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

the generic multiverse of sets 105First Multiverse LawThe set of 2 assertions that are multiverse truths is not recursive in the set of multiversetruths of V δ0+1.The motivation for this multiverse law is that if the set of 2 multiverse truths isrecursive in the set of multiverse truths of V δ0 +1, then as far as evaluating 2 assertionsis concerned, the multiverse is equivalent to the reduced multiverse of just the fragmentsV δ0 +1 of the universes of the multiverse. This amounts to a rejection of the transfinitebeyond V δ0 +1 and constitutes, in effect, the unacceptable brand of formalism alludedto earlier. This claim would be reinforced should the multiverse position also violate asecond multiverse law, which I now formulate.A set Y ⊂ V ω is definable in V δ0 +1 across the multiverse if the set Y is definable inthe structure V δ0 +1 of each universe of the multiverse (possibly by formulas that dependon the parent universe). The second multiverse law is a variation of the first multiverselaw.Second Multiverse LawThe set of 2 assertions that are multiverse truths is not definable in V δ0+1 across themultiverse.Again, by Tarski’s theorem on the undefinability of truth, this multiverse law isobviously a reasonable one if one regards the only possibility for the multiverse to bethe universe of sets such that the set of multiverse truths of V δ0 +1 is simply the set of allsentences that are true in V δ0 +1 and the set of 2 assertions that are multiverse truths issimply the set of 2 assertions that are true in V. Likewise, the second multiverse lawwould have to hold if one modified the law to simply require that the set of 2 assertionsthat are multiverse truths is not uniformly definable in V δ0 +1 across the multiverse (i.e.,by a single formula).Assuming that both Conjecture and the existence of a proper class of Woodincardinals hold in each (or one) universe of the generic multiverse generated by V,both the first multiverse law and the second multiverse law are violated by the genericmultiverse position. This is the basis for the argument I am giving against the genericmultiverse position in this chapter. In fact, the technical details of how the genericmultiverse position violates these multiverse laws provide an even more compellingargument against the generic multiverse position because the analysis shows that,in addition, the generic multiverse position is truly a form of formalism because ofthe connections to logic. The argument also shows that the violation of the firstmultiverse law is explicit; that is, assuming the Conjecture, there is an explicitrecursive reduction of the set of 2 assertions that are generic multiverse truths to theset of generic-multiverse truths of V δ0 +1.There is a special case that I can present without any additional definitions and thatis not contingent on any conjectures.Theorem 12.Suppose that M is a countable transitive setM ZFC + “There is a proper class of Woodin cardinals”and that M ∩ Ord is as small as possible. Then V M violates both multiverselaws.