A fixed point for the jump operator on structures

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.