Feedback hyperjump

Abstract

Feedback is oracle computability when the oracle consists exactly of the con- and divergence information about computability relative to that same oracle. Here we study two possible feedback hyperjumps and characterize each of them as the complete $\varSigma _1$ set relative to a level of Gödel’s constructible hierarchy $L$.

Locally hyperarithmetical induction

“…These are, respectively, $$\omega _1^{ck}$$, the Church–Kleene ordinal; $$\sigma ^1_1$$, the least $$\Sigma ^1_1$$‐reflecting ordinal; and $$\pi ^1_1$$, the least $$\Pi ^1_1$$‐reflecting ordinal. These ordinals have many alternate characterizations; for a compilation of characterizations of $$\sigma ^1_1$$, we refer the reader to [2]. Characterizations of $$|\Pi ^1_2|$$ and $$|\Sigma ^1_2|$$ were carried out by Richter [20] and Cutland [5].…”