Proof Theory as an Analysis of Impredicativity(<Special Section>New Developments in Logic: Proof-Theoretic Ordinals and Set-Theoretic Ordinals)

Abstract

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.