A note on the hyperarithmetical hierarchy

Abstract

The hyperarithmetical hierarchy assigns a degree of unsolvability hγ to each constructive ordinal γ. This assignment has the properties that h0 is the recursive degree and hγ+1 is the jump h′γ of hγ. For a limit ordinal λ < ω1 it is not so easy to define hγ. The original definitions used systems of notations for ordinals, see Spector [6]. There are also later notation-free definitions due to Shoenfield (unpublished) and to Hensel and Putnam [4].