{"paper":{"title":"Godel's Second Incompleteness Theorem for Definable Theories","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Conden Chao, Payam Seraji","submitted_at":"2016-02-07T20:14:42Z","abstract_excerpt":"It is proved that if $T$ is a $\\Sigma_{n+1}$ Definable theory which is $\\Sigma_n$-sound and extends $PA$, then $T$ can not prove the sentence $\\Sigma_n-sound(T)$ that expresses the $\\Sigma_n$-soundness of $T$. Optimality of this result is showed by constructing a $\\Sigma_{n+1}$-definable and $\\Sigma_{n-1}$-sound theory extending $PA$ such that $\\Sigma_n-sound(T)$ is $T$-provable. It is also proved that no R.E. arithmetical theory, evevn very weak theories which are not $\\Sigma_1$-complete, can prove $\\Sigma_1$-soundness of itself."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.02416","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"}