{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T17:21:09Z","timestamp":1709832069638},"reference-count":33,"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":4302,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that the first order theory of the lattice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009774_inline1\" \/><jats:sup>&lt;<jats:italic>\u03c9<\/jats:italic><\/jats:sup>(<jats:italic>S<\/jats:italic>) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System <jats:italic>S<\/jats:italic> has logical complexity at least that of first order number theory and that the first order theory of the lattice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009774_inline1\" \/>(<jats:italic>S<jats:sub>\u221e<\/jats:sub><\/jats:italic>) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System <jats:italic>S<\/jats:italic><jats:sub>\u221e<\/jats:sub> has logical complexity exactly that of first order number theory. Thus, for example, the lattice of finite dimensional subspaces of a standard copy of \u2295<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub><jats:bold><jats:italic>Q<\/jats:italic><\/jats:bold> interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009774_inline1\" \/>(<jats:italic>V<jats:sub>\u221e<\/jats:sub><\/jats:italic>) of c.e. subspaces of a fully effective \u2135<jats:sub>0<\/jats:sub>-dimensional vector space <jats:italic>V<jats:sub>\u221e<\/jats:sub><\/jats:italic> and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field <jats:italic>F<jats:sub>\u221e<\/jats:sub><\/jats:italic> of countably infinite transcendence degree each have logical complexity that of first order number theory.<\/jats:p>","DOI":"10.2178\/jsl\/1190150113","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T14:13:31Z","timestamp":1197555211000},"page":"859-878","source":"Crossref","is-referenced-by-count":1,"title":["More undecidable lattices of Steinitz exchange systems"],"prefix":"10.1017","volume":"67","author":[{"given":"L. 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