The Continuum Hypothesis - Logic at Harvard

3.2 Ω-LogicWe will now recast the above results in terms of a strong logic. We shallmake full use of large cardinal axioms and in this setting we are interested inlogics that are “well-behaved” in the sense that the question of what implieswhat is not radically independent. For example, it is well known that CH isexpressible in full second-order logic. It follows that in the presence of largecardinals one can always use set forcing to flip the truth-value of a purportedlogical validity of full second-order logic. However, there are strong logics—likeω-logicandβ-logic—thatdonothavethisfeature—theyarewell-behavedin the sense that in the presence of large cardinal axioms the question ofwhat implies what cannot be altered by set forcing. We shall introduce avery strong logic that has this feature—Ω-logic. In fact, the logic we shallintroduce can be characterized as the strongest logic with this feature. 113.2.1 Ω-logicDefinition 3.4. Suppose that T is a countable theory in the language of settheory and ϕ is a sentence. ThenT |= Ω ϕif for all complete Boolean algebras B and for all ordinals α,if V B α |= T then V B α |= ϕ.We say that a statement ϕ is Ω-satisfiable if there exists an ordinal αand a complete Boolean algebra B such that V B α |= ϕ, and we say that ϕ isΩ-valid if ∅ |= Ω ϕ. So, the above theorem says that (under our backgroundassumptions), the statement “ϕ is Ω-satisfiable” is generically invariant andin terms of Ω-validity this is simply the following:Theorem 3.5 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Suppose that T is a countable theory in the language ofset theory and ϕ is a sentence. Then for all complete Boolean algebras B,T |= Ω ϕ iff V B |= “T |= Ω ϕ.”Thus thislogicisrobust inthatthequestion ofwhat implieswhat isinvariantunder set forcing.11 See Koellner (2010) for further discussion of strong logics and for a precise statementof this result.15