Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

the generic multiverse of sets 109Theorem 17. Assume there is a proper class of Woodin cardinals. Then forall (T,φ) and for all complete Boolean algebras, B, T ⊢ φ if and only ifV B “T ⊢ φ.”Theorem 18 (Soundness). Assume that there is a proper class of Woodincardinals and that T ⊢ φ. Then T φ.I now come to the Conjecture, which in essence is simply the conjecture that theGödel completeness theorem holds for -logic; see Woodin (2009) for a more detaileddiscussion.Definition 19 ( Conjecture). Suppose that there exists a proper class ofWoodin cardinals. Then for all sentences φ, ∅ φ if and only if ∅⊢ φ.Assuming the Conjecture, one can analyze the generic multiverse view of truth bycomputing the logical complexity of -logic. The key issue, of course, is whether thegeneric multiverse view of truth satisfies the two multiverse laws. This is the subjectof the next section. We end this section with a curious connection between the Conjecture, HOD, and the universally Baire sets. This requires a definition.Definition 20.such thatA set A ⊆ R is OD if there exists an ordinal α and a formula φA = {x ∈ R | V α φ[x]} .Theorem 21. Suppose that there is a proper class of Woodin cardinals and thatfor every set A ⊆ R, ifAisOD, then A is universally Baire. ThenHOD “ Conjecture.”4.4.3 The Complexity of -logicLet V (as defined prior to Theorem 14) be the set of sentences φ such that∅ φ,and (assuming there is a proper class of Woodin cardinals) let V (V δ0 +1) be the set ofsentences, φ, such thatZFC “V δ0 +1 φ.”Assuming there is a proper class of Woodin cardinals, then the set of genericmultiverse truths that are 2 assertions is of the same Turing complexity as V (i.e.,each set is recursive in the other). Further (assuming there is a proper class of Woodincardinals), the set V (V δ0 +1) is precisely the set of generic multiverse truths of V δ0 +1.Thus, the requirement that the generic multiverse position satisfies the first multiverselaw, as discussed previously, reduces to the requirement that V not be recursive inthe set V (V δ0 +1).The following theorem is a corollary of the basic analysis of -logic in the contextthat there is a proper class of Woodin cardinals.