The Continuum Hypothesis - Logic at Harvard

this version of CH holds for all sets of reals in the definability hierarchy weare considering; more precisely, if there is a proper class of Woodin cardinalsthen this version of CH holds for all universally Baire sets of reals.2.1.2 Well-ordering VersionThe second formulation of CH asserts that every well-ordering of the realshas order type less than ℵ 2 . For a given pointclass Γ in the hierarchy, thecorresponding definable well-ordering version of CH asserts that every wellordering(coded by a set) in Γ has order type less than ℵ 2 .Again, axioms of definable determinacy and large cardinal axioms implythis version of CH for richer notions of definability. For example, if AD L(R)holds then this version of CH holds for all sets of real numbers in L(R). Andif there is a proper class of Woodin cardinals then this version of CH holdsfor all universally Baire sets of reals.2.1.3 Surjection VersionThe third version formulation of CH asserts that there is no surjection ρ :R → ℵ 2 , or, equivalently, that there is no prewellordering of R of length ℵ 2 .For a given pointclass Γ in the hierarchy of definability, the correspondingsurjection version of CH asserts that there is no surjection ρ : R → ℵ 2 suchthat (the code for) ρ is in Γ.Here the situation is more interesting. Axioms of definable determinacyand large cardinal axioms have bearing on this version since they place1bounds on how long definable prewellorderings can be. Let ∼δ n be the supremumof the lengths of the Σ ∼n -prewellorderings of reals and let Θ L(R) be the1supremum of the lengths of prewellorderings of reals where the prewellorderingis definable in the sense of being in L(R). It is a classical result that1δ∼ 1 = ℵ 11. Martin showed that δ ∼2 ℵ 2 and that if there is a measurable cardinalthen δ ∼3 ℵ 3 . Kunen and Martin also showed under PD, δ ∼4 ℵ 4 and1 11Jackson showed that under PD, for each n < ω, ∼δ n < ℵ ω . Thus, assumingthat there are infinitely many Woodin cardinals, these bounds hold. Moreover,the bounds continue to hold regardless of the size of 2 ℵ 0. Of course,the question is whether these bounds can be improved to show that theprewellorderings are shorter than ℵ 2 . In 1986 Foreman and Magidor initiateda program to establish this. In the most general form they aimed toshow that large cardinal axioms implied that this version of CH held for all9