{"paper":{"title":"Torsion-Free Abelian Groups are Consistently $a \\Delta^1_2$-complete","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Douglas Ulrich, Saharon Shelah","submitted_at":"2018-04-22T18:32:39Z","abstract_excerpt":"Let $\\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + \\kappa(\\omega)$ exists, then $\\mbox{TFAG}$ is $a \\Delta^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $\\alpha$-ary Schr\\\"{o}der- Bernstein property, and show that $\\mbox{TFAG}$ fails the $\\alpha$-ary Schr\\\"{o}der-Bernstein property for every $\\alpha < \\kappa(\\omega)$. We leave open whether or not $\\mbox{TFAG}$ can have the $\\kappa(\\omega)$-ary Schr\\\"{o}der-Bernstein property; if it did, then it would not be $a \\Delta^1_2$-complete, and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.08152","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"}