2003 Fiscal Year Final Research Report Summary
Ideals on Ρ_κλ and infinitary combinatorics related to large cardinal axiom
| Project/Area Number |
14540142
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kanagawa University |
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Principal Investigator |
ABE Yoshihiro Kanagawa University, Faculty of Engineering, Professor, 工学部, 教授 (10159452)
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| Co-Investigator(Kenkyū-buntansha) |
FUCHINO Sakae Chubu University, Faculty of Engineering, Professor, 工学部, 教授 (30292098)
SHIOYA Masahiro University of Tsukuba, Institute of Mathematics, Full-time Lecture, 数学系, 講師 (30251028)
KAMO Shizuo University of Osaka Prefecture, Faculty of General Science, Professor, 総合科学部, 教授 (30128764)
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| Project Period (FY) |
2002 – 2003
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| Keywords | Ρ_κλ / ineffable / partition property / stationary reflection / saturated ideal / club filter / cardinal invariant / forcing |
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| Research Abstract |
(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).
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Research Products
(12 results)
