The Continuum Hypothesis - Logic at Harvard

forced to fail. The challenge for the advocate of this approach is to modifythe generic-multiverse conception of truth in such a way that it counts ϕ asdeterminate and yet counts CH as indeterminate.In summary: There is evidence that the only way out is the fourth wayoutandthisplaces theburden back onthepluralist—the pluralist must comeup with a modified version of the generic multiverse.Further Reading: FormoreontheconnectionbetweenΩ-logicandthegenericmultiverse and the above criticism of the generic multiverse see Woodin(2011a). For the bearing of recent results in inner model theory on thestatus of the Ω Conjecture see Woodin (2011b).5 The Local Case RevisitedLet us now turn to a second way in which one might resist the local case forthe failure of CH. This involves a parallel case for CH. In Section 5.1 we willreview the main features of the case for ¬CH in order to compare it with theparallel case for CH. In Section 5.2 we will present the parallel case for CH.In Section 5.3 we will assess the comparison.5.1 The Case for ¬CHRecall that there are two basic steps in the case presented in Section 3.3.The first step involves Ω-completeness (and this gives ¬CH) and the secondstep involves maximality (and this gives the stronger 2 ℵ 0= ℵ 2 ). For ease ofcomparison we shall repeat these features here:The first step is based on the following result:Theorem 5.1 (Woodin). Assume that there is a proper class of Woodincardinals and that the Strong Ω Conjecture holds.(1) There is an axiom A such that(i) ZFC+A is Ω-satisfiable and(ii) ZFC+A is Ω-complete for the structure H(ω 2 ).(2) Any such axiom A has the feature thatZFC+A |= Ω “H(ω 2 ) |= ¬CH”.27