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1999 Fiscal Year Final Research Report Summary

INNER MODEL THEORY AND LARGE CARDINALS

Research Project

Project/Area Number 10640118
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKobe University

Principal Investigator

BRENDLE Jorg  Kobe University, Graduate School of Science and Technology, Associate Professor, 大学院・自然科学研究科, 助教授 (70301851)

Co-Investigator(Kenkyū-buntansha) FUCHINO Sakae  Kitami Institute of Technology, Faculty of Engineering, Professor, 工学部, 教授 (30292098)
KAKUDA Yuzuru  Kobe University, Faculty of Engineering, Professor, 工学部, 教授 (50031365)
WELCH Philip  Kobe University, Graduate School of Science and Technology, Professor, 大学院・自然科学研究科, 教授 (90294248)
YOSHINOBU Yasuo  Nagoya University, Graduate School of Human Informatics, Research Associate, 大学院・人間情報学研究科, 助手 (90281063)
MATSUBARA Yo  Nagoya University, School of Informatics and Sciences, Associate Professor, 情報文化学部, 助教授 (30242788)
Project Period (FY) 1998 – 1999
KeywordsINNER MODELS / LARGE CARDINALS / FORCING THEORY / CARDINAL INVARIANTS INFINITARY COMBINATORICS / 無限組合せ論
Research Abstract

Research in this project was devoted to inner model theory, large cardinals, forcing theory, cardinal invariants of the continuum and other subfields of set theory, as well as to the interplay between them and to applications to other areas of pure mathematics. We briefly sketch the main topics and results.
1.Maximality properties of inner models and elementary embeddings. For example, we investigated under which circumstances the Jonsson property holds for a given cardinal in the core model K if it holds in the universe V.
2.Research on infinite time Turning machines. We showed in particular that, if λ is the supremum of the writable ordinals, i.e. ordinals which arise at outputs of computations on input 0, and γ is the supremum of clockable ordinals, i.e. ordinals which are lengths of halting computations on input 0, then λ=γ.
3.We proved the set of reals Cohen over a model of ZFC must either be empty or non-meager.
4.The effect of cardinal invariants of the continuum on combinatorial properties of uncountable cardinals. For example, we proved that additivity of the null ideal implies that Martin's axiom MA holds for any Cohen algebra. On the other hand, we showed it is consistent that the continuum c is large and covering of the null ideal as well as the combinatorial principle * hold.
5.Say that PSP(κ, Γ) holds if every set in the pointclass Γ of size at least κ has a perfect subset. We showed that PSP(ΝィイD21ィエD2, GィイD2ΝィエD2ィイD21ィエD2) holds if and only if σ 【greater than or equal】 ΝィイD22ィエD2 where GィイD2ΝィエD2ィイD21ィエD2 is the class of sets which are intersections of (at most) ΝィイD21ィエD2 many open sets and σ is the dominating number.
6.We proved that every maximal confinitary subgroup of Sym(ω), the permutation group of the natural numbers, has size at least the cardinality of the smallest non-meager set reals.
7.Completing a cycle of results initiated by Shelah and Spinas, we obtained that if c=ΝィイD22ィエD2 then there is a Gross space over every countably infinite field.

  • Research Products

    (30 results)

All Other

All Publications (30 results)

  • [Publications] Jorg Brendle: "How small can the rest of generics be?"Logic Colloquium '98. 109-126 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Jorg Brendle: "Martin's axiom and the dual distributivity number"Mathematical Logic Quarterly. 46 (印刷中). (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Jorg Brendle: "A new method for iterating 6-centered forcing"数理解析研究所講究録. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Jorg Brendle: "On the refinement and countable refinement numbers"Questions and Answers in General Topology. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Jorg Brendle: "Uniformity of the meager ideal and maximal cofinitary groups"Journal of Algebra. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 渕野 昌: "Weak Freese-Nation propertyについて"北見工業大学研究報告. 31. 1-9 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Sakae Fuchino: "On the weak Freese-Nation property of P(w)"Archive for Mathematical Logic. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Sakae Fuchino: "On absolutely divergent series"Fundamenta Mathematicae. 160. 255-268 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Sakae Fuchino: "On a theorem of Helly"Proceedings of the American Mathematical Society. 127. 491-497 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Yo Matsubara: "The extent of strength in the club filters"Israel Journal of Mathematics. 114. 253-263 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 松原 洋: "Non stationary idealとuniverse of sets"数学. 51. 18-33 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Philip Welch: "Minimality arguments for infinite time Turing degrees"Sets and Proofs (Logic Colloquium '97). 425-436 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Philip Welch: "Some remarks on the maximality of inner models"Logic Colloquium '98. 516-540 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Philip Welch: "The length of infinite time Turing machine computations"Bulletin of the London Mathematical Society. 32. 129-136 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Philip Welch: "Eventually infinite time Turing machine degrees"Journal of Symbolic Logic. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Jorg Brendle: "How small can the set of generics be?"Logic Colloquium '98. 109-126 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Jorg Brendle: "Martin's axiom and the dual distributivity number"Math. Logic Quart.. 46, (to appear). (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Jorg Brendle: "A new method for iterating σ-centered forcing"Suri kaiseki kenkyusho kokyuroku. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Jorg Brendle, Winfried Just and Claude Laflamme: "On the refinement and countable refinement numbers"Questions Answers Gen. Topology. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Jorg Brendle, Otmar Spinas and Yi Zhang: "Uniformity of the meager ideal and maximal cofinitary groups"J. Algebra. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Douglas Burke and Yo Matsubara: "The extent of strength in the club filters"Israel J. Math.. 114. 253-263 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sakae Fuchino: "Weak Freese-Nation property ni tsuite"Kitami kogyo daigaku kenkyu hokoku. 31. 1-9 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sakae Fuchino, Stefan Geschke and Lajos Soukup: "On the weak Freese-Nation property of Ρ(ω)"Arch. Math. Logic. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sakae Fuchino, Heike Mildenberger, Peter Vojtas and Saharon Shelah: "On absolutely divergent series"Fund. Math.. 160. 255-268 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Sakae Fuchino and Szymon Plewik: "On a theorem of Helly"Proc. Amer. Math. Soc.. 127. 491-497 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Yo Matsubara: "Non-stationary ideal to universe of sets"Sugaku. 51. 18-33 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Philip D.Welch: "Minimality arguments for infinite time Turing degrees"Sets and Proofs. 425-436 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Philip D.Welch: "Some remarks on the maximality of inner models"Logic Colloquium '98. 516-540 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Philip D.Welch: "The length of infinite time Turing machine computations"Bull. London Math. Soc.. 32. 129-136 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Philip D.Welch: "Eventually infinite time Turing machine degrees : infinite time decidable reals"J. Symbolic Logic. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2001-10-23   Modified: 2021-04-07  

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