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of measurable Woodin cardinals then there is a forcing extension satisfyingall Σ 2 2 sentences ϕ such that ZFC + CH + ϕ is Ω-satisfiable. (See Larson,Ketchersid & Zapletal (2008).) It follows that if the question of existenceis answered positively with an A that is Σ 2 2 then T A must be this maximumΣ 2 2 theory and, consequently, all T A agree when A is Σ 2 2 . So, assuming thatthere is a T A where A is Σ 2 2, then, although not all T A agree (when A isarbitrary) there is one that stands out, namely, the one that is maximum forΣ 2 2 sentences.Thus, if the above conjecture holds, then the case of CH parallels thatof ¬CH, only now Σ 2 2 takes the place of the theory of H(ω 2).5.3 AssessmentAssuming that the conjecture holds the case of CH parallels that of ¬CH,only now Σ 2 2 takes the place of the theory of H(ω 2): Under the backgroundassumptions we have:(1) (a) there are A such that ZFC+A is Ω-complete for H(ω 2 )(b) for every such A the associated T A contains ¬CH, and(c) there is a T A which is maximal, namely, T (∗) and this theory contains2 ℵ 0= ℵ 2 .(2) (a) there are Σ 2 2-axioms A such that ZFC+A is Ω-complete for Σ 2 2(b) for every such A the associated T A contains CH, and(c) there is a T A which is maximal.The two situations are parallel with regard to maximality but in termsof the level of Ω-completeness the first is stronger. For in the first case weare not just getting Ω-completeness with regard to the Π 2 theory of H(ω 2 )(with the additional predicates), rather we are getting Ω-completeness withregard to all of H(ω 2 ). This is arguably an argument in favour of the casefor ¬CH, even granting the conjecture.But there is a stronger point. There is evidence coming from inner modeltheory (which we shall discuss in the next section) to the effect that theconjecture is in fact false. Should this turn out to be the case it would breakthe parallel, strengthening the case for ¬CH.However, one might counter this as follows: The higher degree of Ω-completeness in the case for ¬CH is really illusory since it is an artifact of32
the fact that under (∗) the theory of H(ω 2 ) is in fact mutually interpretablewith that of H(ω 1 ) (by a deep result of Woodin). Moreover, this latter factis in conflict with the spirit of the Transcendence Principles discussed inSection 4.3. Those principles were invoked in an argument to the effect thatCH does not have an answer. Thus, when all the dust settles the real importof Woodin’s work on CH (so the argument goes) is not that CH is false butrather that CH very likely has an answer.It seems fair to say that at this stage the status of the local approachesto resolving CH is somewhat unsettled. For this reason, in the remainder ofthis entry we shall focus on global approaches to settling CH. We shall verybriefly discuss two such approaches—the approach via inner model theoryand the approach via quasi-large cardinal axioms.6 The Ultimate Inner ModelInner model theory aims to produce “L-like” models that contain large cardinalaxioms. For each large cardinal axiom Φ that has been reached by innermodel theory, one has an axiom of the form V=L Φ . This axiom has thevirtue that (just as in the simplest case of V=L) it provides an “effectivelycomplete” solution regarding questions about L Φ (which, by assumption, isV). Unfortunately, it turns out that the axiom V=L Φ is incompatible withstronger large cardinal axioms Φ ′ . For this reason, axioms of this form havenever been considered as plausible candidates for new axioms.But recent developments in inner model theory (due to Woodin) showthat everything changes at the level of a supercompact cardinal. These developmentsshow that if there is an inner model N which “inherits” a supercompactcardinal from V (in the manner in which one would expect, giventhe trajectory of inner model theory), then there are two remarkable consequences:First, N is close to V (in, for example, the sense that for sufficientlylarge singular cardinals λ, N correctly computes λ + ). Second, N inherits allknown large cardinals that exist in V. Thus, in contrast to the inner modelsthat have been developed thus far, an inner model at the level of a supercompactwould provide one with an axiom that could not be refuted by strongerlarge cardinal assumptions.The issue, of course, is whether one can have an “L-like” model (onethat yields an “effectively complete” axiom) at this level. There is reasonto believe that one can. There is now a candidate model L Ω that yields33
Page 1 and 2: The Continuum HypothesisPeter Koell
Page 3 and 4: and Determinacy”, which contains
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: Theorem 5.8 (Koellner and Woodin).
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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