AD, Derived Models, and Σ1-Reflection

Let + be the theory + ℝ + " Every set of reals is ∞-Borel " + " Ordinal Determinacy ". For any Γ ⊆ (ℝ), let Γ = ∪{ ∣ is transitive and ∃, ⊆ ℝ×ℝ (, ∈ Γ and (ℝ//,) ∼ = (, ∈))}. We'll prove the following theorems: Theorem 1. (Woodin) Assume + + = ((ℝ)). Then the following are equivalent: 1. Let us call the statement in (2) above " Σ 1-reflection " to Suslin co-Suslin. Theorem 2. (Woodin) Assume + + + = ((ℝ)), then 1. Σ 2 1 has the scale property. 2. Δ 2 1 ≺ Σ 1. Proof. The theorem follows immediately from Theorem 1 and lemma 7.2 in [3], whose proof is essentially due to Woodin. In the course of proving Theorem 1, we shall prove part of the determinacy-to-large-cardinals direction of the Derived Model Theorem. Let be a limit of Woodin cardinals, and G be V-generic over (, <). We set ℝ * = ∪ << ℝ + + = ((ℝ)). Suppose also that if ℝ holds, then Θ is singular. Then there is a set X in some generic extension of V such that setting = [], then 1. for some , ⊨ + is a limit of Woodins; 2. for some M-generic G over (, <): • = (, ℝ *), and