Stationary sets added when forcing squares
type: Item
instance of: scholarly article
title: Stationary sets added when forcing squares
zbMATH Open document ID: 1446.03078
author: Maxwell Levine
published in: Archive for Mathematical Logic
publication date: 2019-01-14T00:00:00Z
review text: The \(\Box_\kappa\) principle was introduced by R. Jensen. By the way, R. Jensen did not develop \(\Box_\kappa\) to study the fine structure of \(L\) as the author claims. Jensen used this principle to build \(\kappa^+\)-Souslin trees for any cardinal \(\kappa\) and showed that, under \(V=L\), there is a \(\Box_\kappa\)-square sequence for any cardinal \(\kappa\) as an outstanding application of the fine structure theory. Also, appealing to the fine structure theory he showed that in \(V=L\), the cardinal \(\kappa\) is weakly compact if and only if for every stationary \(E\subseteq\kappa\) there exists a regular cardinal \(\lambda<\kappa\) such that \(E\cap \lambda\) is stationary in \(\lambda\), where \(V=L\) is needed only in the right to the left implication. If \(\kappa\) is a cardinal and \(S\subseteq\kappa\) is stationary in \(\kappa\), it is said that \(S\) reflects at \(\alpha<\kappa\) if \(\alpha\) has uncountable cofinality and \(S\cap \alpha\) is stationary as a subset of \(\alpha\). So, when \(\kappa\) is weakly compact, each stationary \(S\subseteq\kappa\) reflects at some \(\alpha<\kappa\). The theme of the paper under review is how square-like principles stop reflection. The author puts forward a family of square-like principles. Let \(\kappa\) and \(\lambda\) be infinite cardinals with \(1<\lambda<\kappa^+\). A \(\Box_{\kappa,<\lambda}\)-sequence is a sequence \(\langle\mathcal{C}_\alpha:\alpha \in\kappa\cap \lim\rangle\) with the following properties. \begin{itemize} \item[1.] \(1\leq|\mathcal{C}_\alpha|<\lambda\); \item[2.] \(\mathcal{C}_\alpha\) consists of clubs in \(\alpha\); \item[3.] If \(C\in\mathcal{C}_\alpha\), then \(ot(C)\leq\kappa\); \item[4.] If \(C\in\mathcal{C}_\alpha\) and \(\beta\) is a limit point of \(C\), then \(C\cap\beta\in\mathcal{C}_\beta\). \end{itemize} The principle \(\Box_{\kappa,\lambda}\) is defined as \(\Box_{\kappa,<\lambda}\) except that (1) is replaced by \(1\leq|\mathcal{C}_\alpha|\leq\lambda\). With this notation,…
reviewed by: Luis Miguel Villegas Silva
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E55
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 7001424
zbMATH Keywords: forcing square-like principles reflection
MaRDI profile type: Publication
full work available at URL: s00153-018-0613-8
OpenAlex ID: W2786286890
cites work: Squares, scales and stationary reflection
cites work: Successors of Singular Cardinals
cites work: The fine structure of the constructible hierarchy
cites work: Making the supercompactness of \(\nu\) indestructible under \(\nu\)-directed closed forcing
cites work: WEAK SQUARES AND VERY GOOD SCALES
cites work: Combinatorial principles in the core model for one Woodin cardinal
DOI: 10.1007/S00153-018-0613-8
description: scientific article; zbMATH DE number 7001424