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

the generic multiverse of sets 103Let V M be the smallest set of countable transitive sets such that M ∈ V M and such thatfor all pairs, (M 1 ,M 2 ), of countable transitive sets such thatM 1 ZFC,and such that M 2 is a generic extension of M 1 , if either M 1 ∈ V M or M 2 ∈ V M , thenboth M 1 and M 2 are in V M . It is easily verified that for each N ∈ V M ,V N = V M ,where V N is defined using N in place of M. V M is the generic multiverse generated inV from M.The generic multiverse conception of truth is the position that a sentence is true ifand only if it holds in each universe of the generic multiverse generated by V. Thiscan be formalized within V in the sense that for each sentence φ there is a sentenceφ ∗ , recursively depending on φ, such that φ is true in each universe of the genericmultiverse generated by V if and only if φ ∗ is true in V. The sentence φ ∗ is explicitgiven φ and does not depend on V. For example, given any countable transitive set, M,such that M ZFC,M φ ∗if and only if N φ for all N ∈ V M (the proof is given in Woodin [in press]). This isan important point in favor of the generic-multiverse position because it shows that, asfar as assessing truth is concerned, the generic-multiverse position is not that sensitiveto the meta-universe in which the generic multiverse is being defined.Is the generic-multiverse position a reasonable one? The refinements of Cohen’smethod of forcing in the decades since his initial discovery of the method and theresulting plethora of problems shown to be unsolvable have, in a practical sense, almostcompelled one to adopt the generic-multiverse position. This has been reinforced bysome rather unexpected consequences of large cardinal axioms, which I discuss laterin this section.The purpose of this section is not to argue against any possible multiverse position,but to examine more carefully the generic-multiverse position within the context ofmodern set theory. In brief, I argue that modulo the Conjecture (which I define inthe next section), the generic-multiverse position outlined earlier is not plausible. Theessence of the argument against the generic-multiverse position is that assuming the Conjecture is true (and that there is a proper class of Woodin cardinals), then thisposition is simply a brand of formalism that denies the transfinite by reducing truthabout the universe of sets to truth about a simple fragment such as the integers or, inthis case, the sets of real numbers. The Conjecture is invariant between V and anygeneric extension of V, and so the generic-multiverse position must either declare the Conjecture to be true or false.It is a fairly common (informal) claim that the quest for truth about the universeof sets is analogous to the quest for truth about the physical universe. However, Iam claiming an important distinction. While physicists would rejoice in the discoverythat the conception of the physical universe reduces to the conception of some simplefragment or model, the Set Theorist rejects this possibility. I claim that by the verynature of its conception, the set of all truths of the transfinite universe (the universe of