A new condensation principle

We generalize ∇(A), which was introduced in [Sch∞], to larger cardinals. For a regular cardinal κ > א0 we denote by ∇κ(A) the statement that A ⊆ κ and for all regular θ > κ, {X ∈ [Lθ[A] ] <κ : X ∩ κ ∈ κ ∧ otp(X ∩ Ord) ∈ Card} is stationary in [Lθ[A] ] . It was shown in [Sch∞] that ∇א1(A) can hold in a set-generic extension of L. We here prove that ∇א2(A) can hold in a set-generic extension of L as well. In both cases we in fact get equiconsistency theorems. This strengthens results of [Rä00] and [Rä∞]. ∇א3(∅) is equivalent with the existence of 0 .