Generalized ordinal analysis and reflection principles in set theory
Logic
2023-12-21 v1
Abstract
It is widely claimed that the natural axiom systemsincluding the large cardinal axiomsform a well-ordered hierarchy. Yet, as is well-known, it is possible to exhibit non-linearity and ill-foundedness by means of \emph{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.
Cite
@article{arxiv.2312.12859,
title = {Generalized ordinal analysis and reflection principles in set theory},
author = {Hanul Jeon and James Walsh},
journal= {arXiv preprint arXiv:2312.12859},
year = {2023}
}
Comments
36 pages