{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T08:39:33Z","timestamp":1648975173536},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":6620,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1995,12]]},"abstract":"In this paper, we sketch the development\n of two important themes of modern set theory, both\n of which can be regarded as growing out of work of\n Kurt G\u00f6del. We begin with a review of some basic\n concepts and conventions of set theory.<\/jats:p>\n \u00a70<\/jats:bold>. The ordinal\n numbers were Georg Cantor's deepest contribution\n to mathematics. After the natural numbers 0, 1, \u2026,\n n<\/jats:italic>, \u2026 comes the first\n infinite ordinal number \u03c9, followed by \u03c9 + 1, \u03c9 +\n 2, \u2026, \u03c9 + \u03c9, \u2026 and so forth. \u03c9 is the first\n limit ordinal<\/jats:italic> as it is\n neither 0 nor a successor ordinal. We follow the\n von Neumann convention, according to which each\n ordinal number \u03b1 is identified with the set {\u03bd \u2223 \u03bd\n \u03b1} of its predecessors. The \u2208 relation on ordinals\n thus coincides with <. We have 0 = \u2205 and \u03b1 + 1\n = \u03b1 \u222a {\u03b1}. According to the usual set-theoretic\n conventions, \u03c9 is identified with the first\n infinite cardinal \u21350<\/jats:sub>,\n similarly for the first uncountable ordinal number\n \u03c91<\/jats:sub> and the first uncountable\n cardinal number \u21351<\/jats:sub>, etc. We\n thus arrive at the following picture:<\/jats:p>\n <\/jats:disp-formula><\/jats:p>\n The von Neumann\n hierarchy<\/jats:italic> divides the class\n V<\/jats:italic> of all sets into a\n hierarchy of sets\n V<\/jats:italic>\u03b1<\/jats:sub>\n indexed by the ordinal numbers. The recursive\n definition reads: \n (where } is\n the power set<\/jats:italic> of\n x<\/jats:italic>);<\/jats:p>\n V<\/jats:italic>\u03bb<\/jats:sub>\n =\n \u222av<\/jats:italic><\u03bb<\/jats:sub>Vv<\/jats:sub><\/jats:italic>\n for limit ordinals \u03bb. We can represent this\n hierarchy by the following picture.<\/jats:p>","DOI":"10.2307\/421129","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:08:20Z","timestamp":1146985700000},"page":"393-407","source":"Crossref","is-referenced-by-count":17,"title":["Inner Models and Large Cardinals"],"prefix":"10.1017","volume":"1","author":[{"given":"Ronald","family":"Jensen","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600008003_ref003","unstructured":"Jensen R. , Measures of\n order 0, unpublished\n manuscript."},{"key":"S1079898600008003_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S1079898600008003_ref001","volume-title":"Constructability","author":"Devon","year":"1984"},{"key":"S1079898600008003_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1989-0955605-X"},{"key":"S1079898600008003_ref009","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"},{"key":"S1079898600008003_ref008","unstructured":"Steel J. R. , The core\n model iterability problem, unpublished\n manuscript."},{"key":"S1079898600008003_ref007","volume-title":"Lecture notes in logic","volume":"3","author":"Mitchell","year":"1994"},{"key":"S1079898600008003_ref004","volume-title":"Set theory: An introduction to independence proofs","author":"Kunen","year":"1980"},{"key":"S1079898600008003_ref002","volume-title":"Set theory: An introduction to large cardinals","author":"Drake","year":"1974"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600008003","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,6,23]],"date-time":"2020-06-23T09:05:01Z","timestamp":1592903101000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600008003\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,12]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1995,12]]}},"alternative-id":["S1079898600008003"],"URL":"http:\/\/dx.doi.org\/10.2307\/421129","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":["Logic","Philosophy"],"published":{"date-parts":[[1995,12]]}}}