{"paper":{"title":"$\\Sigma_1(\\kappa)$-definable subsets of $\\mathrm{H}(\\kappa^+)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Philipp L\\\"ucke, Philipp Schlicht, Ralf Schindler","submitted_at":"2017-10-26T15:42:57Z","abstract_excerpt":"We study $\\Sigma_1(\\omega_1)$-definable sets (i.e. sets that are equal to the collection of all sets satisfying a certain $\\Sigma_1$-formula with parameter $\\omega_1$) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is $\\Sigma_1(\\omega_1)$-definable, the set of all stationary subsets of $\\omega_1$ is not $\\Sigma_1(\\omega_1)$-definable and the complement of every $\\Sigma_1(\\omega_1)$-definable Bernstein subset of ${}^{\\omega_1}\\omega_1$ is not $\\Sigma_1(\\omega_1)$-definable. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.09766","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"}