The Continuum Hypothesis - Logic at Harvard

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Thisconstraint isinthespirit ofthoseprinciples ofset theory—most notably,reflection principles—which aim to capture the pretheoretic idea that theuniverse of sets is so rich that it cannot “be described from below”; moreprecisely, itassertsthatanytenableconception oftruthmust respect theideathat the universe of sets is so rich that truth (or even just Π 2 -truth) cannotbe described in some specifiable fragment. (Notice that by Tarski’s theoremon the undefinability of truth, the truth constraint is trivially satisfied bythe standard conception of truth in set theory which takes the multiverse tocontain a single element, namely, V.)There is also a related constraint concerning the definability of truth. Foraspecifiable cardinal κ, set Y ⊆ ω isdefinable in H(κ + ) across the multiverseif Y is definable in the structure H(κ + ) of each universe of the multiverse(possibly by formulas which depend on the parent universe).Definition 4.3 (DefinabilityConstraint). Anytenablemultiverseconceptionof truth in set theory must be such that the Π 2 -truths (according to thatconception) in the universe of sets are definable in H(κ) across the multiverseuniverse, for any specifiable cardinal κ.Notice again that by Tarski’s theorem on the undefinability of truth, the definabilityconstraint is trivially satisfied by the degenerate multiverse conceptionthat takes the multiverse to contain the single element V. (Notice alsothatifonemodifiesthedefinabilityconstraintbyaddingtherequirement thatthe definition be uniform across the multiverse, then the constraint wouldautomatically be met.)The bearing of the Ω Conjecture on the tenability of the genericmultiverseconception of truth is contained in the following two theorems:Theorem 4.4 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Suppose that the Ω Conjecture holds. Then V Ω is recursivein V Ω (H(δ + 0 )), where δ 0 is the least Woodin cardinal.Theorem 4.5 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Suppose that the Ω Conjecture holds. Then V Ω is definablein H(δ + 0 ), where δ 0 is the least Woodin cardinal.In other words, if there is a proper class of Woodin cardinals and if the ΩConjecture holds then the generic multiverse conception of truth violatesboth the Truth Constraint (at δ 0 ) and the Definability Constraint (at δ 0 ).ThereareactuallysharperversionsoftheaboveresultsthatinvolveH(c + )in place of H(δ + 0 ). 24
Theorem 4.6 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Suppose that the Ω Conjecture holds. Then V Ω is recursivein V Ω (H(c + )).Theorem 4.7 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Suppose that the Ω Conjecture holds and that the AD +Conjecture holds. Then V Ω is definable in H(c + ).In other words, if there is a proper class of Woodin cardinals and if the ΩConjecture holds then the generic-multiverse conception of truth violates theTruth Constraint at the level of third-order arithmetic, and if, in addition,the AD + Conjecture holds, then the generic-multiverse conception of truthviolates the Definability Constraint at the level of third-order arithmetic.4.4 Is There a Way Out?There appear to be four ways that the advocate of the generic multiversemight resist the above criticism.First, one could maintain that the Ω Conjecture is just as problematic asCH and hence like CH it is to be regarded as indeterminate according to thegeneric-multiverse conception of truth. The difficulty with this approach isthe following:Theorem 4.8 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Then, for any complete Boolean algebra B,V |= Ω-conjecture iff V B |= Ω-conjecture.Thus, in contrast to CH, the Ω Conjecture cannot be shown to be independentof ZFC+“There is a proper class of Woodin cardinals” via set forcing.In terms of the generic multiverse conception of truth, we can put the pointthis way: While the generic-multiverse conception of truth deems CH to beindeterminate, it does not deem the Ω Conjecture to be indeterminate. Sothe above response is not available to the advocate of the generic-multiverseconception of truth. The advocate of that conception already deems the ΩConjecture to be determinate.Second, one could grant that the Ω Conjecture is determinate but maintainthat it is false. There are ways in which one might do this but thatdoes not undercut the above argument. The reason is the following: To beginwith there is a closely related Σ 2 -statement that one can substitute for25
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: Theorem 4.1 (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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