The Continuum Hypothesis - Logic at Harvard

and Determinacy”, which contains a general discussion of the independenceresults, the nature of axioms, the nature of justification, and the successes oflarge cardinal axioms in the realm “below CH”. Second, we have not beenabletodiscuss everyapproachtoCHthatisintheliterature. Insteadwehaverestricted ourselves to those approaches that appear most promising from aphilosophical point of view and where the mathematics has been developedto a sufficiently advanced state. In the approaches we shall discuss—forcingaxioms, inner modeltheory, quasi-largecardinals—themathematicshasbeenpressed to a very advanced stage over the course of 40 years. And this hasmade our task somewhat difficult. We have tried to keep the discussion asaccessible as possible and we have placed the more technical items in theendnotes. But the reader should bear in mind that we are presenting a bird’seye view and that for a higher resolution at any point the reader should dipinto the suggested readings that appear at the end of each section. 2There are really two kinds of approaches to new axioms—the local approachand the global approach. On the local approach one seeks axiomsthat answer questions concerning a specifiable fragment of the universe, suchas V ω+1 or V ω+2 , where CH lies. On the global approach one seeks axiomsthat attempt to illuminate the entire structure of the universe of sets. Theglobal approach is clearly much more challenging. In this entry we shall startwith the local approach and toward the end we shall briefly touch upon theglobal approach.Here is an overview of the entry: Section 1 surveys the independenceresults in cardinal arithmetic, covering both the case of regular cardinals(where CH lies) and singular cardinals. Section 2 considers approaches toCH where one successively verifies a hierarchy of approximations to CH, eachof which is an “effective” version of CH. This approach led to the remarkablediscovery of Woodin that it is possible (in the presence of large cardinals)to have an effective failure of CH, thereby showing, that the effective failureof CH is as intractable (with respect to large cardinal axioms) as CH itself.Section 3 continues with the developments that stemmed from this discovery.The centerpiece of the discussion is the discovery of a “canonical” model inwhich CH fails. This formed the basis of a network of results that wascollectively presented by Woodin as a case for the failure of CH. To present2 We have of necessity presupposed much in the way of set theory. The reader seekingadditional detail—for example, the definitions of regular and singular cardinals and otherfundamental notions—is directed to one of the many excellent texts in set theory, forexample Jech (2003).3