If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of { delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing clubs and for each alpha<lambda we have: { C_delta \cap alpha: alpha \in nacc(C_delta)} has cardinality <lambda. Also, we prove that e.g. there is a stationary subset of S_{ <aleph_1}(lambda) of cardinality cf(S_{ <aleph_1}(lambda),subseteq) Then we prove the existence of nice filters when instead being normal filters on omega_1 they are normal filters with larger domains, which can increase during a play. They can help us transfer situation on aleph_1-complete filters to normal ones
Categories: math.LO