{"paper":{"title":"A Direct Proof of Schwichtenberg's Bar Recursion Closure Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Paulo Oliva, Silvia Steila","submitted_at":"2016-07-18T18:46:14Z","abstract_excerpt":"In 1979 Schwichtenberg showed that the System $\\text{T}$ definable functionals are closed under a rule-like version Spector's bar recursion of lowest type levels $0$ and $1$. More precisely, if the functional $Y$ which controls the stopping condition of Spector's bar recursor is $\\text{T}$-definable, then the corresponding bar recursion of type levels $0$ and $1$ is already $\\text{T}$-definable. Schwichtenberg's original proof, however, relies on a detour through Tait's infinitary terms and the correspondence between ordinal recursion for $\\alpha < \\varepsilon_0$ and primitive recursion over f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.05237","kind":"arxiv","version":4},"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"}