SATURATED FILTERS AT SUCCESSORS OF SINGULARS, WEAK REFLECTION AND YETANOTHER WEAK CLUB PRINCIPLE

Suppose that lambda is the successor of a singular cardinal mu whose c ofinality is an uncountable cardinal kappa. We give a sufficient condi tion that the club filter of lambda concentrating on the points of cof inality kappa is not lambda(+)-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 (sic )-principle, which we call (sic)(-)+, and show that if it holds on a s tationary subset S of lambda, then no normal filter on S is lambda(+)- saturated. Under the above assumptions, (sic)(-)(S) is true for any s tationary subset S of lambda which does not contain points of cofinali ty kappa. For stationary sets S which concentrate on points of cofinal ity kappa, we show that (sic)(-)(S) holds module an ideal obtained th rough the weak reflection.