{"paper":{"title":"Bounds on Scott Ranks of Some Polish Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"William Chan","submitted_at":"2019-06-11T02:10:05Z","abstract_excerpt":"If $\\mathcal{N}$ is a proper Polish metric space and $\\mathcal{M}$ is any countable dense submetric space of $\\mathcal{N}$, then the Scott rank of $\\mathcal{N}$ in the natural first order language of metric spaces is countable and in fact at most $\\omega_1^{\\mathcal{M}} + 1$, where $\\omega_1^{\\mathcal{M}}$ is the Church-Kleene ordinal of $\\mathcal{M}$ (construed as a subset of $\\omega$) which is the least ordinal with no presentation on $\\omega$ computable from $\\mathcal{M}$.\n  If $\\mathcal{N}$ is a rigid Polish metric space and $\\mathcal{M}$ is any countable dense submetric space, then the Sc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.04351","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"}