Chance and the Continuum Hypothesis

TL;DR

This paper argues that the continuum hypothesis is false by linking objective chance, probabilistic methods, and set theory, suggesting there are intermediate cardinalities between countable infinity and the continuum.

Contribution

It introduces a novel argument against the continuum hypothesis based on considerations of objective chance and the Banach-Kuratowski theorem.

Findings

Objective chance presupposes intermediate cardinalities.

Probabilistic methods imply the existence of many different cardinalities.

The continuum hypothesis is false under these considerations.

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.