{"paper":{"title":"Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Vassily Lyubetsky, Vladimir Kanovei","submitted_at":"2017-12-03T13:47:31Z","abstract_excerpt":"We present a model of set theory, in which, for a given $n\\ge2$, there exists a non-ROD-uniformizable planar lightface $\\varPi^1_n$ set in $\\mathbb R\\times\\mathbb R$, whose all vertical cross-sections are countable sets (and in fact Vitali classes), while all planar boldface $\\bf\\Sigma^1_n$ sets with countable cross-sections are $\\bf\\Delta^1_{n+1}$-uniformizable. Thus it is true in this model, that the ROD-uniformization principle for sets with countable cross-sections first fails precisely at a given projective level."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00769","kind":"arxiv","version":3},"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"}