On a Question of Jaegers

We show that there exists a positive arithmetical formula $ψ(x,R)$, where $x \in ω$, $R \subseteq ω$, 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.

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