The Continuum Hypothesis - Logic at Harvard

(though this is sensitive to the background theory) but the statement PM(all projective sets are Lebesgue measurable) is classified as indeterminate.For those who are convinced by the arguments (surveyed in the entry“Large Cardinals and Determinacy”) for large cardinal axioms and axiomsofdefinable determinacy, even thesemultiverse conceptions aretooweak. Wewill followthis route. Fortherest of thisentry we will embrace non-pluralismconcerning large cardinal axioms and axioms of definable determinacy andfocus on the question of CH.4.2 The Generic MultiverseThe motivation behind the generic multiverse is to grant the case for largecardinal axioms and definable determinacy but deny that statements such asCHhave a determinate truthvalue. Tobespecific aboutthe background theorylet us take ZFC+“There is a proper class of Woodincardinals” and recallthat this large cardinal assumption secures axioms of definable determinacysuch as PD and AD L(R) .Let the generic multiverse V be the result of closing V under genericextensions and generic refinements. One way to formalize this is by takingan external vantage point and start with a countable transitive model M.The generic multiverse based on M is then the smallest set V M such thatM ∈ V M and, for each pair of countable transitive models (N,N[G]) suchthat N |= ZFC and G ⊆ P is N-generic for some partial order in P ∈ N, ifeither N or N[G] is in V M then both N and N[G] are in V M .Let the generic multiverse conception of truth be the view that a statementis true simpliciter iff it is true in all universes of the generic multiverse.We will call such a statement a generic multiverse truth. A statement is saidto be indeterminate according to the generic multiverse conception iff it isneither true nor false according to the generic multiverse conception. Forexample, granting our large cardinal assumptions, such a view deems PM(and PD and AD L(R) ) true but deems CH indeterminate.4.3 The Ω Conjecture and the Generic MultiverseIs the generic multiverse conception of truth tenable? The answer to thisquestion is closely related to the subject of Ω-logic. The basic connectionbetween generic multiverse truth and Ω-logic is embodied in the followingtheorem:22