{"paper":{"title":"Countable infinitary theories admitting an invariant measure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.LO","authors_text":"Cameron Freer, Nathanael Ackerman, Rehana Patel","submitted_at":"2017-10-17T07:11:29Z","abstract_excerpt":"Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\\Sigma$ of $\\mathcal{L}_{\\omega_1, \\omega}(L)$ for which there exists an $S_\\infty$-invariant probability measure on the collection of models of $\\Sigma$ with underlying set $\\mathbb{N}$. Restricting to $\\mathcal{L}_{\\omega, \\omega}(L)$, this answers an open question of Gaifman from 1964, via a translation between $S_\\infty$-invariant measures and Gaifman's symmetric measure-models with strict equality. It also extends the known characterization in the case where $\\Sigma$ implies a Scott "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.06128","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"}