GENERALIZED ORDINAL ANALYSIS AND REFLECTION PRINCIPLES IN SET THEORY

Abstract It is widely claimed that the natural axiom systems—including the large cardinal axioms—form a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by means of ad hoc constructions. In this article we formulate notions of proof-theoretic strength based on set-theoretic reflection principles. We prove that they coincide with orderings on theories given by the generalized ordinal analysis of Pohlers. Accordingly, these notions of proof-theoretic strength engender genuinely well-ordered hierarchies. The reflection principles considered in this article are formulated relative to Gödel’s constructible universe; we conclude with generalizations to other inner models.