2021 Fiscal Year Final Research Report
Phase transitions of provability and supercritical propositions of reverse mathematics
| Project/Area Number |
15H03634
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
横山 啓太 東北大学, 先端科学技術研究科, 講師 (10534430)
木原 貴行 名古屋大学, 情報学研究科, 准教授 (80722701)
根元 多佳子 北陸先端科学技術大学院大学, 先端科学技術研究科, 助教 (20546155)
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | 逆数学 / 証明論 / 計算可能性理論 / 超準モデル / ゲームの決定性 / ランダム決定木 |
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| Outline of Final Research Achievements |
The purpose of this study is to obtain multidimensional perspectives on the dynamism of "proof" by elucidating new incompleteness phenomena in the fundations of mathematics, e.g. the phase transition between provability and unprovability induced by parameters inserted in theorems, and supercritical propositions such as the determinacy of infinite games which cannot be proved by ordinary axioms.
During the first half of the plan, we proceeded the research in five sub-themes. In the latter half, we rearranged into the following three sub-themes and obtained remarkable results. (1) Reverse mathematics and non-standard methods. (2) Open game and Weihrauch degrees. (3) Query complexity of random decision trees.
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| Free Research Field |
数学基礎論
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| Academic Significance and Societal Importance of the Research Achievements |
毎年５０名程度の国際研究集会CTFMおよび種々の勉強会を開催した．第一線の研究者を海外から招いて本研究の進捗について随時アドバイスを得るとともに、若手研究者が国際舞台に立つ後押しをした．この間，数学基礎論を専門とする３人の博士と１０人強の修士を指導した．そして，本研究の成果を踏まえて，教科書『数学基礎論序説』(裳華房 2019)と『計算理論と数理論理学』(共立出版 2022)を上梓し，また関連の論文を集めた『Computability Theory and Foundations of Mathematics』(World Scientific 2022)を編纂した．
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