On a Question of Jaegers

Abstract

We show that there exists a positive arithmetical formula $\psi (x,R)$, where $x\in \omega$, $R\subseteq \omega$, with no hyperarithmetical fixed point. This answers a question of Gerhard Jäger. As corollaries we obtain results on the proof-theoretic strength of the Kripke-Platek set theory; the fixed points of monotone functions in chain-complete partial orders; the non-Borel uniformization of Borel sets; and the hyperdegrees of fixed points of positive formulae. Further we prove a Suslin-Kleene type result for the specific encoding of the hyperarithmetical sets that we are using.

15 pages