A Recursion-theoretic Characterization of Constructible Reals

Let Ly denote Godel's constructive universe up to level y. A countable ordinal y is said to be an index if Ly+1 contains a real not in L r The notion was introduced by Boolos and Putnam [1] who also initiated the study (from the recursion-theoretic viewpoint) of what is known today as " the fine structure of L". In the set-theoretic context, Jensen [3] later extended their results to all levels of the constructible universe. Still, as all the constructible reals occur in LW IL (CO1 L is the first constructibly uncountable ordinal), a great amount of insight into their structure can be gained by studying this " small" universe. One may cite as examples the fruitful investigations by various people on the Turing and hyperdegrees of these reals, the basis theorems associated with them, and so on. Yet another approach is to study fragments Ly for y < cot . The general program is: study the structure of Ly when (a) y is not an index, and (b) y is an index. Leeds and Putnam [4] and also Marek and Srebrny [5] have taken up (a). Among other things, they showed that if y is the limit of indices but is itself not an index, then Ly is a model of ZF minus the power set axiom, and is hereditarily countable. The first result related to (b) was obtained by Boolos and Putnam [1], to the effect that if y ^ co is an index, then there is an arithmetic copy Ey of Ly occurring in Ly + 1 . What are the other reals that occur at level y? In this paper we provide an answer to this question by way of admissible ordinals. For every real T, coj is the least ordinal a such that La[T] is an admissible set. If y is an ordinal, then y is the least admissible ordinal greater than y. In this paper we show the following.