On reflection principles at small uncountable cardinals

• Project/Area Number: 23740076
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kobe University
• Principal Investigator: SAKAI Hiroshi 神戸大学, システム情報学研究科, 准教授 (70468239)
• Project Period (FY): 2011 – 2013
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2013: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2012: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2011: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 公理的集合論 / 反映原理 / 巨大基数公理 / 無限組み合わせ論 / 国際情報交換 / 国際研究者交流 / 多国籍

Research Abstract

I investigated reflection principles at small uncountable cardinals, in particular, their consistencies and their relationships with infinite combinatorics, forcing axioms and large cardinal axioms. Among other things, the following results were obtained: (1) I clarified how weak square principles are denied by each reflection principle at omega 2. (2) By showing that the forcing axiom MA+ (sigma-closed) does not follow from any variation of stationary reflection principles at omega 2, I clarified the difference between reflection principles and forcing axioms. (3) It was proved that in L there is a correspondence between hierarchies of stationary reflection principles and indescribable cardinals.

Research Products

• [Journal Article] Stationary reflection principles and two cardinal tree properties (2014)
  • Author(s): Hiroshi Sakai and Boban Velickovic
  • Journal Title: Journal of the Institute of Mathematics of Jussieu Volume: 掲載確定 Issue: 1 Pages: 69-85
  • DOI: 10.1017/s1474748013000315
  • NAID: 120005848133
  • Peer Reviewed / Open Access
• [Journal Article] Chang's conjecture and weak square (2013)
  • Author(s): H. Sakai
  • Journal Title: Archive for Mathematical Logic Volume: Vol.52, No.1-2 Pages: 29-45
  • NAID: 120005367491
  • Peer Reviewed
• [Journal Article] Chang's conjecture and weak square (2013)
  • Author(s): Hiroshi Sakai
  • Journal Title: Archive for Mathematical Logic Volume: 52 Issue: 1-2 Pages: 29-45
  • DOI: 10.1007/s00153-012-0305-8
  • NAID: 120005367491
  • Peer Reviewed
• [Journal Article] On proofs of the incompleteness theorems based on Berry's paradox by Vopenka, Chaitin, and Boolos (2012)
  • Author(s): M. Kikuchi, T. Kurahashi and H. Sakai
  • Journal Title: Mathematical Logic Quarterly Volume: Vol.58 ,No.4-5 Pages: 307-316
  • Peer Reviewed
• [Journal Article] On proofs of the incompleteness theorems based on Berry's paradox by Vopenka, Chaitin, and Boolos (2012)
  • Author(s): Makoto Kikuchi, Taishi Kurahashi, Hiroshi Sakai
  • Journal Title: Mathematical Logic Quarterly Volume: 57 Issue: 4-5 Pages: 307-316
  • DOI: 10.1002/malq.201110067
  • Peer Reviewed
• [Journal Article] On proofs of the incompleteness theorem based on Berry's paradox by Vopenka, Boolos and Chaitin (2012)
  • Author(s): 酒井拓史，倉橋太志，菊池誠
  • Journal Title: Mathematical Logic Quarterly Volume: 58
  • Peer Reviewed
• [Journal Article] Partial square at omega_1 is implied by MM but not by PFA (2011)
  • Author(s): H. Sakai
  • Journal Title: Fundamenta Mathematicae Volume: Vol.215,No.2 Pages: 109-131
  • Peer Reviewed
• [Journal Article] On standard model ofMartin's Maximum (2011)
  • Author(s): H. Sakai
  • Journal Title: 京都大学数理解析研究所講究録 Volume: No.1754 Pages: 97-107
• [Journal Article] Partial square at \omega_1 is impied by MM but not by PFA (2011)
  • Author(s): 酒井拓史
  • Journal Title: Fundamenta Mathematicae Volume: 215 Pages: 109-131
  • Peer Reviewed
• [Journal Article] Reflection and indescribability in the constructible universe
  • Author(s): J. Bagaria, M. Magidor and H. Sakai
  • Journal Title: Israel Journal of Mathematics Volume: (掲載決定)
  • Peer Reviewed
• [Journal Article] Stationary reflection principles and two cardinal tree properties
  • Author(s): H. Sakai and B. Velickovic
  • Journal Title: Journal of the Institute of Mathematics of Jussieu Volume: (掲載決定)
