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arXiv 2103.15728math.LO
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树与正则基数双后继的平稳反射

Trees and stationary reflection at double successors of regular cardinals

Thomas Gilton, Maxwell Levine, Šárka Stejskalová

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摘要：

我们得到了一系列关于树和正则基数$κ$的双后继上的平稳反射的一致性结果，更新了一些经典的构造过程。这包括$\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$的模型（有和没有$\mathsf{AP}(κ^{++})$）以及连接词$\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$和$\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$的模型（后者最初是由Krueger和第一作者\cite{GilKru:8fold}的联合工作获得的，这里使用不同的方法给出）。还给出了这些结果与$\mathsf{SH}(κ^{++})$失败的类似情况。最后，我们得到了我们的所有结果与任意大的$2^κ$，应用Honzik和第三作者最近的联合工作。

英文摘要：

We obtain an array of consistency results concerning trees and stationary reflection at double successors of regular cardinals $κ$, updating some classical constructions in the process. This includes models of $\mathsf{CSR}(κ^{++})\wedge \mathsf{TP}(κ^{++})$ (both with and without $\mathsf{AP}(κ^{++})$) and models of the conjunctions $\mathsf{SR}(κ^{++}) \wedge \mathsf{wTP}(κ^{++}) \wedge \mathsf{AP}(κ^{++})$ and $\neg \mathsf{AP}(κ^{++}) \wedge \mathsf{SR}(κ^{++})$ (the latter was originally obtained in joint work by Krueger and the first author \cite{GilKru:8fold}, and is here given using different methods). Analogs of these results with the failure of $\mathsf{SH}(κ^{++})$ are given as well. Finally, we obtain all of our results with an arbitrarily large $2^κ$, applying recent joint work by Honzik and the third author.

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