{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,25]],"date-time":"2023-10-25T18:22:06Z","timestamp":1698258126352},"reference-count":8,"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":9323,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,9]]},"abstract":"<jats:p>Let PA be first order Peano arithmetic, let <jats:italic>\u039b<\/jats:italic> be the lattice of <jats:italic>\u03a0<\/jats:italic><jats:sub>1<\/jats:sub> sentences modulo PA, and let <jats:italic>S<\/jats:italic> be the poset of prime filters of <jats:italic>\u039b<\/jats:italic> ordered by <jats:italic>reverse<\/jats:italic> inclusion. We show there are large convex discrete parts of <jats:italic>S<\/jats:italic>; in particular there are convex parts which form a completed Baire tree or an Aronszajn tree.<\/jats:p><jats:p>The elements of <jats:italic>S<\/jats:italic>, which we call nodes, correspond to the extensions of PA which are complete for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline1\" \/> sentences. Equivalently, for each model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline2\" \/> of PA the <jats:italic>\u03a0<\/jats:italic><jats:sub>1<\/jats:sub>-theory <jats:italic>\u2200<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline2\" \/>) of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline2\" \/> is a node, and every node occurs in this form. Note that the <jats:italic>\u03a0<\/jats:italic><jats:sub>1<\/jats:sub>-theory <jats:italic>\u2200<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline3\" \/>) of the standard model <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline3\" \/> (i.e. the filter of true <jats:italic>\u03a0<\/jats:italic><jats:sub>1<\/jats:sub> sentences) is the unique root of <jats:italic>S<\/jats:italic>.<\/jats:p><jats:p>This poset <jats:italic>S<\/jats:italic>, which is sometimes called the <jats:italic>E<\/jats:italic>-tree, was first studied in [1] where it is shown that:<\/jats:p><jats:p>(1) The poset is tree-like, i.e. the set of predecessors of any node is linearly ordered.<\/jats:p><jats:p>(2) The poset has <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline2\" \/> branches, each of which is closed under unions and intersections; in particular each branch has a maximum member.<\/jats:p><jats:p>(3) There are branches on which <jats:italic>\u2200<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline3\" \/>) does not have an immediate successor. Further properties of the <jats:italic>E<\/jats:italic>-tree are given in [2]\u2212[7]. In particular in [4] Misercque shows that:<\/jats:p><jats:p>(4) There are branches on which <jats:italic>\u2200<\/jats:italic>(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200042195_inline3\" \/>) does have an immediate successor.<\/jats:p><jats:p>(5) There are nodes with both an immediate predecessor and an immediate successor.<\/jats:p><jats:p>The two results (3) and (4) show that there are fundamentally different branches of <jats:italic>S<\/jats:italic>, and (5) shows that parts of branches may be discrete.<\/jats:p>","DOI":"10.2307\/2274587","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:27:13Z","timestamp":1146940033000},"page":"980-984","source":"Crossref","is-referenced-by-count":3,"title":["Large discrete parts of the <i>E<\/i>-tree"],"prefix":"10.1017","volume":"53","author":[{"given":"Harold","family":"Simmons","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200042195_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8601-8"},{"key":"S0022481200042195_ref007","volume-title":"The lattice of universal sentences modulo Peano arithmetic","author":"Simmons","year":"1980"},{"key":"S0022481200042195_ref006","first-page":"325","volume-title":"Logic Colloquium '76","author":"Simmons","year":"1977"},{"key":"S0022481200042195_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/17.6.513"},{"key":"S0022481200042195_ref003","first-page":"573","article-title":"The nonhomogeneity of the E-tree\u2014answer to a problem raised by D. Jensen and A. Ehrenfeucht","volume":"84","author":"Misercque","year":"1982","journal-title":"Proceedings of the American Mathematical Society"},{"key":"S0022481200042195_ref002","first-page":"65","article-title":"Solution de deux probl\u00e8mes pos\u00e9s par H. Simmons","volume":"33","author":"Misercque","year":"1981","journal-title":"Bulletin de la Soci\u00e9t\u00e9 Math\u00e9matique de Belgique, S\u00e9rie B"},{"key":"S0022481200042195_ref001","doi-asserted-by":"publisher","DOI":"10.4064\/fm-92-3-223-245"},{"key":"S0022481200042195_ref005","unstructured":"Misercque D. , Sur le treillis de formules fermes universelles de l'arithm\u00e9tique de Peano, Thesis, Universit\u00e9 Libre de Bruxelles, Brussels, 1985."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200042195","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,20]],"date-time":"2019-05-20T15:16:59Z","timestamp":1558365419000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200042195\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,9]]},"references-count":8,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1988,9]]}},"alternative-id":["S0022481200042195"],"URL":"https:\/\/doi.org\/10.2307\/2274587","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,9]]}}}