## Journal of Symbolic Logic

### Definability and Initial Segments of $c$-Degrees

Robert S. Lubarsky

#### Abstract

We combine two techniques of set theory relating to minimal degrees of constructibility. Jensen constructed a minimal real which is additionally a $\Pi^1_2$ singleton. Groszek built an initial segment of order type $1 + \alpha^\ast$, for any ordinal $\alpha$. This paper shows how to force a $\Pi^1_2$ singleton such that the $c$-degrees beneath it, all represented by reals, are of type $1 + \alpha^\ast$, for many ordinals $\alpha$. We also examine the definability $\alpha$ needs to be so represented by a real.

#### Article information

Source
J. Symbolic Logic, Volume 53, Issue 4 (1988), 1070-1081.

Dates
First available in Project Euclid: 6 July 2007

Lubarsky, Robert S. Definability and Initial Segments of $c$-Degrees. J. Symbolic Logic 53 (1988), no. 4, 1070--1081. https://projecteuclid.org/euclid.jsl/1183742782