{"paper":{"title":"The Destruction of the Axiom of Determinacy by Forcings on $\\mathbb{R}$ when $\\Theta$ is Regular","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Stephen Jackson, William Chan","submitted_at":"2019-03-16T23:57:53Z","abstract_excerpt":"$\\mathsf{ZF + AD}$ proves that for all nontrivial forcings $\\mathbb{P}$ on a wellorderable set of cardinality less than $\\Theta$, $1_{\\mathbb{P}} \\Vdash_{\\mathbb{P}} \\neg\\mathsf{AD}$. $\\mathsf{ZF + AD} + \\Theta$ is regular proves that for all nontrivial forcing $\\mathbb{P}$ which is a surjective image of $\\mathbb{R}$, $1_{\\mathbb{P}} \\Vdash_{\\mathbb{P}} \\neg\\mathsf{AD}$. In particular, $\\mathsf{ZF + AD + V = L(\\mathbb{R})}$ proves that for every nontrivial forcing $\\mathbb{P} \\in L_\\Theta(\\mathbb{R})$, $1_{\\mathbb{P}} \\Vdash_{\\mathbb{P}} \\neg\\mathsf{AD}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.07005","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"}