On the Hierarchy of Natural Theories

Abstract

It is a well-known empirical phenomenon that natural axiomatic theories are pre-well-ordered by consistency strength. Without a precise mathematical definition of "natural," it is unclear how to study this phenomenon mathematically. We will discuss the significance of this problem and survey some strategies that have recently been developed for addressing it. These strategies emphasize the role of reflection principles and ordinal analysis and draw on analogies with research in recursion theory. We will conclude with a discussion of open problems and directions for future research.

An Incompleteness Theorem via Ordinal Analysis

“…Pakhomov and the author proved Theorem 3.6 to provide an explanation of the apparent pre-well-ordering of natural theories by proof-theoretic strength; see [17] for a discussion of this phenomenon. In [8,9], Theorem 3.6 is proved using G ödel's second incompleteness theorem.…”

An incompleteness theorem via ordinal analysis

“…Pakhomov and the author proved Theorem 4.6 to provide an explanation of the apparent pre-well-ordering of natural theories by proof-theoretic strength; see [16] for a discussion of this phenomenon. In [7,8], Theorem 4.6 is proved using Gödel's second incompleteness theorem.…”