{"id":"hexagon/2609.00141","pdf":"https://hexagonmath.org/pdf/2609.00141v1","url":"https://hexagonmath.org/abs/2609.00141v1","tags":["msc:03C62","msc:03E99","msc:03E25","Boolean algebras","Set theory","definability"],"score":0,"title":"Ordinal-definable families in Cohen, random, and collapse extensions","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Elliot Glazer"],"reviews":0,"subject":"math.LO","summary":"We study ordinal-definable families of sets of arbitrary rank in Cohen, random, and collapse extensions. Over $L$, countable OD families have OD enumerations after adding one Cohen or one random real. A single generalized Cohen subset gives the corresponding sharp theorem at every regular uncountable cardinal. When $\\kappa$ is singular strong limit and has uncountable cofinality, adding $\\kappa$ Cohen reals makes every member of a short-parameter definable family of size at most $\\kappa$ definable from a short parameter. Adding $\\kappa$ random reals gives a single short-parameter definable…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex"],"discussed":0,"published":"2026-09-30","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-09-19-000005&version=1"],"human_understanding":"some"}