 Here the situation is more interesting. Axioms of definable determinacy and large cardinal axioms have bearing on this version since they place bounds on how long definable prewellorderings can be. Let  δ̰1n be the supremum of the lengths of the Σ̰1n-prewellorderings of reals and let ΘL(ℝ) be the supremum of the lengths of prewellorderings of reals where the prewellordering is definable in the sense of being in L(ℝ). It is a classical result that δ̰11 = ℵ1. Martin showed that δ̰12 ≤ ℵ2 and that if there is a measurable cardinal then δ̰13 ≤ ℵ3. Kunen and Martin also showed under PD, δ̰14 ≤ ℵ4 and Jackson showed that under PD, for each n &lt; ω, δ̰1n &lt; ℵω. Thus, assuming that there are infinitely many Woodin cardinals, these bounds hold. Moreover, the bounds continue to hold regardless of the size of 2ℵ0. Of course, the question is whether these bounds can be improved to show that the prewellorderings are shorter than ℵ2. In 1986 Foreman and Magidor initiated a program to establish this. In the most general form they aimed to show that large cardinal axioms implied that this version of CH held for all universally Baire sets of reals. 