{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T07:11:36Z","timestamp":1769152296886,"version":"3.49.0"},"reference-count":20,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":2202,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>I investigate versions of the Maximality Principles for the classes of forcings which are &lt;<jats:italic>\u03ba<\/jats:italic>-closed, &lt;<jats:italic>\u03ba<\/jats:italic>-directed-closed, or of the form Col(<jats:italic>\u03ba<\/jats:italic>, &lt;<jats:italic>\u03bb<\/jats:italic>). These principles come in many variants, depending on the parameters which are allowed, I shall write MP<jats:sub>\u0393<\/jats:sub> (<jats:italic>A<\/jats:italic>) for the maximality principle for forcings in \u0393, with parameters from <jats:italic>A<\/jats:italic>. The main results of this paper are:<\/jats:p><jats:p>\u2022 The principles have many consequences, such as &lt;<jats:italic>\u03ba<\/jats:italic>-closed-generic <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200004734_inline1\"\/>(<jats:italic>H<jats:sub>\u03ba<\/jats:sub><\/jats:italic>) absoluteness, and imply, e.g., that \u25ca<jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.<\/jats:p><jats:p>\u2022 The principles can be separated into a hierarchy which is strict, for many <jats:italic>\u03ba<\/jats:italic>.<\/jats:p><jats:p>\u2022 Some of the principles can be combined, in the sense that they can hold at many different <jats:italic>\u03ba<\/jats:italic> simultaneously.<\/jats:p><jats:p>The possibilities of combining the principles are limited, though: While it is consistent that MP<jats:sub>&lt;<jats:italic>\u03ba<\/jats:italic>-closed<\/jats:sub>(<jats:italic>H<jats:sub>\u03ba<\/jats:sub><\/jats:italic> +) holds at all regular <jats:italic>\u03ba<\/jats:italic> below any fixed <jats:italic>\u03b1<\/jats:italic>, the \u201cglobal\u201d maximality principle, stating that MP<jats:sub>&lt;<jats:italic>\u03ba<\/jats:italic>-closed<\/jats:sub> (<jats:italic>H<\/jats:italic><jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub> \u222a {<jats:italic>\u03ba<\/jats:italic>} ) holds at every regular <jats:italic>\u03ba<\/jats:italic>, is <jats:italic>inconsistent<\/jats:italic>. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every <jats:italic>local<\/jats:italic> statement with parameters from <jats:italic>H<jats:sub><jats:italic>\u03ba<\/jats:italic><\/jats:sub>\u22a6<\/jats:italic> that's provably &lt;<jats:italic>\u03ba<\/jats:italic>-closed-forceably necessary is true, for all regular <jats:italic>\u03ba<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1208358754","type":"journal-article","created":{"date-parts":[[2008,10,14]],"date-time":"2008-10-14T15:19:02Z","timestamp":1223997542000},"page":"276-308","source":"Crossref","is-referenced-by-count":11,"title":["Closed maximality principles: implications, separations and combinations"],"prefix":"10.1017","volume":"73","author":[{"given":"Gunter","family":"Fuchs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200004734_ref020","unstructured":"Thomas Simon , The automorphism tower problem, to appear."},{"key":"S0022481200004734_ref016","unstructured":"Leibman George , Consistency strengths of modified maximality principles, Ph.D. thesis, The City University of New York, 2004."},{"key":"S0022481200004734_ref014","volume-title":"Set theory. 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