{"paper":{"title":"The definability of $\\mathbb{E}$ in self-iterable mice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Farmer Schlutzenberg","submitted_at":"2014-11-29T08:35:28Z","abstract_excerpt":"Let $M$ be a fine structural mouse and let $F\\in M$ be such that $M\\models$``$F$ is a total extender'' and $(M||\\mathrm{lh}(F),F)$ is a premouse. We show that it follows that $F\\in\\mathbb{E}^M$, where $\\mathbb{E}^M$ is the extender sequence of $M$. We also prove generalizations of this fact.\n  Let $M$ be a premouse with no largest cardinal and let $\\Sigma$ be a sufficient iteration strategy for $M$. We prove that if $M$ knows enough of $\\Sigma\\upharpoonright M$ then $\\mathbb{E}^M$ is definable over the universe $\\lfloor M\\rfloor$ of $M$, so if also $\\lfloor M\\rfloor\\models\\mathrm{ZFC}$ then $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.0085","kind":"arxiv","version":2},"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"}