Ideals on Ρ_κλ and infinitary combinatorics related to large cardinal axiom

关于 Ρ_κλ 和与大基数公理相关的无限组合学的想法

• 批准号: 14540142
• 负责人: ABE Yoshihiro
• 金额: $ 0.9万
• 依托单位: Kanagawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540142/
• 关键词: Ρ_κλ; ineffable; partition property; stationary reflection; saturated ideal; club filter; cardinal invariant; forcing; Pκλ; stationary set; unbounded set; ineffability; normal ultrafilter; ideal; saturatlon; weak Freese-Nation Property; pr

(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).

(1)证明了Ρ_κλ的任何几乎不可说的子集都是不可说的，只要λ^&lt；&lt；κ&>t；=2^λ。(2)我们证明了κ并不总是λ^&lt；&lt；κ&>；即使它是λ不可说的。(3)如果Ρ_κλ有性质：(A){χ∈Ρ_κλ:χ∩κ不是序数}，{χ∈Ρ_κλ:χ∩κ=|χ|}，和{χ∈Ρ_κλ:χ∩κ&lt；|χ|}(B){χ∈Ρ_κλ:χ∩κ∈Α}其中isΑ是κ的任何无界子集。(4)我们提出了两个强迫概念，使得(A)和(B)在一般扩张中分别成立：(A)存在Ρ_κκ^+的一个平稳子集，它不分裂为κ^+许多平稳集。(B)在κ上存在κ稠密理想。这些方法比Gitik和Woodin关于(A)和(B)的已知方法更简单。(5)我们证明了关于平稳反射(SR)：(A))如果σSR forΡ_κλ成立，且λ^2^&lt；2&lt；κ&>=λ，则σ上的俱乐部过滤器是陡峭的。(B)如果Ρ_κκ上的κSR为Ρ_&lt；ω1&>λHold且λ^&lt；2κ&>；=λ，则κ上的俱乐部过滤器具有弱覆盖性质。(C)如果κ_lt；Ρ；ω1&>λHold，λ^κ^+=λ和2^ω&lt；κ&lt；2^ω_1=^2&lt；&lt；κ&>；(6)我们分别建立了关于(A)~(C)的强迫模型：(A)b=θ^*=Cov(Μ)，(B)d=θ^*ω_1和θ=ω_1和ω_2，(C)Cov(Μ)=ω_1和θ^*=θ=ω_2)。(7)对于偏序集合，我们定义了性质“Μ)=ω”，以证明：(A)非(Ρ(ω时为θ^*=θ=ω_1)有SEP。如果我们进一步假设ω)，则Ρ(ω中几乎不相交的族的最小基数也是ω_1。(B)我们可以分别建立Ρ(ω的强迫模型和Ρ(ω的强迫模型。

项目成果

Sakae Fuchino: "All aspects of internal approachability and its application"Suri Kaiseki Kenkyusho Kokyuroku. 1304. 67-77 (2003)

渊野荣：《内部平易近人的各个方面及其应用》Suri Kaiseki Kenkyusho Kokyuroku。

Masahiro Shioya: "A saturated stationary subset of P_κλ"Mathematical Research Letters. 10・4. 493-500 (2003)

Masahiro Shioya：“P_κλ 的饱和平稳子集”数学研究快报 10・4（2003）。

Shizuo Kamo: "Cardinal invariants associated with some combinatorial statements"京都大学数理解析研究所講究録. 1304. 78-87 (2003)

Shizuo Kamo：“与某些组合语句相关的基数不变量”京都大学数学科学研究所 Kokyuroku. 1304. 78-87 (2003)。

Masahiro Shioya: "A saturated stationary subset of P_κκ^+"Mathematical Research Letters. 10. 493-500 (2003)

Masahiro Shioya：“P_κκ^+ 的饱和平稳子集”《数学研究快报》10. 493-500 (2003)。

Yoshihiro Abe: "Variants of subtlety in P_κλ"数理解析研究所講究録. 1304. 47-66 (2003)

Yoshihiro Abe：“P_κλ 中微妙的变体”数学科学研究所 Kokyuroku。1304. 47-66 (2003)。