A fixed point for the jump operator on structures

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.

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