{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T05:06:33Z","timestamp":1648616793796},"reference-count":16,"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":10054,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,9]]},"abstract":"Suppose \u03ba<\/jats:italic> is a regular uncountable cardinal, \u03bb<\/jats:italic> is a cardinal > 1, and is a \u03ba<\/jats:italic>-complete uniform ideal on \u03ba<\/jats:italic>. This paper deals with a saturation property Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ) of , which is a weakening of usual \u03bb<\/jats:italic>-saturatedness. Roughly speaking, Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ) means that can be densely extended to \u03bb<\/jats:italic>-saturated ideals on small fields of subsets of \u03ba<\/jats:italic>. We will show that some consequences of the existence of a \u03bb<\/jats:italic>-saturated ideal on \u03ba<\/jats:italic> follow from weaker \u2203: Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ), and that \u2203: Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ) is connected with weak compactness and complete ineffability of \u03ba<\/jats:italic> in much the same way as the existence of a saturated ideal on \u03ba<\/jats:italic> is connected with measurability of \u03ba<\/jats:italic>.<\/jats:p>In \u00a72, we define Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ), mention a few results that can be proved by straightforward adaptation of known methods, and discuss generic ultrapowers of ZFC\u2212<\/jats:sup>, which will be used repeatedly in the subsequent sections as a main technical tool. A related concept Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>) is also defined and shown to be equivalent to \u2203: Sat(\u03ba<\/jats:italic>, \u03bb<\/jats:italic>, ) under a certain condition.<\/jats:p>In \u00a73, we show that \u2203: Sat(\u03ba<\/jats:italic>, \u03ba<\/jats:italic>, ) implies that \u03ba<\/jats:italic> is highly Mahlo, improving results in [KT] and [So].<\/jats:p>","DOI":"10.2307\/2274010","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:18:32Z","timestamp":1146939512000},"page":"513-525","source":"Crossref","is-referenced-by-count":0,"title":["A saturation property of ideals and weakly compact cardinals"],"prefix":"10.1017","volume":"51","author":[{"given":"Joji","family":"Takahashi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030735_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-30-1-132-155"},{"key":"S0022481200030735_ref004","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200030735_ref002","article-title":"Structural properties of ideals","volume":"197","author":"Baumgartner","year":"1982","journal-title":"Dissertationes Mathematicae (Rozprawy Matematyczne)"},{"key":"S0022481200030735_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90010-6"},{"key":"S0022481200030735_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90005-5"},{"key":"S0022481200030735_ref010","article-title":"Changing measurable into accessible cardinals","volume":"68","author":"Prikry","year":"1970","journal-title":"Dissertationes Mathematicae (Rozprawy Matematycze)"},{"key":"S0022481200030735_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0001-8708(78)90032-4"},{"key":"S0022481200030735_ref009","unstructured":"Paris J. B. , Boolean extensions and large cardinals, Ph.D. thesis, Manchester University, Manchester, 1969."},{"key":"S0022481200030735_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200030735_ref016","doi-asserted-by":"publisher","DOI":"10.4064\/fm-16-1-140-150"},{"key":"S0022481200030735_ref013","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200030735_ref003","first-page":"347","article-title":"Extending Lebesgue measure by infinitely many sets","volume":"1","author":"Carlson","year":"1980","journal-title":"Abstracts of Papers Presented to the American Mathematical Society"},{"key":"S0022481200030735_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0290961"},{"key":"S0022481200030735_ref014","unstructured":"Takahashi J. , Partition and saturation properties of ideals, Ph.D. thesis, University of Minnesota, Minneapolis, Minnesota, 1982."},{"key":"S0022481200030735_ref005","doi-asserted-by":"publisher","DOI":"10.1017\/S0027763000015142"},{"key":"S0022481200030735_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-53-3-225-308"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030735","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T00:40:40Z","timestamp":1558485640000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030735\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1986,9]]}},"alternative-id":["S0022481200030735"],"URL":"https:\/\/doi.org\/10.2307\/2274010","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,9]]}}}