Applications of forcing axioms and large continuum

• Project/Area Number: 18K03393
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Shizuoka University
• Principal Investigator: Yorioka Teruyuki 静岡大学, 理学部, 准教授 (60432192)
• Project Period (FY): 2018-04-01 – 2024-03-31
• Project Status: Completed (Fiscal Year 2023)
• Budget Amount: ¥4,030,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥930,000); Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 強制法理論 / 無限組合せ論 / 強制法 / 公理的集合論 / 強制法公理 / 基数不変量 / サイドコンディション法 / 強制公理 / 連続体濃度

Outline of Final Research Achievements

The author presented an application of Aspero-Mota iteration with symmetric markers, and gave an example of Aspero-Mota iteration with symmetric markers which may not be proper when the length of the iteration is longer than omega_2.
The author showed that some forcing notions with side condition method satisfy Y-proper, and showed that Y-Proper Forcing Axiom due to Chodounsky-Zapletal implies Mapping reflection Principle, Weak Culb Guessing, and mho.

Academic Significance and Societal Importance of the Research Achievements

集合を形式化することで，その形式化で「ある定理を証明する」ことが定義され，それにより「ある定理はその形式化から証明できない」ことが定式化された．これにより，カントールの連続体仮説をはじめいくつかの数学命題は集合論の形式化から証明も反証もされないことが示されてきた．これは数学に対する見方について問題提起している．本研究はこの問題に一役買うことを目指している．

Research Products

• [Journal Article] Two chain conditions and their Todocevic's fragments of Martin's Axiom (2024)
  • Author(s): Yorioka Teruyuki
  • Journal Title: Annals of Pure and Applied Logic Volume: 175 Issue: 1 Pages: 103320-103320
  • DOI: 10.1016/j.apal.2023.103320
  • Peer Reviewed / Open Access
• [Journal Article] Aspero-Mota iteration and the size of the continuum (2023)
  • Author(s): YORIOKA TERUYUKI
  • Journal Title: The Journal of Symbolic Logic Volume: 88 Issue: 4 Pages: 1387-1420
  • DOI: 10.1017/jsl.2022.37
  • Peer Reviewed / Open Access
• [Journal Article] A fragment of Asper??Mota’s finitely proper forcing axiom and entangled sets of reals (2020)
  • Author(s): Miyamoto Tadatoshi、Yorioka Teruyuki
  • Journal Title: Fundamenta Mathematicae Volume: 251 Issue: 1 Pages: 35-68
  • DOI: 10.4064/fm814-11-2019
  • Peer Reviewed
• [Journal Article] Todorcevic’s Axiom K_2 and ladder system colorings (2020)
  • Author(s): Justin Tatch Moore and Teruyuki Yorioka
  • Journal Title: RIMS Kokyuroku Volume: 2164 Pages: 104-107
• [Presentation] トドロチェビッチによるマーティンの公理の部分公理と様々な梯子系の色付けの一様化 (2023)
  • Author(s): 依岡輝幸
  • Organizer: 日本数学会2023年度秋季総合分科会
• [Presentation] Aspero-Mota iteration and the size of continuum (2022)
  • Author(s): Teruyuki Yorioka
  • Organizer: RIMS Set Theory workshop 2022
  • Int'l Joint Research / Invited
• [Presentation] 梯子系の色付けの$\sigma$-一様化とトドロチェビッチによるマーティンの公理の部分公理 (2022)
  • Author(s): 依岡輝幸
  • Organizer: 2022年度日本数学会年会
• [Presentation] Preservation of Cohen reals (2021)
  • Author(s): Teruyuki Yorioka
  • Organizer: Kobe Set Theory Workshop 2021 -- on the occasion of Sakae Fuchino's Retirement --
  • Int'l Joint Research / Invited
• [Presentation] YPFA implies MRP (2019)
  • Author(s): Teruyuki Yorioka
  • Organizer: Set theory of the Reals (19w5064), at The Casa Matematica Oaxaca (CMO).
  • Int'l Joint Research / Invited
• [Presentation] 梯子系の色付けの一様化とトドロチェビッチによるマーティンの公理の部分公理 (2019)
  • Author(s): 依岡輝幸
  • Organizer: 日本数学会2019年度秋季総合分科会
• [Presentation] Applications of Todorcevic's side condition method (Tutorial) (2019)
  • Author(s): Teruyuki Yorioka
  • Organizer: IPM Set Theory Conference 2019
  • Int'l Joint Research / Invited
• [Presentation] Todorcevic’s fragments of Martin’s Axiom and some weak form of uniformization of a ladder system coloring (2019)
  • Author(s): Teruyuki Yorioka
  • Organizer: RIMS Set Theory Workshop 2019: Set Theory and Infinity
  • Int'l Joint Research
• [Presentation] $(\omega^\omega, <^*)$上の極大飽和直線と実数直線の性質 (2019)
  • Author(s): 依岡輝幸
  • Organizer: 2019年度日本数学会年会
• [Presentation] A fragment of Aspero-Mota's Finitely Proper Forcing Axiom and an entangled set of reals (2018)
  • Author(s): Teruyuki Yorioka
  • Organizer: Novi Sad Conference in Set Theory and General Topology , SETTOP 2018
  • Int'l Joint Research
• [Presentation] ランダム強制法による拡大モデルの中での弱い Martin’s Axiom (2018)
  • Author(s): 依岡輝幸
  • Organizer: 日本数学会 2018年度秋季総合分科会
• [Presentation] The difference between two properties of Aronszajn trees (2018)
  • Author(s): Teruyuki Yorioka
  • Organizer: Workshop on Iterated Forcing Theory and Cardinal Invariants (RIMS workshop)
  • Int'l Joint Research
• [Remarks] Teruyuki YORIOKA
  • URL: https://wwp.shizuoka.ac.jp/yorioka/
• [Remarks] Teruyuki Yorioka/Works
  • URL: https://wwp.shizuoka.ac.jp/yorioka/works/
• [Remarks] Teruyuki Yorioka
  • URL: https://wwp.shizuoka.ac.jp/yorioka/
• [Funded Workshop] Methods in Higher Forcing Axioms (2019)