Disagreement About New Axioms in Mathematics

Abstract

Disagreement is widespread in mathematics. In this paper, I focus on disagreement about new axioms, which in our time means chiefly new axioms for set theory. Set theorists seek new axioms because the currently accepted axioms of Zermelo-Fraenkel set theory (ZFC) do not resolve key questions about sets like the continuum hypothesis. One way to resolve this disagreement would be to find reasons for choosing one or another new axiom. Philosophers classify such reasons as ‘intrinsic” and “extrinsic”, where the former are based on intuition or the content of the concept of set, and the latter are based on the value of the consequences such an axiom would have for set theory. Our modest goal in this article is to argue that this distinction, between intrinsic and extrinsic reasons for resolving a disagreement about new axioms, is not well-defined.