{"paper":{"title":"Completely separably MAD families and the modal logic of $\\beta\\omega$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.LO","authors_text":"Jonathan L. Verner, Tom\\'a\\v{s} L\\'avi\\v{c}ka","submitted_at":"2017-09-20T13:47:58Z","abstract_excerpt":"We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of $\\omega$ implies that the modal logic S4.1.2 is complete with respect to the \\v{C}ech-Stone compactification of the natural numbers, the space $\\beta\\omega$. In the same fashion we prove that the modal logic S4 is complete with respect to the space $\\omega^*=\\beta\\omega\\setminus\\omega$. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions ($\\mathfrak{a}=\\mathfrak{c}$). Our proof is also somewhat simpler."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.06862","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"}