{"paper":{"title":"Structures Associated with Real Closed Fields and the Axiom of Choice","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Merlin Carl","submitted_at":"2014-02-25T11:07:07Z","abstract_excerpt":"An integer part I of a real closed field K is a discretely ordered subring with minimal element 1 such that, for every x in K, I contains some i such that x is between i and i+1 in the ordering of K. Mourgues and Ressayre showed that every real closed field has an integer part. Their construction implicitely uses the axiom of choice.\n  We show that the axiom of choice is actually necessary to obtain the result by constructing a transitive model of ZF (i.e. set theory without the axiom of choice) which contains a real closed field without an integer part. Then we analyze some cases where the ax"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.6130","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"}