{"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":1767198393408,"version":"build-2238731810"},"reference-count":24,"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:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Let\n                    <jats:italic>\u03ba<\/jats:italic>\n                    and\n                    <jats:italic>\u03bb<\/jats:italic>\n                    be infinite cardinals such that\n                    <jats:italic>\u03bb<\/jats:italic>\n                    \u2264\n                    <jats:italic>\u03bb<\/jats:italic>\n                    (we have new information for the case when\n                    <jats:italic>\u03ba<\/jats:italic>\n                    \u2264\n                    <jats:italic>\u03bb<\/jats:italic>\n                    ). Let\n                    <jats:italic>T<\/jats:italic>\n                    be a theory in\n                    <jats:italic>\n                      L\n                      <jats:sub>\u03ba<\/jats:sub>\n                    <\/jats:italic>\n                    +,\n                    <jats:sub>\n                      <jats:italic>\u03c9<\/jats:italic>\n                    <\/jats:sub>\n                    of cardinality at most\n                    <jats:italic>\u03ba<\/jats:italic>\n                    , let\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline1.png\"\/>\n                    . Now define\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_eqnU1.png\"\/>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    Our main concept in this paper is\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline2.png\"\/>\n                    is a theory in\n                    <jats:italic>\n                      L\n                      <jats:sub>\u03ba<\/jats:sub>\n                    <\/jats:italic>\n                    +,\n                    <jats:sub>\n                      <jats:italic>\u03c9<\/jats:italic>\n                    <\/jats:sub>\n                    of cardinality\n                    <jats:italic>\u03ba<\/jats:italic>\n                    at most, and\n                    <jats:italic>\u03c6(x, y)<\/jats:italic>\n                    \u03f5\n                    <jats:italic>\n                      L\n                      <jats:sub>\u03ba<\/jats:sub>\n                    <\/jats:italic>\n                    +,\n                    <jats:sub>\n                      <jats:italic>\u03c9<\/jats:italic>\n                    <\/jats:sub>\n                    }. This concept is interesting because of\n                  <\/jats:p>\n                  <jats:p>\n                    Theorem 1.\n                    <jats:italic>\n                      Let T \u2286 L\n                      <jats:sub>\u03ba<\/jats:sub>\n                      +,\n                      <jats:sub>\u03c9<\/jats:sub>\n                      of cardinality \u2264 \u03ba, and\n                    <\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline1.png\"\/>\n                    .\n                    <jats:italic>If<\/jats:italic>\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_eqnU2.png\"\/>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:italic>then (\u2200\u03c7 &gt; \u03ba)I(\u03c7, T)<\/jats:italic>\n                    = 2\n                    <jats:italic>\n                      <jats:sup>\u03c7<\/jats:sup>\n                      (where I(\u03c7, T) stands for the number of isomorphism types of models of T of cardinality \u03c7\n                    <\/jats:italic>\n                    ).\n                  <\/jats:p>\n                  <jats:p>\n                    Many years ago the second author proved that\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline3.png\"\/>\n                    . Here we continue that work by proving\n                  <\/jats:p>\n                  <jats:p>\n                    Theorem 2.\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline4.png\"\/>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    Theorem 3.\n                    <jats:italic>For every<\/jats:italic>\n                    <jats:italic>\u03ba<\/jats:italic>\n                    \u2264\n                    <jats:italic>\u03bb<\/jats:italic>\n                    <jats:italic>we have<\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline5.png\"\/>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    For some\n                    <jats:italic>\u03ba<\/jats:italic>\n                    or\n                    <jats:italic>\u03bb<\/jats:italic>\n                    we have better bounds than in Theorem 3, and this is proved via a new two cardinal theorem.\n                  <\/jats:p>\n                  <jats:p>\n                    Theorem 4.\n                    <jats:italic>\n                      For every T \u2286 L\n                      <jats:sub>\u03ba<\/jats:sub>\n                      +,\n                      <jats:sub>\u03c9<\/jats:sub>\n                      , and any set of formulas\n                    <\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline6.png\"\/>\n                    \u2286\n                    <jats:italic>\n                      L\n                      <jats:sub>\u03ba<\/jats:sub>\n                      +,\n                      <jats:sub>\u03c9<\/jats:sub>\n                      such that\n                    <\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline6.png\"\/>\n                    <jats:italic>\n                      T \u2287 L\n                      <jats:sub>\u03ba<\/jats:sub>\n                      +,\n                      <jats:sub>\u03c9<\/jats:sub>\n                      , if T is\n                    <\/jats:italic>\n                    (\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline6.png\"\/>\n                    ,\n                    <jats:italic>\u03bc)-unstable for \u03bc satisfying<\/jats:italic>\n                    <jats:italic>\n                      \u03bc\n                      <jats:sup>\u03bc*(\u03bb,\u03ba)<\/jats:sup>\n                      = \u03bc then T is\n                    <\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline6.png\"\/>\n                    -\n                    <jats:italic>unstable (i.e. for every \u03c7<\/jats:italic>\n                    \u2265\n                    <jats:italic>\u03bb, T is<\/jats:italic>\n                    (\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200031169_inline6.png\"\/>\n                    ,\n                    <jats:italic>\n                      \u03c7)-unstable). Moreover, T is L\n                      <jats:sub>\u03ba<\/jats:sub>\n                      +,\n                      <jats:sub>\u03c9<\/jats:sub>\n                      -unstable\n                    <\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    In the second part of the paper, we show that always in the applications it is possible to replace the function\n                    <jats:italic>I<\/jats:italic>\n                    (\n                    <jats:italic>\u03c7<\/jats:italic>\n                    ,\n                    <jats:italic>T<\/jats:italic>\n                    ) by the function\n                    <jats:italic>IE(\u03c7<\/jats:italic>\n                    ,\n                    <jats:italic>T<\/jats:italic>\n                    ), and we give an application of the theorems to Boolean powers.\n                  <\/jats:p>","DOI":"10.2307\/2274053","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:17:35Z","timestamp":1146939455000},"page":"302-322","source":"Crossref","is-referenced-by-count":8,"title":["On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. 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Part B, submitted to this Journal."}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200031169","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:39:05Z","timestamp":1679467145000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200031169\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,6]]},"references-count":24,"aliases":["10.1017\/s0022481200031169"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1986,6]]}},"alternative-id":["S0022481200031169"],"URL":"https:\/\/doi.org\/10.2307\/2274053","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,6]]}}}