{"paper":{"title":"A fixed point for the jump operator on structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Antonio Montalban","submitted_at":"2011-06-05T15:17:59Z","abstract_excerpt":"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 $\\mathcal A$.\n  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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.0908","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}