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

106 the realm of the infinite4.4.1 -logicThe generic multiverse conception of truth declares the continuum hypothesis to beneither true nor false and declares, granting large cardinals, assertion,L(R) AD,to be true (see Theorem 7). I note that for essentially all current large cardinal axioms,the existence of a proper class of large cardinals holds in V if and only if it holds in V Bfor all complete Boolean algebras, B. In other words, in the generic multiverse positionthe existence of a proper class of, say, Woodin cardinals is either true or false becauseit either holds in every universe of the generic multiverse or holds in no universe of thegeneric multiverse (Hankins and Woodin 2000).I am going to analyze the generic multiverse position from the perspective of-logic, which I first briefly review. I will use the standard modern notation for Cohen’smethod of forcing; potential extensions of the universe, V, are given by completeBoolean algebras B, V B denotes the corresponding Boolean valued extension, and foreach ordinal α, V B α denotes V α as defined in that extension.Definition 13. Suppose that T is a countable theory in the language of set theoryand φ is a sentence. ThenT φif for all complete Boolean algebras, B, for all ordinals, α, ifVαB T , thenVα B φ.If there is a proper class of Woodin cardinals, then the relation T φ is genericallyabsolute. This fact, which arguably was a completely unanticipated consequence oflarge cardinals, makes -logic interesting from a metamathematical point of view. Forexample, the setV = {φ |∅ φ}is generically absolute in the sense that for a given sentence, φ, the question whetheror not φ is logically -valid i.e., whether or not φ ∈ V is absolute between V and allof its generic extensions. In particular, the method of forcing cannot be used to showthe formal independence of assertions of the form ∅ φ.Theorem 14. Suppose that T is a countable theory in the language of set theory,φ is a sentence, and there exists a proper class of Woodin cardinals. Then for allcomplete Boolean algebras, B, V B “T φ” if and only if T φ.There are a variety of technical theorems that show that one cannot hope to provethe generic invariance of -logic from any large cardinal hypothesis weaker thanthe existence of a proper class of Woodin cardinals – for example, if V = L, thendefinition of V is not absolute between V and V B ,forany nonatomic complete Booleanalgebra, B.It follows easily from the definition of -logic that for any 2 -sentence, φ,∅ φ