{"paper":{"title":"Does $\\mathsf{DC}$ imply $\\mathsf{AC}_\\omega$, uniformly?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alessandro Andretta, Lorenzo Notaro","submitted_at":"2023-05-11T09:23:47Z","abstract_excerpt":"The Axiom of Dependent Choice $\\mathsf{DC}$ and the Axiom of Countable Choice $\\mathsf{AC}_\\omega$ are two weak forms of the Axiom of Choice that can be stated for a specific set: $\\mathsf{DC}(X)$ asserts that any total binary relation on $X$ has an infinite chain, while $\\mathsf{AC}_\\omega (X)$ asserts that any countable collection of nonempty subsets of $X$ has a choice function. It is well-known that $\\mathsf{DC} \\Rightarrow \\mathsf{AC}_\\omega$. We study for which sets and under which hypotheses $\\mathsf{DC}(X) \\Rightarrow \\mathsf{AC}_\\omega (X)$, and then we show it is consistent with $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.06676","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.06676/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}