Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Abstract

Suppose that λ is the successor of a singular cardinal μ whose cofinality is an uncountable cardinal κ. We give a sufficient condition that the club filter of λ concentrating on the points of cofinality κ is not λ +-saturated. 1 1 Added in proof: M. Gitik and S. Shelah have subsequently and by a different technique shown that the club filter on such λ is never saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection. We introduce a weak version of the ♣-principle, which we call ♣ ∗ −, and show that if it holds on a stationary subset S of λ, then no normal filter on S is λ +-saturated. Under the above assumptions, ♣ ∗ −(S) is true for any stationary subset S of λ which does not contain points of cofinality κ. For stationary sets S which concentrate on points of cofinality κ, we show that ♣ ∗ −(S) holds modulo an ideal obtained through the weak reflection.