{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T15:38:17Z","timestamp":1649000297887},"reference-count":7,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":14256,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1975,3]]},"abstract":"<jats:p>Takeuti [6] has suggested the need for higher axioms for set theory which are analogous to higher axioms of infinity, but which claim that power sets are in some sense large. In this paper we investigate a reflection axiom of this sort (Axiom T).<\/jats:p><jats:p>In \u00a71, we introduce Axiom T and explore some related axioms. A technical lemma involving an elementary embedding is developed in \u00a72 which allows us, in \u00a73, to prove the relative consistency of Axiom T.<\/jats:p><jats:p>The reader is assumed to be familiar with ramified forcing languages and the usual techniques of forcing. A suitable treatment of these subjects is given by Takeuti and Zaring [7].<\/jats:p><jats:p>ZFC, GBC, ZF and GB are the set theories of Zermelo-Fraenkel and of G\u00f6del-Bernays, with and without the axiom of choice (AC). CH is the continuum hypothesis.<\/jats:p><jats:p>For \u03b1 an ordinal, define<\/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=\"S0022481200054256_eqnU01\" \/><\/jats:disp-formula><\/jats:p><jats:p>For <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub> \u2208 <jats:italic>p<\/jats:italic><jats:sub>\u03b1<\/jats:sub> let <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub> \u2264 <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub> mean that <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub> \u2287 <jats:italic>p<\/jats:italic><jats:sub>2<\/jats:sub> (thus <jats:italic>p<\/jats:italic><jats:sub>1<\/jats:sub> is the stronger forcing condition). It is well known that <jats:italic>P<\/jats:italic><jats:sub>\u03b1<\/jats:sub> satisfies the countable chain condition [5].<\/jats:p>","DOI":"10.2307\/2272269","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:33:55Z","timestamp":1146936835000},"page":"48-54","source":"Crossref","is-referenced-by-count":0,"title":["A large power set axiom"],"prefix":"10.1017","volume":"40","author":[{"given":"Paul E.","family":"Cohen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200054256_ref006","first-page":"439","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Takeuti","year":"1971"},{"key":"S0022481200054256_ref005","first-page":"357","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Shoenfield","year":"1971"},{"key":"S0022481200054256_ref003","volume-title":"Set theory and the continuum hypothesis","author":"Cohen","year":"1966"},{"key":"S0022481200054256_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-8751-0"},{"key":"S0022481200054256_ref004","first-page":"445","article-title":"Classifying inaccessible cardinals","volume":"8","author":"Hanf","year":"1961","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200054256_ref002","unstructured":"Cohen P. E. , Some applications of forcing in set theory, Doctoral Dissertation, University of Illinois, Champaign-Urbana, 1972."},{"key":"S0022481200054256_ref001","first-page":"579","volume":"39","author":"Cohen","year":"1974","journal-title":"Models of set theory with more real numbers than ordinals"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200054256","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T16:06:04Z","timestamp":1559145964000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200054256\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1975,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1975,3]]}},"alternative-id":["S0022481200054256"],"URL":"https:\/\/doi.org\/10.2307\/2272269","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1975,3]]}}}