{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,7,25]],"date-time":"2024-07-25T12:37:51Z","timestamp":1721911071808},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":1653,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2009,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals <jats:italic>\u03ba<\/jats:italic> (even only allowing <jats:italic>\u03ba<\/jats:italic> itself as a parameter in the Maximality Principle for &lt;<jats:italic>\u03ba<\/jats:italic>-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal <jats:italic>\u03ba<\/jats:italic> (instead of <jats:italic>\u221e<\/jats:italic>), and if so, how strong it is. It turns out that it is consistent in many cases, but the consistency strength is quite high.<\/jats:p>","DOI":"10.2178\/jsl\/1245158097","type":"journal-article","created":{"date-parts":[[2009,6,16]],"date-time":"2009-06-16T13:16:09Z","timestamp":1245158169000},"page":"1015-1046","source":"Crossref","is-referenced-by-count":9,"title":["Combined Maximality Principles up to large cardinals"],"prefix":"10.1017","volume":"74","author":[{"given":"Gunter","family":"Fuchs","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000342X_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(75)90009-1"},{"key":"S002248120000342X_ref011","volume-title":"The higher infinite","author":"Kanamori","year":"2003"},{"key":"S002248120000342X_ref010","first-page":"315","volume":"74","author":"Jensen","year":"2009","journal-title":"Stacking mice"},{"key":"S002248120000342X_ref005","unstructured":"Fuchs Gunter , Generic embeddings associated to an indestructibly weakly compact cardinal, submitted, ca. 30 pages."},{"key":"S002248120000342X_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-5764-9"},{"key":"S002248120000342X_ref002","doi-asserted-by":"publisher","DOI":"10.1002\/1521-3870(200111)47:4<563::AID-MALQ563>3.0.CO;2-#"},{"key":"S002248120000342X_ref001","first-page":"119","article-title":"Universal indestructibility","volume":"16","author":"Apter","year":"1999","journal-title":"Kobe Journal of Mathematics"},{"key":"S002248120000342X_ref014","volume-title":"Reflection principles for the continuum","author":"Stavi","year":"2001"},{"key":"S002248120000342X_ref004","first-page":"276","volume":"73","author":"Fuchs","year":"2008","journal-title":"Closed maximality principles: Implications, separations and combinations"},{"key":"S002248120000342X_ref008","volume-title":"Set theory","author":"Jech","year":"2003"},{"key":"S002248120000342X_ref012","unstructured":"Leibman George , Consistency strengths of modified maximality principles, Ph.D. thesis, The City University of New York, 2004."},{"key":"S002248120000342X_ref006","doi-asserted-by":"publisher","DOI":"10.4064\/fm180-3-4"},{"key":"S002248120000342X_ref009","first-page":"294","volume":"31","author":"Jensen","year":"1966","journal-title":"Independence of the axiom of dependent choices from the countable axiom of choice"},{"key":"S002248120000342X_ref007","first-page":"527","volume":"68","author":"Hamkins","year":"2003","journal-title":"A simple maximality principle"},{"key":"S002248120000342X_ref015","doi-asserted-by":"publisher","DOI":"10.1515\/9783110857818"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000342X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T11:24:20Z","timestamp":1556623460000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000342X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2009,9]]}},"alternative-id":["S002248120000342X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1245158097","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9]]}}}