa }\lambda$, which were first defined by Baumgartner. First we show that under reasonable assumptions the ${\mathrm{\Pi }}_0^1$-indescribability ideal on $P_{\kappa }\lambda$ equals the minimal strongly normal ideal ${\text{NSS}}_{\kappa ,\lambda }$ on $P_{\kappa }\lambda$, which is different from ${\mathrm{NS}}_{\kappa ,\lambda }$. We then formulate a notion of n-club subset of $P_{\kappa }\lambda$ and prove that a set $S\subseteq P_{\kappa }\lambda$ is ${\mathrm{\Pi }}_n^1$-indescribable if and only if $S\cap C\ne \varnothing$ for every n-club $C\subseteq P_{\kappa }\lambda$. We also prove that elementary embeddings considered by Schanker [26] 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–${\mathrm{\Pi }}_1^1$-indescribable subsets of $P_{\kappa }\lambda$. Additionally, as applications of these results we answer a question of Cox-Lücke [10] about $\mathcal{F}$-layered posets, provide a characterization of ${\mathrm{\Pi }}_n^m$-indescribable subsets of $P_{\kappa }\lambda$ in terms of generic elementary embeddings and prove several results involving a two-cardinal weakly compact diamond principle.

Hellsten [15]给出了 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-a的不可描述子集 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-根据自然过滤器基数可描述的基数：当κ为${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-难以描述的主教，一套 $\mathrm{小号}\subseteq \kappa$ 是 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-仅当且仅当- $\mathrm{小号}\cap C\ne \varnothing$每个n俱乐部$C\subseteq \kappa$。我们将Hellsten的特征概括为${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$的-无法描述的子集 $P_{\kappa }\lambda$，最初由包姆加特纳（Baumgartner）定义。首先，我们证明在合理的假设下${\mathrm{\Pi }}_0^{\mathrm{1个}}$-难以刻画的理想 $P_{\kappa }\lambda$等于最小强法线理想${\text{新高中}}_{\kappa ，\lambda }$ 上 $P_{\kappa }\lambda$，这与 ${\mathrm{NS}}_{\kappa ，\lambda }$。然后，我们提出n -club子集的概念$P_{\kappa }\lambda$ 并证明一套 $\mathrm{小号}\subseteq P_{\kappa }\lambda$ 是 ${\mathrm{\Pi }}_{ñ}^{\mathrm{1个}}$-仅当且仅当- $\mathrm{小号}\cap C\ne \varnothing$每个n俱乐部$C\subseteq P_{\kappa }\lambda$。我们还证明，由Schanker [26]考虑的基本嵌入见证了近超紧致性，从而导致了对标准理想的定义。$P_{\kappa }\lambda$，实际上，这个理想等于鲍姆加特纳（Baumgartner）的非理想${\mathrm{\Pi }}_{\mathrm{1个}}^{\mathrm{1个}}$的-无法描述的子集 $P_{\kappa }\lambda$。另外，作为这些结果的应用，我们回答了Cox-Lücke[10]关于$\mathcal{F}$层状的姿势，可以表征 ${\mathrm{\Pi }}_{ñ}^{米}$的-无法描述的子集 $P_{\kappa }\lambda$ 在通用基本嵌入方面，并证明了涉及两个基本弱密实钻石原理的几个结果。