Provability algebras and proof-theoretic ordinals, I

Abstract

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic (syntax-sensitive) information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to €0 can be recovered from the corresponding algebra canonical way. This method also establishes links between prooftheoretic ordinal analysis and the work which has been done in the last two decades on provability, logic and reflection principles. Because of its abstract algebraic nature, we hope it will also be of interest for nonprooftheorists.