The Continuum Hypothesis - Logic at Harvard

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So the trouble is with (A).This illustrates an interesting contrast between our three versions of theeffective continuum hypothesis, namely, that they can come apart. For whilelarge cardinals rule out definable counterexamples of the first two kinds, theycannot rule out definable counterexamples of the third kind. But again wemust stress that they cannot prove that there are such counterexamples.But there is animportant point: Assuming largecardinal axioms (AD L(R)suffices), although one can produce outer models in which δ ∼13 > ℵ 2 it is notcurrently known how to produce outer models in which δ ∼13 > ℵ 3 or evenΘ L(R) > ℵ 3 . Thus it is an open possibility that from ZFC+AD L(R) one canprove Θ L(R) ℵ 3 . Were this to be the case, it would follow that althoughlarge cardinals cannot rule out the definable failure of CH they can rule outthe definable failure of 2 ℵ 0= ℵ 2 . This could provide some insight into thesize of the continuum, underscoring the centrality of ℵ 2 .Further Reading: For more on the three effective versions of CH see Martin(1976); for more on the Foreman-Magidor program see Foreman & Magidor(1995) and the introduction to Woodin (1999).3 The Case for ¬CHThe above results led Woodin to the identification of a “canonical” modelin which CH fails and this formed the basis of his an argument that CH isfalse. In Section 3.1 we will describe the model and in the remainder of thesection we will present the case for the failure of CH. In Section 3.2 we willintroduce Ω-logic and the other notions needed to make the case. In Section3.3 we will present the case.3.1 P maxThe goal is to find a model in which CH is false and which is canonical in thesense that its theory cannot be altered by set forcing in the presence of largecardinals. The background motivation is this: First, we know that in thepresence of large cardinal axioms the theory of second-order arithmetic andeven the entire theory of L(R) is invariant under set forcing. The importanceof this is that it demonstrates that our main independence techniques cannotbe used to establish the independence of questions about second-order12
arithmetic (orabout L(R)) inthepresence of largecardinals. Second, experiencehas shown that the large cardinal axioms in question seem to answer allof the major known open problems about second-order arithmetic and L(R)and the set forcing invariance theorems give precise content to the claim thatthese axioms are “effectively complete”. 9It follows that if P is any homogeneous partial order in L(R) then thegeneric extension L(R) P inherits the generic absoluteness of L(R). Woodindiscovered that there isavery special partial order P max thathas thisfeature.Moreover, the model L(R) Pmax satisfies ZFC+¬CH. The key feature of thismodel is that it is “maximal” (or “saturated”) with respect to sentences thatare of a certain complexity and which can be shown to be consistent via setforcing over the model; in other words, if these sentences can hold (by setforcing over the model) then they do hold in the model. To state this moreprecisely we are going to have to introduce a few rather technical notions.There are two ways of stratifying the universe of sets. The first is interms of 〈V α | α ∈ On〉, the second is in terms of 〈H(κ) | κ ∈ Card〉, whereH(κ) is the set of all sets which have cardinality less than κ and whosemembers have cardinality less than κ, and whose members of members havecardinality less than κ, and so on. For example, H(ω) = V ω and the theoriesof the structures H(ω 1 ) and V ω+1 are mutually interpretable. This latterstructureisthestructureofsecond-orderarithmeticand, asmentionedabove,large cardinal axioms give us an “effectively complete” understanding of thisstructure. We should like to be in the same position with regard to largerand larger fragments of the universe and the question is whether we shouldproceed in terms of the first or the second stratification.The second stratification is potentially more fine-grained. Assuming CHone has that the theories of H(ω 2 ) and V ω+2 are mutually interpretable andassuming larger and larger fragments of GCH this correspondence continuesupward. But if CH is false then the structure H(ω 2 ) is less rich than thestructure V ω2 . In this event the latter structure captures full third-orderarithmetic, while the former captures only a small fragment of third-orderarithmetic but is nevertheless rich enough to express CH. Given this, inattempting to understand the universe of sets by working up through it levelby level, it is sensible to use the potentially more fine-grained stratification.9 For more on the topic of invariance under set forcing and the extent to which this hasbeen established in the presence of large cardinal axioms, see §4.4 and §4.6 of the entry“Large Cardinals and Determinacy”.13
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: more generally, all universally Bai
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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