Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength
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.
Keywords
Cite
@article{arxiv.2208.07445,
title = {Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength},
author = {Joel David Hamkins},
journal= {arXiv preprint arXiv:2208.07445},
year = {2022}
}
Comments
38 pages. Commentary can be made about this article on the author's blog at http://jdh.hamkins.org/nonlinearity-in-the-hierarchy-of-consistency-strength. v2 includes several relevant references to work of James Walsh