Stationary sets added when forcing squares
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, \(\Box_{\kappa,1}\) is \(\Box_\kappa\) and \(\Box_{\kappa,\kappa}\) is \(\Box^*_\kappa\). There are several instances of \(\Box_{\kappa,<\lambda}\) which preclude reflection to hold. The forcing involved in such results use variations of two posets introduced by the author to force square-like sequences. These forcing use posets called \(\mathbb{S}(\kappa,<\lambda)\) and CMB, the collapses-mod-bounded poset, where \(\mathbb{C}= \text{CMB}\) is defined in terms of a singular strong limit cardinal \(\kappa\) of cofinality \(\mu\). Associated to these two parameters we have the set \(\operatorname{cof}(\mu)=\{\alpha\leq\kappa:\operatorname{cf}(\alpha)=\mu\}\). Both posets are \((\kappa+1)\)-strategically closed, \(\Vdash_{\mathbb{S}(\kappa,<\lambda)}\Box_{\kappa,<\lambda}\), and CMB adds \(\Box^*_\kappa\). Moreover, if \(\tau\) is supercompact and \(\mathbb{C}\) is defined in terms of \(\kappa>\tau\) with \(\operatorname{cf}(\kappa)<\tau\), then the generic extension by \(\mathbb{C}\) contains no very good scales on \(\kappa\). The poset \(\mathbb{S}(\kappa,<\lambda)\) adds non-reflecting stationary subsets of \(\kappa^+\) in every cofinality. The poset \(\mathbb{C}\) adds non-reflecting stationary subsets of \(\kappa^+\) in cofinality \(\operatorname{cf}(\kappa)\), as described next. The first relevant result points up that if we force with \(\mathbb{S}(\kappa,<\lambda)\), then in the generic extension there exists a non-reflecting stationary subset of \(\kappa^+\cap \operatorname{cof}(\mu)\) for every regular \(\mu\leq\kappa\). Then, the author conducts the proof that \(\mathbb{C}\) adds a non-reflecting stationary set in \(\kappa^+\cap \operatorname{cof}(\mu)\). Finally, it is proved that the poset \(\mathbb{C}\) does not necessarily add non-reflecting stationary sets in all cofinalities. Namely, the author shows that if \(\lambda\) is indestructibly supercompact and \(\omega<\mu<\lambda<\kappa\), then in the generic extension by \(\mathbb{C}\), for every \(\tau<\mu\) each stationary subset of \(\kappa^+\cap \operatorname{cof}(\tau)\) reflects. This is a well-written paper easy to follow when the reader is acquainted with forcing.
forcing square-like principles reflection
scientific article; zbMATH DE number 7001424