  • NAID: 120005848133
  • Peer Reviewed
• [Presentation] Separation of MA^+ (sigma-closed) from reflection principles (2013)
  • Author(s): H. Sakai
  • Organizer: ESI Workshop Forcing and Large Cardinals
  • Place of Presentation: Erwin Schrodinger Institute,ウィーン(オーストリア)
  • Year and Date: 2013-09-24
• [Presentation] n-stationary and Pi^1_n-1-indescribable sets (2013)
  • Author(s): H. Sakai
  • Organizer: RIMS共同利用研究集会反映原理と巨大基数の集合論
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2013-09-11
• [Presentation] Separation of MA^+ (sigma-closed) from reflection principles (2013)
  • Author(s): H. Sakai
  • Organizer: International Conference on Topology and Geometry2013
  • Place of Presentation: 島根大学
  • Year and Date: 2013-09-05
  • Invited
• [Presentation] 定常性反映原理と半定常性反映原理(特別講演) (2013)
  • Author(s): 酒井拓史
  • Organizer: 日本数学会年会
  • Place of Presentation: 京都大学
  • Year and Date: 2013-03-21
• [Presentation] Consequences of Martin's Maximum and weak square (2012)
  • Author(s): H. Sakai
  • Organizer: Workshop of Forcing Axioms and their Applications
  • Place of Presentation: Fields Institute,トロント(カナダ)
  • Year and Date: 2012-10-24
  • Invited
• [Presentation] Fragments of Martin's Maximum and weak square (2012)
  • Author(s): H. Sakai
  • Organizer: Association for Symbolic Logic 2012 North American Annual Meeting
  • Place of Presentation: University of Wisconsin-Madison,マディソン(アメリカ)
  • Year and Date: 2012-03-31
  • Invited
• [Presentation] Fragments of Martin's Maximum and weak square (2012)
  • Author(s): 酒井拓史
  • Organizer: ASL 2012 North American Annual Meeting(招待講演)
  • Place of Presentation: University of Wisconsin-Madison (United States of America)
• [Presentation] Chang's conjecture and weak square (2011)
  • Author(s): H. Sakai
  • Organizer: the 12th Asian Logic Conference
  • Place of Presentation: Victoria University of Wellington,ウェリントン(ニュージーランド)
  • Year and Date: 2011-12-17
• [Presentation] Chang's conjecture and weak square (2011)
  • Author(s): H. Sakai
  • Organizer: RIMS共同利用研究集会記述集合論の展望
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2011-10-20
• [Presentation] Chang's Conjecture and weak square (2011)
  • Author(s): 酒井拓史
  • Organizer: 京都大学数理解析研究所 共同利用研究集会「記述集合論の展望」
  • Place of Presentation: 京都大学数理解析研究所（京都府）
• [Presentation] Chang's Conjecture and weak square (2011)
  • Author(s): 酒井拓史
  • Organizer: 12th Asian Logic Conference
  • Place of Presentation: Victoria University of Wellington (New Zealand)
• [Presentation] Separation of MA^+ (sigma-closed) from reflection principles
  • Author(s): Hiroshi Sakai
  • Organizer: International Conference on Topology and Geometry 2013
  • Place of Presentation: 島根大学
  • Invited
• [Presentation] n-stationary and Pi^1_n-1-stationary sets
  • Author(s): Hiroshi Sakai
  • Organizer: RIMS Workshop "Reflection principles and set theory of large cardinals"
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Separation of MA^+ (sigma-closed) from reflection principles
  • Author(s): Hiroshi Sakai
  • Organizer: Erwin Schrodinger Institute Workshop "Forcing and Large Cardinals"
  • Place of Presentation: Erwin Schrodinger Institute, ウィーン（オーストリア）
  • Invited
• [Presentation] Fragments of Martin's Maximum and weak square
  • Author(s): Hiroshi Sakai
  • Organizer: ASL 2012 North American Annual Meeting
  • Place of Presentation: University of Wisconsin-Madison
  • Invited
• [Presentation] Consequences of Martin's Maximum and weak square
  • Author(s): Hiroshi Sakai
  • Organizer: Workshop on Forcing Axioms and their Applications
  • Place of Presentation: Fields Institute
  • Invited
• [Presentation] 定常性反映原理と半定常性反映原理
  • Author(s): 酒井 拓史
  • Organizer: 日本数学会年会
  • Place of Presentation: 京都大学
  • Invited
• [Remarks]
  • URL: http://www2.kobe-u.ac.jp/~hsakai/
• [Remarks]
  • URL: http://researchmap.jp/hsakai/
• [Remarks] Research Map 個人ページ
  • URL: http://researchmap.jp/hsakai/