∏^1_2 singletons and O^#

Abstract

A conjecture of Solovay states: Assuming that for every real ɑ, ɑ^# exists, the constructibility degrees of ∏^1_2 singletons are wellordered and the successor steps in this wellordering are given by the sharps. In this paper we prove among others things that (assuming ∀ɑ (ɑ^# exists)) for every ∏^1_2 singleton a either O^# is constructible from ɑ or ɑ^# is constructible from O^#. From a relativized version of this result it follows that the constructibility degrees of O^#, O^(##), O^(###),... are the first ω constructibility degrees of sharps of ∏^1_2 singletons.

Additional details