Write an explanation of the continuum hypothesis, the question of whether there is a set whose size is strictly between that of the integers and the real numbers. Cover Cantor's work on different sizes of infinity, Gödel's proof that the continuum hypothesis is consistent with Zermelo-Fraenkel set theory, and Cohen's proof that the negation of the continuum hypothesis is also consistent. Discuss the philosophical implications of this independence result: what does it mean for a mathematical statement to be neither provable nor disprovable within our standard axioms?