The Continuum Hypothesis - Logic at Harvard

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together showed that CH cannot be resolved on the basis of the axioms thatmathematicians were employing; in modern terms, CH is independent ofZFC.This independence result was quickly followed by many others. The independencetechniques were so powerful that set theorists soon found themselvespreoccupied with the meta-theoretic enterprise of proving that certainfundamental statements could not be proved or refuted within ZFC. Thequestion then arose as to whether there were ways to settle the independentstatements. The community of mathematicians and philosophers of mathematicswas largely divided on this question. The pluralists (like Cohen)maintained that the independence results effectively settled the question byshowing that it had no answer. On this view, one could adopt a system inwhich, say CH was an axiom and one could adopt a system in which ¬CHwas an axiom and that was the end of the matter—there was no questionas to which of two incompatible extensions was the “correct” one. The nonpluralists(like Gödel) held that the independence results merely indicatedthe paucity of our means for circumscribing mathematical truth. On thisview, what was needed were new axioms, axioms that are both justified andsufficient for the task. Gödel actually went further in proposing candidatesfor new axioms—large cardinal axioms—and he conjectured that they wouldsettle CH.Gödel’s program for large cardinal axioms proved to be remarkably successful.Over the course of the next 30 years it was shown that large cardinalaxioms settle many of the questions that were shown to be independent duringthe era of independence. However, CH was left untouched. The situationturned out to be rather ironic since in the end it was shown (in a sensethat can be made precise) that although the standard large cardinal axiomseffectively settle all question of complexity strictly below that of CH, theycannot (by results of Levy and Solovay and others) settle CH itself. Thus,in choosing CH as a test case for his program, Gödel put his finger preciselyon the point where it fails. It is for this reason that CH continues to play acentral role in the search for new axioms.In this entry we shall give an overview of the major approaches to settlingCH and we shall discuss some of the major foundational frameworks whichmaintain that CH does not have an answer. The subject is a large one andwe have had to sacrifice full comprehensiveness in two dimensions. First, wehave not been able to discuss the major philosophical issues that are lying inthe background. For this the reader is directed to the entry “Large Cardinals2
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
Page 1: The Continuum HypothesisPeter Koell
Page 5 and 6: infinite cardinals: Assuming GCH, i
Page 7 and 8: Theorem 1.6 (Shelah, 1978). If ℵ
Page 9 and 10: this version of CH holds for all se
Page 11 and 12: more generally, all universally Bai
Page 13 and 14: arithmetic (orabout L(R)) inthepres
Page 15 and 16: 3.2 Ω-LogicWe will now recast the
Page 17 and 18: We will need a strong form of this
Page 19 and 20: and T A ≠ T B .How then shall one
Page 21 and 22: Now to arrive at this conclusion on
Page 23 and 24: Theorem 4.1 (Woodin). Assume ZFC an
Page 25 and 26: Theorem 4.6 (Woodin). Assume ZFC an
Page 27 and 28: forced to fail. The challenge for t
Page 29 and 30: So, of the various theories T A inv
Page 31 and 32: Theorem 5.8 (Koellner and Woodin).
Page 33 and 34: the fact that under (∗) the theor
Page 35 and 36: in his work.Now, we have a great de
Page 37 and 38: Koellner, P. (2010). Strong logics

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