The Continuum Hypothesis - Logic at Harvard

and T A ≠ T B .How then shall one select from among these theories? Woodin’s work inthis area goes a good deal beyond Theorem 5.1. In addition to isolating anaxiomthatsatisfies (1) ofTheorem 5.1(assuming Ω-satisfiability), heisolatesa very special such axiom, namely, the axiom (∗) (“star”) mentioned earlier.This axiom can be phrased in terms of (the provability notion of) Ω-logic:Theorem 3.14 (Woodin). Assume ZFC and that there is a proper class ofWoodin cardinals. Then the following are equivalent:(1) (∗).(2) For each Π 2 -sentence ϕ in the language for the structureif〈H(ω 2 ),∈,I NS ,A | A ∈ P(R)∩L(R)〉ZFC+“〈H(ω 2 ),∈,I NS ,A | A ∈ P(R)∩L(R)〉 |= ϕ”is Ω-consistent, then〈H(ω 2 ),∈,I NS ,A | A ∈ P(R)∩L(R)〉 |= ϕ.It follows that of the various theories T A involved in Theorem 5.1, thereis one that stands out: The theory T (∗) given by (∗). This theory maximizesthe Π 2 -theory of the structure 〈H(ω 2 ),∈,I NS ,A | A ∈ P(R)∩L(R)〉.The continuum hypothesis fails in this theory. Moreover, in the maximaltheory T (∗) given by (∗) the size of the continuum is ℵ 2 . 14To summarize: Assuming the Strong Ω Conjecture, there is a “good”theory of H(ω 2 ) and all such theories imply that CH fails. Moreover, (again,assuming the Strong Ω Conjecture) there is a maximal such theory and inthat theory 2 ℵ 0= ℵ 2 .Further Reading: For the mathematics concerning P max see Woodin (1999).For an introduction to Ω-logic see Bagaria, Castells & Larson (2006). Formore on incompatible Ω-complete theories see Koellner & Woodin (2009).For more on the case against CH see Woodin (2001a), Woodin (2001b),Woodin (2005a), and Woodin (2005b).14 As mentioned at the end of Section 2.2 it could be the case (given our present knowledge)that large cardinal axioms imply that Θ L(R) < ℵ 3 and, more generally, rule out thedefinable failure of 2 ℵ0 = ℵ 2 . This would arguably further buttress the case for 2 ℵ0 = ℵ 2 .19