The Consistency Hierarchy Thesis

Abstract

Set theorists often claim that natural theories are well-ordered by their consistency strength. We call this claim the Consistency Hierarchy Thesis. The goal of this paper is to unpack the philosophical and mathematical significance of this thesis; and to develop an understanding of how it is defended and, more particularly, how one might refute it. We shall see that the thesis involves a curious admixture of mathematics and philosophy that makes it difficult to pin down. We investigate some intriguing attempts to refute the thesis that are hampered by the problem of understanding what makes a theory natural. We then develop a thought experiment exploring the idea of what the ideal scenario for refutation would look like. And we show that a counterexample is impossible if we insist that the counterexample uses respectable (i.e., transitive) models. Finally, we reflect on how these hurdles affect our understanding of the significance of the thesis by drawing a parallel with a more famous claim: the Church-Turing thesis.