A robust proof-theoretic well-ordering

Abstract

It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of ${\Pi }_1^0$ theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions -- such as ${\Pi }_1^1$-reflection strength and inclusion of ${\Pi }_1^1$ theorems -- in the presence of an oracle for ${\Sigma }_1^1$ truths. In this context these notions coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.

arXiv commentThis article has been merged with 2112.04980 to form 2209.09765