RECENT PROGRESS ON THE CONTINUUM HYPOTHESIS (AFTER WOODIN)

Anumber of conceptually deep and technically hard results were accumulated in Set Theory since the methods of forcing and of fine structure appeared in the 1960’s. This report is devoted to Woodin’s recent results.Not only are these results technical breakthroughs, but they also renew the conceptual framework, making the theory more globally intelligible and emphasizing its unity. For the first time, there appear a global explanation for the hierarchy of large cardinals, and, chiefly, a realistic perspective to decide the Continuum Hypothesis—namely in the negative: Conjecture 1 (Woodin, 1999). Every set theory that is compatible with the existence of large cardinals and makes the properties of sets with hereditary cardinality at most ℵ1 invariant under forcing implies that the Continuum Hypothesis be false. Woodin’s results come very close to this conjecture, establishing it for a substantial part of the hierarchy of large cardinals. The remaining question is whether this substantial part actually is the whole hierarchy of large cardinals. In any case—and that is what makes it legitimous to discuss these results now, without waiting for a possible solution of the above conjecture—Woodin’s results contribute to show that the Continuum Problem and, more generally, the concept of uncountable infinity are not intrinsically vague and inaccessible to analysis, but that they can be the object of a genuine conceptual theory that goes far beyond simply exploring the formal consequences of more or less arbitrary axioms. The current text owes much to the expository papers [19, 20]. It aims at describing and explaining four recent results of Woodin, here appearing as Theorems 30, 35, 38, and, chiefly, 40. It seems out of reach to give an idea of the proofs. The published part fills a significant proportion of the 900 pages of [18], and the most recent part is only alluded to in [21]. I thank all set theorists who helped me with comments and suggestions, in particular Joa