ay 2 00 1 ∇ κ , remarkable cardinals , and 0

We generalize ∇(A), which was introduced in [2], 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 do we have that {X ∈ [Lθ[A]] <κ : X ∩ κ ∈ κ ∧ otp(X ∩ OR) ∈ Card} is stationary in [Lθ[A]] . It was shown in [2] that ∇א1(A) can hold in a set-generic extension of L. We here prove that ∇א2(A) can hold in a semi-proper set-generic extension of L, whereas ∇א3(∅) is equivalent with the existence of 0. Let A ⊂ ω1. In [2] we introduced the following assertion, denoted by ∇(A): {X ∈ [Lω2 [A]] : ∃α < β ∈ Card ∃π π:Lβ[A ∩ α] ∼= X ≺ Lω2 [A]} is stationary in [Lω2 [A]] . The present note is concerned with generalizations of ∇(A) to larger cardinals. Definition 1 Let κ and θ both be regular cardinals, א0 < κ < θ. Then by ∇κ(A) we denote the statement that A ⊂ κ and {X ∈ [Lθ[A]] <κ : X ∩ κ ∈ κ ∧ otp(X ∩OR) ∈ Card} is stationary in [Lθ[A]] . By ∇κ(A) we denote the statement that ∇ θ κ(A) holds for all regular θ > κ. Moreover, we write ∇κ for ∇ θ κ(∅), and ∇κ for ∇κ(∅). ∗ 1991 Mathematics Subject Classification. Primary 03E55, 03E15. Secondary 03E35, 03E60.