A global version of a theorem of Ben-David and Magidor
In J. Symb. Log. 51, 1029-1033 (1986; Zbl 0621.03035), \textit{S. Ben-David} and \textit{M. Magidor} showed, using a \(\kappa^+\)-supercompact cardinal~\(\kappa\), that \(\square_{\aleph_\omega}^*\) may hold while \(\square_{\aleph_\omega}\) fails. In the present paper the authors start with a~\(\kappa\) that is \(\kappa^{+5}\)-supercompact and construct a model in which \(\kappa\)~is still \(\kappa^{+5}\)-supercompact and for \textit{every} singular cardinal~\(\lambda\) below~\(\kappa\) the principle~\(\square_{\lambda,\text{cf}(\lambda)}\) holds while \(\square_\lambda\)~fails. Thus, in this model the set~\(V_\kappa\) is a model for the global version of the theorem of Ben-David and Magidor that was promised in the title. Note \(\square_{\lambda,\theta}\) says that there is a sequence \(\langle\mathcal C_\alpha:\alpha<\lambda^+, \alpha\)~a limit\(\rangle\) such that \(1\leq|\mathcal C_\alpha|\leq\theta\), \(\mathcal C_\alpha\)~consists of club subsets of~\(\alpha\) of order type at most~\(\lambda\), and if \(C\in\mathcal C_\alpha\) and \(\beta\in\lim(C)\) then \(C\cap\beta\in\mathcal C_\beta\) -- so \(\square_\lambda=\square_{\lambda,1}\) and \(\square_\lambda^*=\square_{\lambda,\lambda}\).
scientific article; zbMATH DE number 1423281
scientific article