Abstract

We combine two techniques of set theory relating to minimal degrees of constructibility. Jensen constructed a minimal real which is additionally a Π 1 2 singleton. Groszek built an initial segment of order type 1 + α * , for any ordinal α. This paper shows how to force a Π 1 2 singleton such that the c-degrees beneath it, all represented by reals, are of type 1 + α * , for many ordinals α. We also examine the definability α needs to be so represented by a real