Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength

TL;DR

This paper challenges the common belief in a linear hierarchy of large cardinal consistency strengths by providing natural examples of nonlinearity and illfoundedness, urging a shift from vague notions of naturality to precise mathematical criteria.

Contribution

It introduces natural instances of nonlinearity and illfoundedness in the large cardinal hierarchy, and critiques the philosophical notion of naturality in set theory.

Findings

Counterexamples of nonlinearity in the hierarchy

Instances of illfoundedness in consistency strength

Arguments for precise mathematical criteria over naturality

Abstract

Many set theorists point to the linearity phenomenon in the hierarchy of consistency strength, by which natural theories tend to be linearly ordered and indeed well ordered by consistency strength. Why should it be linear? In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength, as natural or as nearly natural as I can make them. I present diverse cautious enumerations of ZFC and large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength, and yet, I argue, are natural. I consider the philosophical role played by "natural" in the linearity phenomenon, arguing ultimately that we should abandon empty naturality talk and aim instead to make precise the mathematical and logical features we had found desirable.