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Characterizations of the weakly compact ideal on Pκλ
Annals of Pure and Applied Logic ( IF 0.6 ) Pub Date : 2020-02-19 , DOI: 10.1016/j.apal.2020.102791
Brent Cody

Hellsten [15] gave a characterization of Πn1-indescribable subsets of a Πn1-indescribable cardinal in terms of a natural filter base: when κ is a Πn1-indescribable cardinal, a set Sκ is Πn1-indescribable if and only if SC for every n-club Cκ. We generalize Hellsten's characterization to Πn1-indescribable subsets of Pκλ, which were first defined by Baumgartner. First we show that under reasonable assumptions the Π01-indescribability ideal on Pκλ equals the minimal strongly normal ideal NSSκ,λ on Pκλ, which is different from NSκ,λ. We then formulate a notion of n-club subset of Pκλ and prove that a set SPκλ is Πn1-indescribable if and only if SC for every n-club CPκλ. We also prove that elementary embeddings considered by Schanker [26] witnessing near supercompactness lead to the definition of a normal ideal on Pκλ, and indeed, this ideal is equal to Baumgartner's ideal of non–Π11-indescribable subsets of Pκλ. Additionally, as applications of these results we answer a question of Cox-Lücke [10] about F-layered posets, provide a characterization of Πnm-indescribable subsets of Pκλ in terms of generic elementary embeddings and prove several results involving a two-cardinal weakly compact diamond principle.



中文翻译:

弱紧凑理想的上刻划P κ λ

Hellsten [15]给出了 Πñ1个-a的不可描述子集 Πñ1个-根据自然过滤器基数可描述的基数:当κΠñ1个-难以描述的主教,一套 小号κΠñ1个-仅当且仅当- 小号C每个n俱乐部Cκ。我们将Hellsten的特征概括为Πñ1个的-无法描述的子集 Pκλ,最初由包姆加特纳(Baumgartner)定义。首先,我们证明在合理的假设下Π01个-难以刻画的理想 Pκλ等于最小法线理想新高中κλPκλ,这与 NSκλ。然后,我们提出n -club子集的概念Pκλ 并证明一套 小号PκλΠñ1个-仅当且仅当- 小号C每个n俱乐部CPκλ。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。Pκλ,实际上,这个理想等于鲍姆加特纳(Baumgartner)的非理想Π1个1个的-无法描述的子集 Pκλ。另外,作为这些结果的应用,我们回答了Cox-Lücke[10]关于F层状的姿势,可以表征 Πñ的-无法描述的子集 Pκλ 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。

更新日期:2020-02-19
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