{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T07:38:43Z","timestamp":1776152323146,"version":"3.50.1"},"reference-count":9,"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":10238,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,3]]},"abstract":"<jats:p>It was J. E. Baumgartner who in [1] proved that when a weakly compact cardinal is L\u00e9vy-collapsed to <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub> the new <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub> inherits some of the large cardinal properties; e.g. if <jats:italic>S<\/jats:italic> is a stationary set of <jats:italic>\u03c9<\/jats:italic>-limits in <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub> then for some <jats:italic>\u03b1<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>S<\/jats:italic> \u2229 <jats:italic>\u03b1<\/jats:italic> is stationary in <jats:italic>\u03b1<\/jats:italic>. Later S. Shelah extended this to the following theorem: if a supercompact cardinal <jats:italic>\u03ba<\/jats:italic> is L\u00e9vy-collapsed to <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>, then in the resulting model the following holds: if <jats:italic>S<\/jats:italic> \u2286 <jats:italic>\u03bb<\/jats:italic> is a stationary set of <jats:italic>\u03c9<\/jats:italic>-limits and cf(<jats:italic>\u03bb<\/jats:italic>) \u2265 <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub> then there is an <jats:italic>\u03b1<\/jats:italic>. &lt; <jats:italic>\u03bb<\/jats:italic> such that <jats:italic>S<\/jats:italic> \u2229 <jats:italic>\u03b1<\/jats:italic> is stationary in <jats:italic>\u03b1<\/jats:italic>, i.e. stationary reflection holds for countable cofinality (see [1] and [3]). These theorems are important prototypes of small cardinal compactness theorems; many applications and generalizations can be found in the literature. One might think that these results are true for sets with an uncountable cofinality <jats:italic>\u03bc<\/jats:italic> as well, i.e. when an appropriate large cardinal is collapsed to <jats:italic>\u03bc<\/jats:italic><jats:sup>++<\/jats:sup>. Though this is true for Baumgartner's theorem, there remains a problem with Shelah's result. The point is that the lemma stating that a stationary set of <jats:italic>\u03c9<\/jats:italic>-limits remains stationary after forcing with an <jats:italic>\u03c9<\/jats:italic><jats:sub>2<\/jats:sub>-closed partial order may be false in the case of <jats:italic>\u03bc<\/jats:italic>-limits in a cardinal of the form <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup> with cf(<jats:italic>\u03bb<\/jats:italic>) &lt; <jats:italic>\u03bc<\/jats:italic>, as was shown in [8] by Shelah. The problem has recently been solved by Baumgartner, who observed that if a universal box-sequence on the class of those ordinals with cofinality \u2264 <jats:italic>\u03bc<\/jats:italic> exists, the lemma still holds, and a universal box-sequence of the above type can be added without destroying supercompact cardinals beyond <jats:italic>\u03bc<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2273951","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:16:32Z","timestamp":1146953792000},"page":"147-151","source":"Crossref","is-referenced-by-count":2,"title":["Stationary reflections for uncountable cofinality"],"prefix":"10.1017","volume":"51","author":[{"given":"P\u00e9ter","family":"Komj\u00e1th","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031637_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1976-0540771-8"},{"key":"S0022481200031637_ref004","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200031637_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90001-2"},{"key":"S0022481200031637_ref005","unstructured":"Komj\u00e1th P. , Miller's theorem revisited (to appear)."},{"key":"S0022481200031637_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761379"},{"key":"S0022481200031637_ref008","first-page":"357","volume-title":"Logic Colloquium '78","author":"Shelah","year":"1979"},{"key":"S0022481200031637_ref002","unstructured":"Baumgartner J. E. , unpublished manuscript on stationary reflection."},{"key":"S0022481200031637_ref006","first-page":"755","volume":"47","author":"Magidor","year":"1982","journal-title":"Reflecting stationary sets"},{"key":"S0022481200031637_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90031-1"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031637","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,22]],"date-time":"2019-05-22T07:34:36Z","timestamp":1558510476000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031637\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,3]]},"references-count":9,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1986,3]]}},"alternative-id":["S0022481200031637"],"URL":"https:\/\/doi.org\/10.2307\/2273951","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,3]]}}}