The Continuum Hypothesis - Logic at Harvard

We will need a strong form of this conjecture which we shall call theStrong Ω Conjecture. It is somewhat technical and so we will pass over it insilence. 133.2.3 Ω-Complete TheoriesRecall that one key virtue of large cardinal axioms is that they “effectivelysettle” the theory of second-order arithmetic (and, in fact, the theory ofL(R) and more) in the sense that in the presence of large cardinals onecannot use the method of set forcing to establish independence with respectto statements about L(R). This notion of invariance under set forcing playedakey roleinSection 3.1. Wecannowrephrase thisnotionintermsofΩ-logic.Definition 3.9. A theory T is Ω-complete for a collection of sentences Γ iffor each ϕ ∈ Γ, T |= Ω ϕ or T |= Ω ¬ϕ.The invariance of the theory of L(R) under set forcing can now be rephrasedas follows:Theorem 3.10 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Then ZFC is Ω-complete for the collection of sentences ofthe form “L(R) |= ϕ”.Unfortunately, it follows from a series of results originating with workof Levy and Solovay that traditional large cardinal axioms do not yield Ω-complete theories at the level of Σ 2 1 since one can always use a “small” (andhence large cardinal preserving) forcing to alter the truth-value of CH.Theorem 3.11. Assume L is a standard large cardinal axiom. Then ZFC+Lis not Ω-complete for Σ 2 1.13 Here are the details: First we need another conjecture: (The AD + Conjecture) Supposethat A and B are sets of reals such that L(A,R) and L(B,R) satisfy AD + . Supposeevery setX ∈ P(R)∩ ( L(A,R)∪L(B,R) )is ω 1 -universally Baire. Then eitheror(∆ ∼21 ) L(A,R) ⊆ (∆ ∼21 ) L(B,R)(∆ ∼21 ) L(B,R) ⊆ (∆ ∼21 ) L(A,R) .(Strong Ω conjecture) Assume there is a proper class of Woodin cardinals. Then the ΩConjecture holds and the AD + Conjecture is Ω-valid.17