The Continuum Hypothesis - Logic at Harvard
The Continuum Hypothesis - Logic at Harvard
The Continuum Hypothesis - Logic at Harvard
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So the trouble is with (A).This illustr<strong>at</strong>es an interesting contrast between our three versions of theeffective continuum hypothesis, namely, th<strong>at</strong> 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 th<strong>at</strong> they cannot prove th<strong>at</strong> 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 th<strong>at</strong> from ZFC+AD L(R) one canprove Θ L(R) ℵ 3 . Were this to be the case, it would follow th<strong>at</strong> 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 <strong>The</strong> Case for ¬CH<strong>The</strong> above results led Woodin to the identific<strong>at</strong>ion of a “canonical” modelin which CH fails and this formed the basis of his an argument th<strong>at</strong> 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 max<strong>The</strong> goal is to find a model in which CH is false and which is canonical in thesense th<strong>at</strong> its theory cannot be altered by set forcing in the presence of largecardinals. <strong>The</strong> background motiv<strong>at</strong>ion is this: First, we know th<strong>at</strong> in thepresence of large cardinal axioms the theory of second-order arithmetic andeven the entire theory of L(R) is invariant under set forcing. <strong>The</strong> importanceof this is th<strong>at</strong> it demonstr<strong>at</strong>es th<strong>at</strong> our main independence techniques cannotbe used to establish the independence of questions about second-order12