Type-2 feedback computability

Feedback computability studies oracles that provide convergence and divergence information about computability relative to that same oracle. We consider strict and loose forms of feedback computability relative to Kleene’s 2 E , the Type-2 object embodying quantification over natural numbers. We prove that the sets of integers semi-computable this way are precisely those Σ 1 -definable over L α I and L σ 1 1 , respectively, where α I denotes the least recursively inaccessible ordinal and σ 1 1 the least Σ 1 1 -reflecting ordinal. Furthermore, our construction of higher-type feedback computability through ordinal inductive fix-points serves as a natural and general framework for the study of feedback computability.