{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,12]],"date-time":"2025-09-12T18:18:33Z","timestamp":1757701113494},"reference-count":12,"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":8685,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,6]]},"abstract":"<jats:p>Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> Suslin trees can be extended to <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> Suslin trees for any regular cardinal <jats:italic>\u03ba<\/jats:italic>. Some of these are given in \u00a71.<\/jats:p><jats:p>In many such constructions, sequences of Suslin trees are used. In \u00a7II we show, in various ways, that the generalization to sequences, even <jats:italic>\u03c9<\/jats:italic>-sequences, of <jats:italic>\u03ba<\/jats:italic><jats:sup>+<\/jats:sup> Suslin trees cannot be done.<\/jats:p><jats:p>In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in \u00a7III.<\/jats:p><jats:p>Our notation is standard. If <jats:italic>T<\/jats:italic> is a tree and <jats:italic>x \u2208 T<\/jats:italic>, then \u2223<jats:italic>x<\/jats:italic>\u2223 is the height of <jats:italic>x<\/jats:italic> in <jats:italic>T<\/jats:italic>. We define <jats:italic>T<\/jats:italic><jats:sub>\u03b1<\/jats:sub> (or <jats:italic>T<\/jats:italic>(\u03b1)) = {<jats:italic>x<\/jats:italic> \u2208 <jats:italic>T<\/jats:italic>: \u2223<jats:italic>x<\/jats:italic>\u2223 = \u03b1} and <jats:italic>T<\/jats:italic>\u2223\u03b1 = {<jats:italic>x<\/jats:italic> \u2208 <jats:italic>T<\/jats:italic>: \u2223<jats:italic>x<\/jats:italic>\u2223 &lt; \u03b1}.<\/jats:p><jats:p>If <jats:italic>p<\/jats:italic> and <jats:italic>q<\/jats:italic> are forcing conditions, <jats:italic>p<\/jats:italic> \u2264 <jats:italic>q<\/jats:italic> means that <jats:italic>p<\/jats:italic> has more information than <jats:italic>q<\/jats:italic>.<\/jats:p><jats:p>If (<jats:italic>T<\/jats:italic><jats:sub>\u03b1<\/jats:sub>: \u03b1 \u2208 <jats:italic>I<\/jats:italic>) is a sequence of trees, \u03a0 <jats:italic>T<\/jats:italic><jats:sub>\u03b1<\/jats:sub> will always mean the set of (<jats:italic>x<\/jats:italic><jats:sub>\u03b1<\/jats:sub>: \u03b1 \u2208 <jats:italic>I<\/jats:italic>) such that <jats:italic>x<\/jats:italic><jats:sub>\u03b1<\/jats:sub> \u2208 <jats:italic>T<\/jats:italic><jats:sub>\u03b1<\/jats:sub> and \u2223<jats:italic>x<jats:sub>\u03b1<\/jats:sub><\/jats:italic>\u2223 = \u2223<jats:italic>x<jats:sub>\u03b2<\/jats:sub><\/jats:italic>\u2223 for <jats:italic>\u03b1, \u03b2<\/jats:italic> \u2208 <jats:italic>I<\/jats:italic>.<\/jats:p><jats:p>For functions <jats:italic>b, T<\/jats:italic>,\u2026 we denote by <jats:italic>b<\/jats:italic> \u2223\u03b1, <jats:italic>T<\/jats:italic>\u2223 \u03b1,\u2026 their restriction to \u03b1.<\/jats:p><jats:p>If <jats:italic>x<\/jats:italic> is a sequence of ordinals and \u03b1 is an ordinal, <jats:italic>x<\/jats:italic><jats:sup>\u2227<\/jats:sup>\u03b1 is the sequence obtained by concatenating \u03b1 at the end of <jats:italic>x<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/2274644","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:35:29Z","timestamp":1146940529000},"page":"526-536","source":"Crossref","is-referenced-by-count":3,"title":["Some results on higher Suslin trees"],"prefix":"10.1017","volume":"55","author":[{"given":"R.","family":"David","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025949_ref012","unstructured":"Stanley L. , Another absolute singleton, handwritten notes, 1983."},{"key":"S0022481200025949_ref010","doi-asserted-by":"publisher","DOI":"10.1007\/BF02761942"},{"key":"S0022481200025949_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-81-4-279-290"},{"key":"S0022481200025949_ref006","unstructured":"Jensen R. B. , Coding the universe by a real, handwritten notes, 1975."},{"key":"S0022481200025949_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511629198"},{"key":"S0022481200025949_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90002-X"},{"key":"S0022481200025949_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21543-2"},{"key":"S0022481200025949_ref005","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S0022481200025949_ref008","first-page":"411","article-title":"The \u21352-Souslin hypothesis","volume":"264","author":"Laver","year":"1981","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200025949_ref011","first-page":"50","article-title":"A nonconstructible  set of integers","volume":"127","author":"Solovay","year":"1967","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200025949_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0065979"},{"key":"S0022481200025949_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(82)90001-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025949","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:12:59Z","timestamp":1558199579000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025949\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,6]]},"references-count":12,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1990,6]]}},"alternative-id":["S0022481200025949"],"URL":"https:\/\/doi.org\/10.2307\/2274644","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,6]]}}}