Covering properties and square principles

Abstract

Covering matrices were used by Viale in his proof that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom and later by Sharon and Viale to investigate the impact of stationary reflection on the approachability ideal. In the course of this work, they isolated two reflection principles, CP and S, which may hold of covering matrices. In this paper, we continue previous work of the author investigating connections between failures of CP and S and variations on Jensen’s square principle. We prove that, for a regular cardinal λ > ω 1, assuming large cardinals, □(λ, 2) is consistent with CP(λ, θ) for all θ with θ + < λ. We demonstrate how to force nice θ-covering matrices for λ which fail to satisfy CP and S. We investigate normal covering matrices, showing that, for a regular uncountable κ, □ κ implies the existence of a normal ω-covering matrix for κ + but that cardinal arithmetic imposes limits on the existence of a normal θ-covering matrix for κ + when θ is uncountable. We introduce the notion of a good point for a covering matrix, in analogy with good points in PCF-theoretic scales. We develop the basic theory of these good points and use this to prove some non-existence results about covering matrices. Finally, we investigate certain increasing sequences of functions which arise from covering matrices and from PCF-theoretic considerations and show that a stationary reflection hypothesis places limits on the behavior of these sequences.