{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:26:33Z","timestamp":1767198393467,"version":"build-2238731810"},"reference-count":2,"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":10146,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1986,6]]},"abstract":"<jats:p>\n                    This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not\n                    <jats:italic>almost decidable<\/jats:italic>\n                    . On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven\u2014if every complete extension, in finitely many additional constant symbols, of a theory\n                    <jats:italic>T<\/jats:italic>\n                    fails to have a decidable prime model, then\n                    <jats:italic>T<\/jats:italic>\n                    has 2\n                    <jats:sup>\n                      <jats:italic>\u03c9<\/jats:italic>\n                    <\/jats:sup>\n                    nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness\u2014almost decidable.\n                  <\/jats:p>\n                  <jats:p>\n                    A tree Tr will be a subset of\n                    <jats:italic>\n                      \u03c9\n                      <jats:sup>&lt;\u03c9<\/jats:sup>\n                    <\/jats:italic>\n                    that is closed under predecessor. For elements\n                    <jats:italic>f, g<\/jats:italic>\n                    in\n                    <jats:italic>\n                      \u03c9\n                      <jats:sup>&lt;\u03c9<\/jats:sup>\n                    <\/jats:italic>\n                    \u222a\n                    <jats:italic>\n                      \u03c9\n                      <jats:sup>\u03c9<\/jats:sup>\n                    <\/jats:italic>\n                    ,\n                    <jats:italic>\u0192<\/jats:italic>\n                    \u22b2\n                    <jats:italic>g<\/jats:italic>\n                    iff\n                    <jats:sub>df<\/jats:sub>\n                    \u2200\n                    <jats:italic>i<\/jats:italic>\n                    &lt; lh(\n                    <jats:italic>\u0192<\/jats:italic>\n                    )[\n                    <jats:italic>\u0192<\/jats:italic>\n                    (\n                    <jats:italic>i<\/jats:italic>\n                    ) =\n                    <jats:italic>g<\/jats:italic>\n                    (\n                    <jats:italic>i<\/jats:italic>\n                    )].\n                  <\/jats:p>","DOI":"10.2307\/2274064","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:17:35Z","timestamp":1146939455000},"page":"412-420","source":"Crossref","is-referenced-by-count":6,"title":["Prime models and almost decidability"],"prefix":"10.1017","volume":"51","author":[{"given":"Terrence","family":"Millar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200031273_bib002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(84)90009-5"},{"key":"S0022481200031273_bib001","first-page":"171","volume":"48","author":"Millar","year":"1983","journal-title":"Omitting types, type spectrums, and decidability"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031273","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:39:15Z","timestamp":1679467155000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031273\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":2,"aliases":["10.1017\/s0022481200031273"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["S0022481200031273"],"URL":"https:\/\/doi.org\/10.2307\/2274064","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}