{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,11]],"date-time":"2026-05-11T23:59:27Z","timestamp":1778543967343,"version":"3.51.4"},"reference-count":10,"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":9873,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1987,3]]},"abstract":"<jats:p>For several old problems in combinatorial set theory A. Hajnal and the present author [2] showed that on collapsing a sufficiently Mahlo cardinal to <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> by the L\u00e9vy-collapse one gets a model where these problems are solved in the \u201ccounter-example\u201d direction. The authors of [2] have speculated that the theorems of that paper should hold in <jats:italic>L<\/jats:italic>, and this, in fact, was shown for some of the results by Todor\u010devi\u0107 and Velleman [7,8]. The observation that collapsing a large cardinal to <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> may give rise to <jats:italic>L<\/jats:italic>-like constructions is not new. As it was shown long ago by Silver and Rowbottom, there is a Kurepa-tree if a strongly inaccessible cardinal is L\u00e9vy-collapsed to <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>. In [5] it is proved that even Silver's <jats:italic>W<\/jats:italic> holds in that model. Here we show that even a quagmire exists there, but not necessarily a morass. To be more exact, we show that if <jats:italic>\u03ba<\/jats:italic> &lt; <jats:italic>\u03bb<\/jats:italic> are the first two strongly inaccessible cardinals, first <jats:italic>\u03bb<\/jats:italic> is L\u00e9vy-collapsed to <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup>, and then <jats:italic>\u03ba<\/jats:italic> is L\u00e9vy-collapsed to then there is no <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-morass with built-in diamond in the resulting model (GCH is assumed). If <jats:italic>\u03bb<\/jats:italic> is Mahlo, there is not even a morass.<\/jats:p><jats:p>Our notations are standard. For excellent survey papers on morass-like principles and their uses in combinatorial set theory see [4,5,6].<\/jats:p>","DOI":"10.2307\/2273865","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:20:33Z","timestamp":1146954033000},"page":"111-115","source":"Crossref","is-referenced-by-count":4,"title":["Morasses and the L\u00e9vy-collapse"],"prefix":"10.1017","volume":"52","author":[{"given":"P.","family":"Komj\u00e1th","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200030024_bib009","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90005-5"},{"key":"S0022481200030024_bib003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200030024_bib005","first-page":"153","volume-title":"Logic Colloquium '80","author":"Kanamori","year":"1982"},{"key":"S0022481200030024_bib002","unstructured":"Hajnal, A. and Komj\u00e1th, P. , Some higher-gap examples in combinatorial set theory, Annals of Pure and Applied Logic (to appear)."},{"key":"S0022481200030024_bib010","first-page":"1001","volume":"49","author":"Velleman","year":"1984","journal-title":"Simplified morasses with linear limits"},{"key":"S0022481200030024_bib007","doi-asserted-by":"publisher","DOI":"10.1007\/BF02776079"},{"key":"S0022481200030024_bib008","first-page":"1056","volume":"51","author":"Velleman","year":"1986","journal-title":"On a combinatorial principle of Hajnal and Komj\u00e1th"},{"key":"S0022481200030024_bib001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90001-2"},{"key":"S0022481200030024_bib004","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(08)71372-7"},{"key":"S0022481200030024_bib006","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511758867.007"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200030024","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,11]],"date-time":"2022-04-11T08:12:46Z","timestamp":1649664766000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200030024\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1987,3]]},"references-count":10,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1987,3]]}},"alternative-id":["S0022481200030024"],"URL":"https:\/\/doi.org\/10.2307\/2273865","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1987,3]]}}}