Higher analogies of reflection principles and cardinal arithmetic

2022 Fiscal Year Final Research Report

• Project/Area Number: 18K03397
• 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: 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): 2018-04-01 – 2023-03-31
• Keywords: 反映原理 / 基数算術 / 巨大基数

Outline of Final Research Achievements

Set theorists have been interested in reflection principles at aleph_2 since they have many interesting consequences on cardinal arithmetic and infinite combinatorics. So far, many reflection principles at aleph_2 have been formulated, and the relationships among them and their consequences are studied extensively. These reflection principles can be naturally generalized to those at larger cardinals, which we call higher analogues. In this research, I studied these higher analogues of reflection principles at aleph_2 such as the stationary reflection principle and the Rado conjecture. Especially, I focused on the relationships among them and their consequences on cardinal arithmetic. As a result, it turned out that the relationships among higher analogues are similar to those among reflection principles at aleph_2. Also, I could prove that many of higher analogues do not have similar consequences on cardinal arithmetic to those of reflection principles at aleph_2.

Free Research Field

公理的集合論

Academic Significance and Societal Importance of the Research Achievements

定常性反映原理やフォドア型反映原理などのアレフ２レベルの反映原理は，マルティンの極大強制法公理という強い強制法公理から帰結され，この関係からも興味が持たれている．近年，強制法の手法の開発により，強制法公理の高レベルへの一般化が考察されている．これらの高レベルの強制法公理を考察する上で，本研究の研究成果は重要な知見となる．また，反映原理や強制法公理に限らず，アレフ１やアレフ２のレベルの集合や数学的構造については理解が進んでいるが，より高レベルの集合や数学的構造についての理解は集合論の大きな課題になっている．本研究の成果は，この集合論の課題の克服に貢献するものである．