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

108 the realm of the infiniteBut on close inspection, one realizes that this is not really a justification at all. Thesentence above is meaningful in the generic multiverse view of truth, but there is noexplanation of why it is true. This is exactly as is the case for the sentence that assertsthat the formal theoryZF + “There is a weak Reinhardt cardinal.”,is consistent.To evaluate more fully the generic multiverse position, one must understand thelogical relation T φ. In particular, a natural question arises: is there a correspondingproof relation?4.4.2 The ConjectureI define the proof relation, T ⊢ φ. This requires a preliminary notion that a set ofreals be universally Baire (Feng, Magidor, and Woodin 1992). In fact, I shall defineT ⊢ φ, assuming the existence of a proper class of Woodin cardinals and exploitingthe fact that there are a number of (equivalent) definitions. Without the assumptionthat there is a proper class of Woodin cardinals, the definition is a bit more technical(Woodin 2004). Recall that if S is a compact Hausdorff space, then a set X ⊆ S has theproperty of Baire in the space S if there exists an open set O ⊆ S such that symmetricdifference,X△O,is meager in S (contained in a countable union of closed sets with empty interior).Definition 15. A set A ⊂ R is universally Baire if for all compact Hausdorffspaces, S, and for all continuous functions,F : S → R,the preimage of A by F has the property of Baire in the space S.Suppose that A ⊆ R is universally Baire. Suppose that M is a countable transitivemodel of ZFC. Then M is strongly A-closed if for all countable transitive sets N suchthat N is a generic extension of M,A ∩ N ∈ N.Definition 16. Suppose there is a proper class of Woodin cardinals. Supposethat T is a countable theory in the language of set theory and φ is a sentence.Then T ⊢ φ if there exists a set A ⊂ R such that(1) A is universally Baire,(2) for all countable transitive models, M,ifM is strongly A-closed and T ∈ M, thenM “T φ.”Assuming there is a proper class of Woodin cardinals, the relation, T ⊢ φ,isgenerically absolute. Moreover, Soundness holds as well.