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）我们提供了ρ_κλ的几乎无法释放的子集，但前提是，只要λ^<<κ> = 2^λ。（2）我们表明κ并非总是λ^<<κ>不可延迟，即使λ不可卸载。 χ∩κ= |χ|}和{χ∈ρ_κλ：χ∩κ<|χ|}（b）{χ∈ρ_κλ：χ∩κ<|χ|}（b）{χ∈ρ_κλ：χ∩X_κλ：χ∩κ<|χ| |}，χ∩κ∈α} euste nisty nisty nisty nose nisty nose nisted nisted nisted nisted nisted nisted nisted n.und of und und insset. （a）和（b）分别保持一般延伸：（a）存在ρ_κκ^+的固定子集，该子集未分为κ^+许多固定组。（b）在κ.的κ.的理想之上存在于κ.的理想之处。对于ρ_κλ持有和λ^2^<2 <κ> =λ，则ρ_κκ上的σ杆过滤器是陡峭的。 λ^κ^+ =λ，2^ω<κ<2^ω____________________________________2<<κ>，然后ρ_κκ^+将2^ω___________________________________________________________________________________________________________________________________和（B）和（a） ω_2, (c)cov(Μ)=ω_1 and θ^*=θ=ω_2.(7)For partially ordered sets we define the property "SEP" to show :(a)non(Μ)=ω_1 when Ρ(ω) has SEP.如果我们进一步假设□_<Ω1>，则ρ（ω）中几乎不连接家族的最小基数也为ω_1。

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)。

Masahiro Shioya: "A saturated stationary subset of P_κκ^+"Mathematical Research Letters. (to appear).

Masahiro Shioya：“P_κκ^+ 的饱和平稳子集”数学研究快报（即将出现）。