{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,31]],"date-time":"2022-03-31T15:35:59Z","timestamp":1648740959775},"reference-count":4,"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":14164,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,6]]},"abstract":"<jats:p>The following is a classical result:<\/jats:p><jats:p>Theorem 1.1. <jats:italic>A complete atomic Boolean algebra is isomorphic to a power set algebra<\/jats:italic> [2, <jats:italic>p<\/jats:italic>. 70].<\/jats:p><jats:p>One of the consequences of [3] is: If <jats:italic>M<\/jats:italic> is a countable standard model of ZF and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120005372X_inline1\" \/> is a countable (in <jats:italic>M<\/jats:italic>) model of a complete \u2135<jats:sub>0<\/jats:sub>-categorical theory <jats:italic>T<\/jats:italic>, then there is a countable standard model <jats:italic>N<\/jats:italic> of ZF and a \u039b \u2208 <jats:italic>N<\/jats:italic> such that the Boolean algebra of definable (in <jats:italic>T<\/jats:italic> with parameters from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120005372X_inline2\" \/>) subsets of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120005372X_inline3\" \/> is isomorphic to the power set algebra of \u039b in <jats:italic>N<\/jats:italic>. In particular if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120005372X_inline4\" \/> and <jats:italic>T<\/jats:italic> the theory of equality with additional axioms asserting the existence of at least <jats:italic>n<\/jats:italic> distinct elements for each <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9<\/jats:italic>, then there is an <jats:italic>N<\/jats:italic> and \u039b \u2208 <jats:italic>N<\/jats:italic> with \u3008<jats:italic>P<jats:sup>N<\/jats:sup><\/jats:italic>(\u039b), \u2286\u3009 isomorphic to the countable, atomic, incomplete Boolean algebra of the finite and cofinite subsets of <jats:italic>\u03c9<\/jats:italic>.<\/jats:p><jats:p>From the above we see that some incomplete Boolean algebras can be realized as power sets in standard models of ZF.<\/jats:p><jats:p>Definition 1.1. A countable Boolean algebra \u3008B, \u2264\u3009 is a pseudo-power set if there is a countable standard model of ZF, <jats:italic>N<\/jats:italic> and a set \u039b \u2208 <jats:italic>N<\/jats:italic> such that<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S002248120005372X_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>It is clear that a pseudo-power set is atomic.<\/jats:p>","DOI":"10.2307\/2271897","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:35:13Z","timestamp":1146936913000},"page":"167-170","source":"Crossref","is-referenced-by-count":0,"title":["An algebraic characterization of power set in countable standard models of ZF"],"prefix":"10.1017","volume":"40","author":[{"given":"George","family":"Metakides","sequence":"first","affiliation":[]},{"given":"J. M.","family":"Plotkin","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120005372X_ref003","first-page":"388","volume":"34","author":"Plotkin","year":"1969","journal-title":"Generic embeddings"},{"key":"S002248120005372X_ref002","volume-title":"Lectures on Boolean algebras","author":"Halmos","year":"1963"},{"key":"S002248120005372X_ref001","volume-title":"Lecture Notes in Mathematics","volume":"223","author":"Felgner","year":"1971"},{"key":"S002248120005372X_ref004","first-page":"201","article-title":"A model of set theory  over a given Boolean algebra","volume":"17","author":"Weglorz","year":"1969","journal-title":"Bulletin de L'Acad\u00e9mie Polonaise des Sciences"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120005372X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T15:40:37Z","timestamp":1559144437000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120005372X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,6]]},"references-count":4,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1975,6]]}},"alternative-id":["S002248120005372X"],"URL":"https:\/\/doi.org\/10.2307\/2271897","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,6]]}}}