{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,20]],"date-time":"2024-03-20T02:55:13Z","timestamp":1710903313820},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2013,6,20]],"date-time":"2013-06-20T00:00:00Z","timestamp":1371686400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2013,9]]},"abstract":"Abstract<\/jats:title>In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4\u2019s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space.<\/jats:p>","DOI":"10.1017\/s1755020313000087","type":"journal-article","created":{"date-parts":[[2013,6,20]],"date-time":"2013-06-20T09:41:23Z","timestamp":1371721283000},"page":"545-570","source":"Crossref","is-referenced-by-count":24,"title":["STRONG COMPLETENESS OF S4 FOR ANY DENSE-IN-ITSELF METRIC SPACE"],"prefix":"10.1017","volume":"6","author":[{"given":"PHILIP","family":"KREMER","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2013,6,20]]},"reference":[{"key":"S1755020313000087_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s10992-010-9161-3"},{"key":"S1755020313000087_ref21","doi-asserted-by":"publisher","DOI":"10.4064\/fm-31-1-103-134"},{"key":"S1755020313000087_ref16","doi-asserted-by":"publisher","DOI":"10.2307\/2267105"},{"key":"S1755020313000087_ref14","unstructured":"Lando T. , & Sarenac D . (2011) Fractal Completeness Techniques in Topological Modal Logic: Koch Curve, Limit Tree, and the Real Line. Available from: http:\/\/philosophy.berkeley.edu\/file\/698\/FractalCompletenessTechniques.pdf."},{"key":"S1755020313000087_ref3","unstructured":"Blok W . (1976). Varieties of Interior Algebras. 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(2012). Simple completeness proofs for some spatial logics of the real line. 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