非可述性の分析としての証明論(<特集>あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

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概要

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• Proof theory was initiated by Hilbert for the foundation of mathematics, especially for carrying out "Hilbert's program". After Godel's incompleteness theorems, proof theory (ordinal analysis) has been developed as pure mathematics. It has been widely held that it is quite difficult to explain what has been gained in proof theory (ordinal analysis). In this paper we review the development of proof theory and its recent results for impredicative subsystems of analysis from the viewpoint of "analysis of impredicativity". We point out the main difficulties to explain conceptual significance of proof theory. Our conclusion is that such difficulties should be addressed by developing (not only proof theory as pure mathematics) philosophy of mathematics in future.

• 2012-03-25

著者

• 秋吉 亮太

  慶應義塾大学

関連論文

• An Ordinal-Free Proof of the Cut-elimination Theorem for a Subsystem of $\Pi^1_1$-Analysis with $\omega$-rule (Proof theoretical study of the structure of logic and computation)
• 非可述性の分析としての証明論(あたらしい数理論理学の揺籃:証明論的な順序数と集合論的な順序数)

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