Reverse mathematical analysis of intuitionistic mathematics

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14354
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Tokyo University of Science (2022-2023); Meiji University (2020-2021)
• Principal Investigator: Fujiwara Makoto 東京理科大学, 理学部第一部応用数学科, 助教 (20779095)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 連続性 / 一様連続性定理 / 直観主義算術 / 論理公理 / 冠頭標準形定理 / 保存拡大性定理

Outline of Final Research Achievements

We investigated the notion of continuity in Brouwer's intuitionistic mathematics for functions from the Baire space to the set of natural numbers from the perspective constructive reverse mathematics, and characterized it by the notion of pointwise continuity with having a modulus which possesses a bar recursor.
We also investigated the uniform continuity theorem which is from the intuitionistic mathematics, and showed that the decidable fan theorem for the complete binary tree is equivalent to the statement that every pointwise continuous real-valued function on the unit interval with a continuous modulus is uniformly continuous over intuitionistic finite-type arithmetic containing a weak choice principle only.
In addition, as related to the reverse mathematical analysis of intuitionistic mathematics, we investigated the relation between the hierarchy of logical axioms over intuitionistic arithmetic, semi-classical prenex normal form theorems, and semi-classical conservation theorems.

Free Research Field

数学基礎論

Academic Significance and Societal Importance of the Research Achievements

本研究は20世紀初等にブラウアーとその弟子たちによって直観主義数学における主要概念である関数の連続性概念や直観主義数学に端を発し現代数学に通じる一様連続性定理を現代的立場から再考察したものである．
本研究の学術的意義は，直観主義数学における強い連続性の概念や通常の数学では測れない連続性概念の間の差異を通常の数学の部分公理系である現代的構成的数学の立場から基礎付けた点にある．
また，直観主義算術上の論理公理の階層構造と冠頭標準形定理及び保存拡大性定理の関係性の研究の成果は，通常の数学と現代的構成的数学の中間に位置する数学的理論を調べるための足掛かりとなるものである．