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Assuming that 0# exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure such that where is the set of Turing degrees which compute a copy of More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full “nth-order arithmetic for all n, cannot prove the existence of such a structure.

A fixed point for the jump operator on structures