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

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 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, View the MathML source 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 View the MathML source holds modulo an ideal obtained through the weak reflection.