Some results on the nonstationary ideal
The author investigates further the consistency strengths of hypotheses on \(\text{NS}_\kappa\) (the nonstationary ideal over cardinal \(\kappa\)) and \(\text{NS}^\lambda_\kappa\) (the nonstationary ideal over \(\kappa\) restricted to sets of ordinals of cofinality \(\lambda\)); the three properties studied are precipitousness, presaturatedness and saturatedness. Among the many interesting results in the paper are the following: (1) The exact strength of ``\(\text{NS}^\lambda_{\mu^+}\) is precipitous for a regular \(\mu > \max (\lambda, \aleph_1)\)'' is an \((\omega,\mu)\)-repeat point; (2) The exact strength of ``\(\text{NS}_\kappa\) is presaturated over inaccessible \(\kappa\)'' is an up-repeat point; (3) ``\(\text{NS}_\kappa\) is saturated over inaccessible \(\kappa\)'' implies an inner model with \(\exists \alpha (o(\alpha) = \alpha^{++})\). The notion of a repeat point was originally introduced by \textit{L. B. Radin} [Ann. Math. Logic 22, 243-261 (1982; Zbl 0502.03028)] in his paper introducing what has since become known as Radin forcing. Related notions are defined in the paper under review. For example, with \(\vec {\mathcal F}\) a coherent sequence of ultrafilters, an ordinal \(\alpha\) less than the length of \(\vec {\mathcal F}\) is defined to be an up-repeat point if for each set \(A\) in \(\vec {\mathcal F} (\kappa,\alpha)\) there is \(\beta > \alpha\) such that \(A \in \vec{\mathcal F} (\kappa,\beta)\).
scientific article; zbMATH DE number 836156
scientific article