Generalisations of Stationarity, Closed and Unboundedness, and of Jensen's \Box

2019/08/16 by Hazel Brickhill, Brickhill, H., Philip Welch +1
Mathematics · #Mathematics and Applications #Advanced Combinatorial Mathematics #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1908.05920

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. We use these notions to define generalisations of Jensen's combinatorial principles \Box and \diamondsuit. We define Π¹_γ-indescribability and use the new \Box^γ-sequences to extend the result of Jensen that in the constructible universe a regular cardinal is stationary reflecting if and only if it is Π¹₁-indescribable: we show that in L a cardinal is Π¹_γ -indescribable iff it reflects γ-stationary sets. More particularly (stating only the special case of n finite): Theorem (V=L) Let n

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