{"paper":{"title":"Weakly 2-randoms and 1-generics in Scott sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Linda Brown Westrick","submitted_at":"2017-11-01T00:49:39Z","abstract_excerpt":"Let $S$ be a Scott set, or even an $\\omega$-model of $\\mathsf{WWKL}$. Then for each $A\\in S$, either there is $X \\in S$ that is weakly 2-random relative to $A$, or there is $X\\in S$ that is 1-generic relative to $A$. It follows that if $A_1,\\dots, A_n \\in S$ are non-computable, there is $X \\in S$ such that each $A_i$ is Turing incomparable with $X$, answering a question of Ku\\v{c}era and Slaman. More generally, any $\\forall\\exists$ sentence in the language of partial orders that holds in $\\mathcal D$ also holds in $\\mathcal D_S$, where $\\mathcal D_S$ is the partial order of Turing degrees of e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00153","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"}