{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T21:25:09Z","timestamp":1775856309963,"version":"3.50.1"},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13433,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1977,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:italic>\u03ba<\/jats:italic> denote a regular uncountable cardinal and <jats:italic>NS<\/jats:italic> the normal ideal of nonstationary subsets of <jats:italic>\u03ba<\/jats:italic>. Our results concern the well-known open question whether <jats:italic>NS<\/jats:italic> fails to be <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-saturated, i.e., are there <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> stationary subsets of <jats:italic>\u03ba<\/jats:italic> with pairwise intersections nonstationary? Our first observation is:<\/jats:p><jats:p>Theorem. <jats:italic>NS is<\/jats:italic><jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-<jats:italic>saturated iff for every normal ideal J on <jats:italic>\u03ba<\/jats:italic> there is a stationary set A<\/jats:italic> \u2286 <jats:italic>\u03ba<\/jats:italic><jats:italic>such that J<\/jats:italic> = NS\u2223A = {<jats:italic>X<\/jats:italic> \u2286 <jats:italic>\u03ba<\/jats:italic>: <jats:italic>X<\/jats:italic> \u2229 <jats:italic>A<\/jats:italic> \u2208 <jats:italic>NS<\/jats:italic>}.<\/jats:p><jats:p>Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define \u201cgreatly Mahlo\u201d cardinals and obtain the following:<\/jats:p><jats:p>Theorem.   <jats:italic>If  <jats:italic>\u03ba<\/jats:italic> is greatly Mahlo then NS is not<\/jats:italic><jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-saturated.<\/jats:p><jats:p>Theorem. <jats:italic>If \u03ba is ordinal<\/jats:italic> \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup>-<jats:italic>indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a<\/jats:italic><jats:italic>\u03ba<\/jats:italic>-<jats:italic>saturated ideal, then<\/jats:italic><jats:italic>\u03ba<\/jats:italic><jats:italic>is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal<\/jats:italic> \u03a0<jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">1<\/jats:sup>-<jats:italic>indescribable cardinal<\/jats:italic>.<\/jats:p><jats:p>These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal <jats:italic>\u03ba<\/jats:italic> is not <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>-saturated.<\/jats:p>","DOI":"10.2307\/2272121","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:44:08Z","timestamp":1146951848000},"page":"203-214","source":"Crossref","is-referenced-by-count":35,"title":["On splitting stationary subsets of large cardinals"],"prefix":"10.1017","volume":"42","author":[{"given":"James E.","family":"Baumgartner","sequence":"first","affiliation":[]},{"given":"Alan D.","family":"Taylor","sequence":"additional","affiliation":[]},{"given":"Stanley","family":"Wagon","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200050374_ref014","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0281606"},{"key":"S0022481200050374_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1974-0332481-5"},{"key":"S0022481200050374_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200050374_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1976-14121-3"},{"key":"S0022481200050374_ref018","unstructured":"Wagon Stanley , Decompositions of saturated ideals, Doctoral dissertation, Dartmouth College, 1975."},{"key":"S0022481200050374_ref009","unstructured":"Jensen Ronald B. and Kunen Kenneth , Some combinatorial properties of L and V, mimeographed."},{"key":"S0022481200050374_ref002","first-page":"137","volume-title":"Colloquia Mathematica Societatis J\u00e1nos Bolyai 10, Infinite and finite sets, Keszthely 1973","author":"Baumgartner"},{"key":"S0022481200050374_ref003","first-page":"665","article-title":"Splitting large cardinals into stationary sets","volume":"22","author":"Baumgartner","year":"1975","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200050374_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90014-4"},{"key":"S0022481200050374_ref001","first-page":"109","volume-title":"Colloquia Mathematica Societatis J\u00e1nos Bolyai 10, Infinite and finite sets, Keszthely 1973","author":"Baumgartner"},{"key":"S0022481200050374_ref011","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200050374_ref012","unstructured":"Kunen Kenneth , Saturated Ideals (to appear)."},{"key":"S0022481200050374_ref005","first-page":"139","article-title":"Eine Bemerkung zur Theorie der regressiven Funktionen","volume":"17","author":"Fodor","year":"1956","journal-title":"Acta Scientiarum Mathematicarum (Szeged)"},{"key":"S0022481200050374_ref016","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/013.1\/0290961"},{"key":"S0022481200050374_ref004","volume-title":"Set theory","author":"Drake","year":"1974"},{"key":"S0022481200050374_ref015","article-title":"Changing measurable into accessible cardinals","volume":"68","author":"Prikry","year":"1970","journal-title":"Dissertationes Mathematicae (Rozprawy Matematyczne)"},{"key":"S0022481200050374_ref013","first-page":"33","article-title":"On the closed unbounded ideal of ordinal numbers","volume":"22","author":"Namba","year":"1974","journal-title":"Commentarli Mathematici Universitatis Sancii Pauli"},{"key":"S0022481200050374_ref017","doi-asserted-by":"crossref","first-page":"51","DOI":"10.4064\/fm-33-1-51-65","article-title":"Ideale in vollst\u00e4ndigen Mengenk\u00f6rpern II","volume":"33","author":"Tarski","year":"1945","journal-title":"Fundamenta Mathematicae"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200050374","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T21:35:29Z","timestamp":1558992929000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200050374\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,6]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1977,6]]}},"alternative-id":["S0022481200050374"],"URL":"https:\/\/doi.org\/10.2307\/2272121","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,6]]}}}