A fixed point for the jump operator on structures

TL;DR

This paper constructs a structure that can interpret its own jump under the assumption of $0^#$, revealing complex properties that higher-order arithmetic cannot prove to exist.

Contribution

It demonstrates the existence of a structure with a fixed point for the jump operator, showing its complexity exceeds higher-order arithmetic's proof capabilities.

Findings

Existence of a structure interpreting its own jump under $0^#$

The structure's degree spectrum is fixed under the jump operator

Higher-order arithmetic cannot prove the existence of such a structure

Abstract

Assuming that $0^#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $\mathcal A$ such that \[ Sp({\mathcal A}) = \{{\bf x}':{\bf x}\in Sp ({\mathcal A})\}, \] where $Sp ({\mathcal A})$ is the set of Turing degrees which compute a copy of $\mathcal A$. It turns out that, more interesting than the result itself, is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.