Generalisations of stationarity, closed and unboundedness, and of Jensen's □

Abstract

The concepts of closed unbounded (club) and stationary sets are generalised to γ-club and γ-stationary sets, which are closely related to stationary reflection principles. We use these notions to define generalisations of Jensen's combinatorial principles □ as □γ and □<γ sequences. We define Πγ1-indescribability and show first that in L if γ<κ is an ordinal and κ is Σγ1-indescribable but not Πγ1-indescribable, and A⊆κ is γ-stationary, then there is EA⊆A and a □<γ sequence S on κ such that EA is γ-stationary in κ and S avoids EA. This generalises a result of Jensen for γ=1. As a corollary we also extend the result of Jensen that in L a regular cardinal is stationary reflecting if and only if it is Π11-indescribable by showing that such a κ as above is not γ-reflecting, yielding a different proof of a result appearing in [3]. Thus in L a cardinal is Πγ1-indescribable iff it reflects γ-stationary sets. We define □γ(κ), as stating that there is an unthreadable□γ-sequence at κ; we show this implies that κ is not γ+1-reflecting. Certain assumptions on the γ-club filter allow us to prove that γ-stationarity is downwards absolute to L, and allows for splitting of γ-stationary sets. We define γ-ineffability, and look into the relation between γ-ineffability and various ⋄γ principles.