The Continuum Hypothesis, Part I, Volume 48, Number 6

This problem belongs to an ever-increasing list of problems known to be unsolvable from the (usual) axioms of set theory. However, some of these problems have now been solved. But what does this actually mean? Could the Continuum Hypothesis be similarly solved? These questions are the subject of this article, and the discussion will involve ingredients from many of the current areas of set theoretical investigation. Most notably, both Large Cardinal Axioms and Determinacy Axioms play central roles. For the problem of the Continuum Hypothesis, I shall focus on one specific approach which has developed over the last few years. This should not be misinterpreted as a claim that this is the only approach or even that it is the best approach. However, it does illustrate how the various, quite distinct, lines of investigation in modern set theory can collectively yield new, potentially fundamental, insights into questions as basic as that of the Continuum Hypothesis. The generally accepted axioms for set theory— but I would call these the twentieth-century choice—are the Zermelo-Fraenkel Axioms together with the Axiom of Choice, ZFC. For a discussion of these axioms and related issues, see [Kanamori, 1994]. The independence of a proposition φ from the axioms of set theory is the arithmetic statement: ZFC does not prove φ, and ZFC does not prove ¬φ . Of course, if ZFC is inconsistent, then ZFC proves anything, so independence can be established only by assuming at the very least that ZFC is consistent. Sometimes, as we shall see, even stronger assumptions are necessary. The first result concerning the Continuum Hypothesis, CH, was obtained by Gödel.