Generalized ordinal analysis and reflection principles in set theory
Abstract
It is widely claimed that the natural axiom systemsx2013including the large cardinal axiomsx2013form 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 paper 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 paper are formulated relative to G\"odel's constructible universe; we conclude with generalizations to other inner models.
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