Ideals on Ρ_κλ and infinitary combinatorics related to large cardinal axiom

(1)We证明了若λ^<<κ>=2^λ，则P_κλ的几乎不可描述子集是不可描述的. (2)We证明了κ不总是λ^<<κ>不可形容的，即使它是λ不可形容的。（3）若P_κλ具有以下性质：（a）{χ∈ P_κλ：χ_κ κ不是序数}，{χ∈ P_κλ：χ_κ=| χ|}，且{χ∈ P_κλ：χ ∈ κ<|χ|}（B）{x ∈ P_κλ：x ∈ κ∈ A}其中A是κ的任意无界子集。(4)We提出了两个强制概念，使得（a）和（B）在一般扩张中分别成立：（a）存在P_κκ^+的一个平稳子集，它不分裂成κ^+个平稳集。（B）存在κ上的κ稠密理想，这比Gitik对（a）和Woodin对（B）的已知方法简单. (5)We证明了：（a）若σ SR对P_κλ成立且λ^2^<2<κ>=λ，则P_κκ上的σ俱乐部滤波器是陡峭的。(b)If当λ ^<2κ>=λ时，κ SR成立，则κ上的俱乐部滤子具有弱覆盖性质. (c)If若对P_<ω1>λ的κ SR成立，λ^κ^+=λ，且2^ω<κ<2^ω_1=^2<<κ>，则P_κκ^+分裂为2^ω_1个平稳集. (6)We（a）B=θ^*=cov（M），（B）d=θ^*ω_1和θ=ω_1和ω_2，（c）cov（M）=ω_1和θ^*=θ=ω_2。（7）对于偏序集，我们定义性质“SEP”，表明：（a）当P（ω）有SEP时，non（M）=ω_1，若进一步假定□ _<ω_1>，则P（ω）中几乎不交族的最小基数也是ω_1。(b)We可以建立SEP为P（ω）和-SEP为P（ω）的强迫模型。

(1)We proved any almost ineffable subset of Ρ_κλis ineffable provided that λ^<<κ>=2^λ.(2)We showed κ is not always λ^<<κ> ineffable even if it is λ ineffable.(3)The following sets were shown to have the weak partition property if Ρ_κλ has the property :(a){χ∈Ρ_κλ : χ∩κ is not an ordinal}, {χ∈Ρ_κλ : χ∩κ=|χ|}, and{χ∈Ρ_κλ : χ∩κ<|χ|}(b){χ∈Ρ_κλ : χ∩κ∈Α}where is Α any unbounded subset of κ.(4)We presented two forcing notions such that (a) and (b) holds in the generic extension respectively :(a)There exists a stationary subset of Ρ_κκ^+ which does not split into κ^+ many stationary sets.(b)There exists κ dense ideal on κ.These are simpler than known methods by Gitik for (a) and Woodin for (b).(5)We proved the following on stationary reflection (SR) :(a)If σ SR for Ρ_κλ holds and λ^2^<2<κ>=λ, then the σ club filter on Ρ_κκ is precipitous.(b)If κ SR for Ρ_<ω1>λ holds and λ^<2κ>=λ, then the club filter on κ has a weak covering property.(c)If κ SR for Ρ_<ω1>λ holds, λ^κ^+=λ, and 2^ω<κ<2^ω_1=^2<<κ>, then Ρ_κκ^+ splits into 2^ω_1 many stationary sets.(6)We built forcing models for (a)〜(c) respectively :(a)b=θ^*=cov(Μ), (b)d=θ^*ω_1 and θ=ω_1 and ω_2, (c)cov(Μ)=ω_1 and θ^*=θ=ω_2.(7)For partially orderd sets we define the property "SEP" to show :(a)non(Μ)=ω_1 when Ρ(ω) has SEP. If we further assume □ _<ω1>, then the minimal cardinality of almost disjoint family in Ρ(ω) is also ω_1.(b)We can build forcing models each of SEP for Ρ(ω) and -SEP for Ρ(ω).