Is the dream solution to the continuum hypothesis attainable?

TL;DR

This paper argues that achieving a dream solution to the continuum hypothesis through a new fundamental axiom is impossible, emphasizing the limitations of settling CH via such axiomatic discoveries.

Contribution

The paper provides a philosophical and logical argument against the attainability of a dream solution to the continuum hypothesis through new axioms.

Findings

Dream solution to CH is unattainable.

Limits of axiomatic approaches to CH.

Supports multiverse perspective in set theory.

Abstract

The dream solution of the continuum hypothesis (CH) would be a solution by which we settle the continuum hypothesis on the basis of a newly discovered fundamental principle of set theory, a missing axiom, widely regarded as true. Such a dream solution would indeed be a solution, since we would all accept the new axiom along with its consequences. In this article, however, I argue that such a dream solution to CH is unattainable. The article is adapted from and expands upon material in my article, "The set-theoretic multiverse", to appear in the Review of Symbolic Logic (see arXiv:1108.4223).