{"paper":{"title":"Disjoint Borel Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Dan Hathaway","submitted_at":"2014-08-19T03:24:14Z","abstract_excerpt":"For each $a \\in \\mathbb{R}$, we define a Borel function $f_a : \\mathbb{R} \\to \\mathbb{R}$ which encodes $a$ in a certain sense. We show that for each Borel $g : \\mathbb{R} \\to \\mathbb{R}$, $f_a \\cap g = \\emptyset$ implies $a \\in \\Delta^1_1(c)$ where $c$ is any code for $g$. We generalize this theorem for $g$ in larger pointclasses $\\Gamma$. Specifically, if $\\Gamma = \\mathbf{\\Delta}^1_2$, then $a \\in L[c]$. Also for all $n \\in \\omega$, if $\\Gamma = \\mathbf{\\Delta}^1_{3 + n}$, then $a \\in \\mathcal{M}_{1 + n}(c)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.4200","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"}