Chance and the Continuum Hypothesis

Abstract

This paper presents and defends an argument that the continuum hypothesis is false, based on considerations about objective chance and an old theorem due to Banach and Kuratowski. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. Since it is possible to randomly pick out a point on a continuum, for instance using a roulette wheel or by flipping a countable infinity of fair coins, it follows, given the axioms of ZFC, that there are many different cardinalities between countable infinity and the cardinality of the continuum. Quine told us that there are no islands in science, and that even our mathematical beliefs have to face the tribunal of experience. The examples usually adduced in support of his view are non-Euclidean geometry and quantum probability, mathematical theories whose development was in part inspired by physical discoveries. But in those cases, it is unclear whether our beliefs about the mathematics itself were revised, or just our beliefs about the applicability of certain mathematical theories. In this