Abstract

In this paper we prove the independence of δ 1 n for n ≥ 3. We show that δ 1 4 can be forced to be above any ordinal of L using set forcing. For δ 1 3 we prove that it can be forced, using set forcing, to be above any L cardinal κ such that κ is Π 1 definable without parameters in L. We then show that δ 1 3 cannot be forced by a set forcing to be above every cardinal of L. Finally we present a class forcing construction to make δ 1 3 greater than any given L cardinal