The Continuum Hypothesis - Logic at Harvard

Let us rephrase this as follows: For each A satisfying (1), letT A = {ϕ | ZFC+A |= Ω “H(ω 2 ) |= ¬ϕ”}.The theorem says that if there is a proper class of Woodin cardinals and theStrong Ω Conjecture holds, then there are (non-trivial) Ω-complete theoriesT A of H(ω 2 ) and all such theories contain ¬CH. In other words, under theseassumptions, there is a “good” theory and all “good” theories imply ¬CH.The second step begins with the question of whether there is greateragreement among the Ω-complete theories T A . Ideally, there would be justone. However, this is not the case.Theorem 5.2 (Koellner and Woodin). Assume that there is a proper classof Woodin cardinals. Suppose that A is an axiom such that(i) ZFC+A is Ω-satisfiable and(ii) ZFC+A is Ω-complete for the structure H(ω 2 ).Then there is an axiom B such that(i ′ ) ZFC+B is Ω-satisfiable and(ii ′ ) ZFC+B is Ω-complete for the structure H(ω 2 )and T A ≠ T B .This raises the issue as to how one is to select from among these theories?It turns out that there is a maximal theory among the T A and this is givenby the axiom (∗).Theorem 5.3 (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.28