The Continuum Hypothesis - Logic at Harvard

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the Ω Conjecture in the above arguments. This is the statement that theΩ Conjecture is (non-trivially) Ω-satisfiable, that is, the statement: Thereexists an ordinal α and a universe V ′ of the multiverse such thatandV ′ α |= ZFC+“There is a proper class of Woodin cardinals”V ′ α |= “The Ω Conjecture”.This Σ 2 -statement is invariant under set forcing and hence is one adherentsto the generic multiverse view of truth must deem determinate. Moreover,the key arguments above go through with this Σ 2 -statement instead of theΩ Conjecture. The person taking this second line of response would thusalso have to maintain that this statement is false. But there is substantialevidence that this statement is true. The reason is that there is no knownexample of a Σ 2 -statement that is invariant under set forcing relative to largecardinal axioms and which cannot be settled by large cardinal axioms. (Sucha statement would be a candidate for an absolutely undecidable statement.)So it is reasonable to expect that this statement is resolved by large cardinalaxioms. However, recent advances in inner model theory—in particular,those in Woodin (2011b)—provide evidence that no large cardinal axiom canrefute this statement. Putting everything together: It is very likely that thisstatement is in fact true; so this line of response is not promising.Third, one could reject either the Truth Constraint or the DefinabilityConstraint. The trouble is that if one rejects the Truth Constraint then onthis view (assuming the Ω Conjecture) Π 2 truth in set theory is reducible inthe sense of Turing reducibility to truth in H(δ 0 ) (or, assuming the StrongΩ Conjecture, H(c + )). And if one rejects the Definability Constraint thenon this view (assuming the Ω Conjecture) Π 2 truth in set theory is reduciblein the sense of definability to truth in H(δ 0 ) (or, assuming the Strong ΩConjecture, H(c + )). On either view, the reduction is in tension with theacceptance of non-pluralism regarding the background theory ZFC+“Thereis a proper class of Woodin cardinals”.Fourth, onecouldembracethecriticism, rejectthegenericmultiverse conceptionof truth, and admit that there are some statements about H(δ + 0 ) (orH(c + ), granting, in addition, the AD + Conjecture) that are true simpliciterbut not true in the sense of the generic-multiverse, and yet nevertheless continueto maintain that CH is indeterminate. The difficulty is that any suchsentence ϕ is qualitatively just like CH in that it can be forced to hold and26
forced to fail. The challenge for the advocate of this approach is to modifythe generic-multiverse conception of truth in such a way that it counts ϕ asdeterminate and yet counts CH as indeterminate.In summary: There is evidence that the only way out is the fourth wayoutandthisplaces theburden back onthepluralist—the pluralist must comeup with a modified version of the generic multiverse.Further Reading: FormoreontheconnectionbetweenΩ-logicandthegenericmultiverse and the above criticism of the generic multiverse see Woodin(2011a). For the bearing of recent results in inner model theory on thestatus of the Ω Conjecture see Woodin (2011b).5 The Local Case RevisitedLet us now turn to a second way in which one might resist the local case forthe failure of CH. This involves a parallel case for CH. In Section 5.1 we willreview the main features of the case for ¬CH in order to compare it with theparallel case for CH. In Section 5.2 we will present the parallel case for CH.In Section 5.3 we will assess the comparison.5.1 The Case for ¬CHRecall that there are two basic steps in the case presented in Section 3.3.The first step involves Ω-completeness (and this gives ¬CH) and the secondstep involves maximality (and this gives the stronger 2 ℵ 0= ℵ 2 ). For ease ofcomparison we shall repeat these features here:The first step is based on the following result:Theorem 5.1 (Woodin). Assume that there is a proper class of Woodincardinals and that the Strong Ω Conjecture holds.(1) There is an axiom A such that(i) ZFC+A is Ω-satisfiable and(ii) ZFC+A is Ω-complete for the structure H(ω 2 ).(2) Any such axiom A has the feature thatZFC+A |= Ω “H(ω 2 ) |= ¬CH”.27
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: Theorem 4.6 (Woodin). Assume ZFC an
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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