Reflection and the priority method in E-recursion theory
A central issue in E-recursion theory is the relative status of E-recursive enumerability and Σ1-definability in an E-closed structure. In most initial segments of L these two are not the same. However, as is shown here, every E-closed initial segment of L is canonically represented as the union of Π1-absolute admissible sets with gaps, in which sets the two notions are identical. This representation is used to prove a meta-theorem which provides a translation of Friedberg-style finite injury constructions from classical recursion theory into successful E-recursion theoretic constructions for initial segments of L. The key to the method is to construe an E-recursive enumeration as a direct limit of Σ1-enumerations, in each of which, requirements are satisfied using techniques from α-recursion theory. Suppose that Lk is E-closed; a subset of κ is said to be scattered if its order type is less than ρk1, the E-recursively enumerable projectum of Lk. As an application of this method, it is shown that there is an E-recursively enumerable degree on Lk which is not scattered.
The difference between E-recursive enumerability and Σ1-definability can be used to construct new varieties of recursively enumerable sets. For example, for any given Lk there is a complete E-recursively enumerable subset of κ which has order type the Σ1-cofinality of κ; this shows that there is a complete scattered set whenever possible. In addition, it is shown that if the two notions of definability are different in Lk then the diamond lattice can be embedded in the E-recursively enumerable degrees on Lk preserving meet, join, 0 and 1. This result contrasts with the Lachlan non-diamond theorem, which states that no such embedding exists for the recursively enumerable degrees on ω. As Lachlan's theorem is proven using the finite injury method, this indicates that there is no method for adapting a general finite injury construction from the classical setting to E-recursion on an arbitrary Lk.
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