{"paper":{"title":"Relative Definability of $n$-Generics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Wei Wang","submitted_at":"2015-11-28T07:07:17Z","abstract_excerpt":"A set $G \\subseteq \\omega$ is $n$-generic for a positive integer $n$ if and only if every $\\Sigma^0_n$ formula of $G$ is decided by a finite initial segment of $G$ in the sense of Cohen forcing. It is shown here that every $n$-generic set $G$ is properly $\\Sigma^0_n$ in some $G$-recursive $X$. As a corollary, we also prove that for every $n > 1$ and every $n$-generic set $G$ there exists a $G$-recursive $X$ which is generalized ${\\rm low}_n$ but not generalized ${\\rm low}_{n-1}$. Thus we confirm two conjectures of Jockusch."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08875","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"}