{"paper":{"title":"Computing the Number of Types of Infinite Length","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Will Boney","submitted_at":"2013-09-17T21:33:32Z","abstract_excerpt":"We show that the number of types of sequences of tuples of a fixed length can be calculated from the number of 1-types and the length of the sequences. Specifically, if $\\kappa \\leq \\lambda$, then $$\\sup_{|A| = \\lambda} |S^\\kappa(A)| = (\\sup_{|A| = \\lambda} |S^1(A)|)^\\kappa$$ We show that this holds for any abstract elementary class with $\\lambda$ amalgamation, but it is new for first order theories when $\\kappa$ is infinite. No such calculation is possible for nonalgebraic types. We introduce a generalization of nonalgebraic types for which the same upper bound holds."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4485","kind":"arxiv","version":2},"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"}