{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T13:03:18Z","timestamp":1773406998540,"version":"3.50.1"},"reference-count":22,"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":6767,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We continue the investigation of the Laver ideal \u2113<jats:sup>0<\/jats:sup> and Miller ideal <jats:italic>m<\/jats:italic><jats:sup>0<\/jats:sup> started in [GJSp] and [GRShSp]; these are the ideals on the Baire space associated with Laver forcing and Miller forcing. We solve several open problems from these papers. The main result is the construction of models for <jats:italic>t<\/jats:italic> &lt; add(\u2113<jats:sup>0<\/jats:sup>), <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline1\"\/> &lt; add(<jats:italic>m<\/jats:italic><jats:sup>0<\/jats:sup>), where add denotes the additivity coefficient of an ideal. For this we construct amoeba forcings for these forcings which do not add Cohen reals. We show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline2\"\/> = <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub> implies add(<jats:italic>m<\/jats:italic><jats:sup>0<\/jats:sup>) \u2264 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline3\"\/>. We show that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline4\"\/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline5\"\/> implies cov(\u2113<jats:sup>0<\/jats:sup>) \u2264 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline3\"\/><jats:sup>+<\/jats:sup>, cov(<jats:italic>m<\/jats:italic><jats:sup>0<\/jats:sup>) \u2264 <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline3\"\/><jats:sup>+<\/jats:sup> respectively. Here cov denotes the covering coefficient. We also show that in the Cohen model cov(<jats:italic>m<\/jats:italic><jats:sup>0<\/jats:sup>) &lt; <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200018235_inline6\"\/> holds. Finally we prove that Cohen forcing does not add a superperfect tree of Cohen reals.<\/jats:p>","DOI":"10.2307\/2275753","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:56:10Z","timestamp":1146956170000},"page":"705-726","source":"Crossref","is-referenced-by-count":16,"title":["Generic trees"],"prefix":"10.1017","volume":"60","author":[{"given":"Otmar","family":"Spinas","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200018235_ref004","first-page":"449","volume":"35","author":"Baumgartner","year":"1987","journal-title":"Adjoining dominating functions"},{"key":"S0022481200018235_ref003","first-page":"401","volume":"58","author":"Bartoszy\u0144ski","year":"1993","journal-title":"The Cicho\u0144 diagram"},{"key":"S0022481200018235_ref007","unstructured":"Brendle J. , Mutually generic sets and perfect free subsets, preprint."},{"key":"S0022481200018235_ref015","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1977-0450070-1"},{"key":"S0022481200018235_ref020","unstructured":"Repick\u00fd M. , Handwritten notes."},{"key":"S0022481200018235_ref017","doi-asserted-by":"crossref","first-page":"17","DOI":"10.4064\/fm-24-1-17-34","article-title":"Sur une classe de fonctions de W. Sierpi\u0144ski et la classe correspondante d'ensembles","volume":"24","author":"Marczewski","year":"1935","journal-title":"Fundamenta Mathematicae"},{"key":"S0022481200018235_ref009","unstructured":"Eisworth T. , Groupwise density and tree ideals, preprint."},{"key":"S0022481200018235_ref019","doi-asserted-by":"publisher","DOI":"10.4064\/fm-127-2-127-132"},{"key":"S0022481200018235_ref022","first-page":"279","article-title":"CCC posets of perfect trees","volume":"79","author":"Veli\u010dkovi\u0107","year":"1991","journal-title":"Compositio Mathematica"},{"key":"S0022481200018235_ref018","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/031\/763899"},{"key":"S0022481200018235_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0097329"},{"key":"S0022481200018235_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50006-9"},{"key":"S0022481200018235_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-110-1-11-24"},{"key":"S0022481200018235_ref013","doi-asserted-by":"publisher","DOI":"10.1007\/BF01269943"},{"key":"S0022481200018235_ref010","first-page":"305","volume-title":"Set theory of the reals (Ramat Gan, 1991)","volume":"6","author":"Goldstern","year":"1993"},{"key":"S0022481200018235_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392416"},{"key":"S0022481200018235_ref002","first-page":"333","volume-title":"Handbook of Boolean algebra","author":"Balcar","year":"1989"},{"key":"S0022481200018235_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1994-1284459-3"},{"key":"S0022481200018235_ref014","first-page":"909","volume":"55","author":"Judah","year":"1990","journal-title":"The Kunen-Miller chart"},{"key":"S0022481200018235_ref005","doi-asserted-by":"publisher","DOI":"10.4064\/fm-114-2-149-157"},{"key":"S0022481200018235_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21543-2"},{"key":"S0022481200018235_ref012","volume-title":"Proceedings of the American Mathematical Society","author":"Goldstern"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200018235","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T21:13:08Z","timestamp":1557781988000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200018235\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,9]]},"references-count":22,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1995,9]]}},"alternative-id":["S0022481200018235"],"URL":"https:\/\/doi.org\/10.2307\/2275753","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,9]]}}}