# On a Question of Jaegers

## TL;DR

This paper constructs a positive arithmetical formula with no hyperarithmetical fixed point, impacting various areas such as proof theory, set theory, and the structure of hyperdegrees, and answers a question posed by Jäger.

## Contribution

It introduces a positive arithmetical formula lacking a hyperarithmetical fixed point, providing new insights into fixed points and proof-theoretic strength.

## Key findings

- Existence of a positive arithmetical formula with no hyperarithmetical fixed point
- Implications for proof-theoretic strength of Kripke-Platek set theory
- Results on hyperdegrees and non-Borel uniformization

## 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 J\"{a}ger. 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.