{"paper":{"title":"On the order dimension of locally countable partial orderings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Diip Raghavan, Frank Stephan, Kojiro Higuchi, Steffen Lempp","submitted_at":"2019-02-16T02:55:25Z","abstract_excerpt":"We show that the order dimension of the partial order of all finite subsets of $\\kappa$ under set inclusion is ${\\log}_{2}({\\log}_{2}(\\kappa))$ whenever $\\kappa$ is an infinite cardinal.\n  We also show that the order dimension of any locally countable partial ordering $(P, <)$ of size $\\kappa^+$, for any $\\kappa$ of uncountable cofinality, is at most $\\kappa$. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.06030","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"}