{"paper":{"title":"The Axiom of Determinacy Implies Dependent Choices in Mice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Sandra M\\\"uller","submitted_at":"2019-07-05T10:18:11Z","abstract_excerpt":"We show that the Axiom of Dependent Choices, $\\operatorname{DC}$, holds in countably iterable, passive premice $\\mathcal{M}$ construced over their reals which satisfy the Axiom of Determinacy, $\\operatorname{AD}$, in a $\\operatorname{ZF}+\\operatorname{DC}_{\\mathbb{R}^{\\mathcal{M}}}$ background universe. This generalizes an argument of Kechris for $L(\\mathbb{R})$ using Steel's analysis of scales in mice. In particular, we show that for any $n \\leq \\omega$ and any countable set of reals $A$ so that $M_n(A) \\cap \\mathbb{R} = A$ and $M_n(A) \\vDash \\operatorname{AD}$, we have that $M_n(A) \\vDash \\o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.02755","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"}