Applications of pcf to ideals on P_κλ and infinitary combinatorics, and independence proof

点击翻译按钮获取中文摘要

(1)We give a simple proof for Kamo's theorem that there exists a normal ultrafilter on P_κλ if κ is λ supercompact.(2)We show the ideal of the subsets of P_κλ」a without the weak partition property is not κ^+ saturated.(3)We showed the minimal ideal on P_κλ with the partition property is not λ^+ saturated.(4)We proved the minimal normal ideal on P_κλ with the partition priperty is a proper extension of the ideal of noncompletely ineffable subsets.(5)We showed stationary reflection principle for P_κλ fails if ω_1<κ<λ.(6)We present a forcing model in which, for every regular μ【less than or equal】κ, the restriction of the nonstationary ideal on P_μκ to {x:cf(sup(x))=ω} is weakly presaturated.(7)We show that the restriction of the nonstationary ideal on P_κ to {x:cf(sup(x))=ω} is weakly presaturated under the σ stationary reflection in P_κ2^2^2^<<κ>.(8)We give a correct and simple proof for Shelah's theorem that the minimal cardinality of unbounded subset of P_κλ is equal to that of stationary subset.(9)We prove the existence of a diamond sequence for P_κλ provided that 2^ω=2^<<κ>, and present a forcing model such that P_<ω2>λ carries a diamond sequence whereas 2^ω<2^<ω_1>.(10)We showed that the additivity of the ideal Ι_f) is not, greater than the bounded number, and the cofinality of Ι_f) is not smaller than the dominating number.(11)We present a function f∈^ωω such that T_f is not an ideal.