Superstationary and \(ineffable^ n\) cardinals (Q1263581)

In this paper the author considers a generalization of stationary sets and ineffable cardinals, based on certain operations performed on ideals. The concept of n-superstationary for subsets of the cardinal \(\kappa\) is defined by: \(A\subseteq \kappa\) is 0-superstationary if and only if A is stationary. \(A\subseteq \kappa\) is \((n+1)\)-superstationary if and only if for every function f with domain A and always f(\(\alpha)\) a not n- superstationary subset of \(\alpha\), there is \(B\subseteq A\) which is n- superstationary with \(\cup \{f(\beta)\); \(\beta\in B\}\) not n- superstationary. The cardinal \(\kappa\) is said to be n-superstationary if \(\kappa\) is n-superstationary as a subset of itself. It is shown that the hierarchy of n-superstationary cardinals is nontrivial, and several relative consistency results concerning the existence strength of a 1-superstationary cardinal are given. The generalization of ineffable cardinals goes thus: for \(\kappa\) a regular uncountable cardinal and \(A\subseteq \kappa\), a (1,A)-sequence is a sequence of sets \((S_{\alpha})_{\alpha \in A}\) with always \(S_{\alpha}\subseteq \alpha\). Then \(A\subseteq \kappa\) is said to be \(ineffable^ 0\) if and only if A is stationary. \(A\subseteq \kappa\) is \(ineffable^{n+1}\) if and only if every (1,A)-sequence has an \(ineffable^ n\) homogeneous set H (that is, \(S_{\alpha}=\alpha \cap S_{\beta}\) for all \(\alpha\),\(\beta\in H\) with \(a<\beta)\).