The independence of δ¹_n

In this paper we prove the independence of δ¹_n for n ≥ 3. We show that δ¹₄ can be forced to be above any ordinal of L using set forcing. For δ¹₃ we prove that it can be forced, using set forcing, to be above any L cardinal κ such that κ is Π₁ definable without parameters in L. We then show that δ¹₃ cannot be forced by a set forcing to be above every cardinal of L. Finally we present a class forcing construction to make δ¹₃ greater than any given L cardinal.