{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T22:32:27Z","timestamp":1781562747244,"version":"3.54.5"},"reference-count":6,"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":5889,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1997,12]]},"abstract":"<jats:p>We present here an approach to the fine structure of <jats:italic>L<\/jats:italic> based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in <jats:italic>L<\/jats:italic>. We thereby avoid the L\u00e9vy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called \u201dHyperfine Structure Theory\u201d, as we make use of a hierarchy of structures and hull operations which refines the traditional <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub> -or <jats:italic>J<\/jats:italic><jats:sub>\u03b1<\/jats:sub>-sequences with their \u03a3<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>-hull operations.<\/jats:p><jats:p><jats:bold>\u00a71. Introduction<\/jats:bold>. In 1938, K. G\u00f6del defined the model <jats:italic>L<\/jats:italic> of set theory to show the relative consistency of Cantor's Continuum Hypothesis. <jats:italic>L<\/jats:italic> is defined as a union<\/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=\"S1079898600007423_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>of initial segments which satisfy: <jats:italic>L<\/jats:italic><jats:sub>0<\/jats:sub> = \u2205, <jats:italic>L<\/jats:italic><jats:sub>\u03bb<\/jats:sub> = \u222a<jats:sub>\u03b1&lt;\u03bb<\/jats:sub><jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub> for limit ordinals \u03bb, and, crucially, <jats:italic>L<\/jats:italic><jats:sub>\u03b1 + 1<\/jats:sub> = the collection of 1st order definable subsets of <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>. Since every transitive model of set theory must be closed under 1st order definability, <jats:italic>L<\/jats:italic> turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.<\/jats:p><jats:p>The proof of the continuum hypothesis in <jats:italic>L<\/jats:italic> is based on the very uniform hierarchical definition of the <jats:italic>L<\/jats:italic>-hierarchy. The <jats:italic>Condensation Lemma<\/jats:italic> states that if \u03c0 : <jats:italic>M<\/jats:italic> \u2192 <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub> is an elementary embedding, <jats:italic>M<\/jats:italic> transitive, then <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007423_inline1\"\/> some <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007423_inline2\"\/>; the lemma can be proved by induction on \u03b1. If a real, i.e., a subset of \u03c9, is definable over some <jats:italic>L<\/jats:italic><jats:sub>\u03b1<\/jats:sub>,then by a L\u00f6wenheim-Skolem argument it is definable over some countable <jats:italic>M<\/jats:italic> as above, and hence over some <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007423_inline3\"\/>, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1079898600007423_inline2\"\/> &lt; \u03c9<jats:sub>1<\/jats:sub>. This allows one to list the reals in <jats:italic>L<\/jats:italic> in length \u03c9<jats:sub>1<\/jats:sub> and therefore proves the Continuum Hypothesis in <jats:italic>L<\/jats:italic>.<\/jats:p>","DOI":"10.2307\/421099","type":"journal-article","created":{"date-parts":[[2006,5,7]],"date-time":"2006-05-07T07:10:58Z","timestamp":1146985858000},"page":"453-468","source":"Crossref","is-referenced-by-count":10,"title":["An Elementary Approach to the Fine Structure of <i>L<\/i>"],"prefix":"10.1017","volume":"3","author":[{"given":"Sy D.","family":"Friedman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"Koepke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600007423_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S1079898600007423_ref002","volume-title":"Fundamenta Mathematicae","author":"Friedman"},{"key":"S1079898600007423_ref006","unstructured":"Silver Jack H. , How to eliminate the fine structure from the work of Jensen, handwritten manuscript, 197?"},{"key":"S1079898600007423_ref005","unstructured":"Richardson Thomas L. , Silver machine approach to the constructible universe, Ph.D. thesis , University of California, Berkeley, 1978."},{"key":"S1079898600007423_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1990-0939805-5"},{"key":"S1079898600007423_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21723-8"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600007423","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,11]],"date-time":"2019-05-11T19:41:59Z","timestamp":1557603719000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600007423\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1997,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1997,12]]}},"alternative-id":["S1079898600007423"],"URL":"https:\/\/doi.org\/10.2307\/421099","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1997,12]]}}}