The Continuum Hypothesis - Logic at Harvard

strong case for new axioms completing the axioms of ZFC and large cardinalaxioms.Unfortunately, this optimistic scenario fails: Assuming the existence ofone such theory one can construct another which differs on CH:Theorem 5.5 (Koellner and Woodin). Assume ZFC and that there is aproper class of Woodin cardinals. Suppose V λ is a specifiable fragment of theuniverse (that is sufficiently large) and suppose that there is a large cardinalaxiom L and axioms ⃗ A such thatThen there are axioms ⃗ B such thatZFC+L+ ⃗ A is Ω-complete for Th(V λ ).ZFC+L+ ⃗ B is Ω-complete for Th(V λ )and the first theory Ω-implies CH if and only if the second theory Ω-implies¬CH.This still leaves us with the question of existence and the answer to thisquestion is sensitive to the Ω Conjecture and the AD + Conjecture:Theorem 5.6 (Woodin). Assume that there is a proper class of Woodincardinals and that the Ω Conjecture holds. Then there is no recursive theory⃗A such that ZFC+ ⃗ A is Ω-complete for the theory of V δ0 +1, where δ 0 is theleast Woodin cardinal.In fact, under a stronger assumption, the scenario must fail at a much earlierlevel.Theorem 5.7 (Woodin). Assume that there is a proper class of Woodincardinals and that the Ω Conjecture holds. Assume that the AD + Conjectureholds. Then there is no recursive theory ⃗ A such that ZFC+ ⃗ A is Ω-completefor the theory of Σ 2 3 .It is open whether there can be such a theory at the level of Σ 2 2. It isconjectured that ZFC + ♦ is Ω-complete (assuming large cardinal axioms)for Σ 2 2.Let us assume that it is answered positively and return to the questionof uniqueness. For each such axiom A, let T A be the Σ 2 2 theory computed byZFC+A in Ω-logic. The question of uniqueness simply asks whether T A isunique.30