proof-theoretic investigations of operations on sets

2017 Fiscal Year Final Research Report

• Project/Area Number: 25400193
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Chiba University
• Principal Investigator: Arai Toshiyasu 千葉大学, 大学院理学研究院, 教授 (40193049)
• Project Period (FY): 2013-04-01 – 2018-03-31
• Keywords: 証明論

Outline of Final Research Achievements

We characterized the existence of weakly compact cardinals in terms of iterations of weakly Mahlo operations. We described ZF-provably existing countable ordinals by means of iterations of Mostowski collapsings. We introduced a class of set functions, which are computable in polynomial time, and introduced a theory in which $\Sigma_{1}$-definable functions are exactly polynomial time computable set functions. We showed that intuitionistic fixed point theories over a weak set theory is a conservative extension of the weak set theory. We showed that the wellordering principle of the derivative of a normal function on ordinals is equivalent to the existence of arbitrarily large countable coded $\omega$-models of the normal function. We gave a finitary procedure by which cut inferences are eliminated from derivations for set theory of $\omega_{1}$. We determined a line between predicative and impredicative fragments of the theory of positive elementary inductive definitions.

Free Research Field

数学基礎論