{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T05:13:17Z","timestamp":1782537197463,"version":"3.54.5"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,1,15]],"date-time":"2014-01-15T00:00:00Z","timestamp":1389744000000},"content-version":"unspecified","delay-in-days":6895,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bull. symb. log."],"published-print":{"date-parts":[[1995,3]]},"abstract":"<jats:p>In this paper we shall answer some questions in the set theory of <jats:italic>L<\/jats:italic>(\u211d), the universe of all sets constructible from the reals. In order to do so, we shall assume AD<jats:sup><jats:italic>L<\/jats:italic>(\u211d)<\/jats:sup>, the hypothesis that all 2-person games of perfect information on \u03c9 whose payoff set is in <jats:italic>L<\/jats:italic>(\u211d) are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of <jats:italic>L<\/jats:italic>(\u211d), and for reasons we cannot discuss here, ZFC + AD<jats:italic>L<\/jats:italic>(\u211d) yields the most interesting \u201ccompletion\u201d of the ZFC-theory of <jats:italic>L<\/jats:italic>(\u211d).<\/jats:p><jats:p>AD<jats:italic>L<\/jats:italic>(\u211d) implies that <jats:italic>L<\/jats:italic>(\u211d) satisfies \u201cevery wellordered set of reals is countable\u201d, so that the axiom of choice fails in <jats:italic>L<\/jats:italic>(\u211d). Nevertheless, there is a natural inner model of <jats:italic>L<\/jats:italic>(\u211d), namely HOD<jats:italic>L<\/jats:italic>(\u211d), which satisfies ZFC. (HOD is the class of all hereditarily ordinal definable sets, that is, the class of all sets <jats:italic>x<\/jats:italic> such that every member of the transitive closure of <jats:italic>x<\/jats:italic> is definable over the universe from ordinal parameters (i.e., \u201cOD\u201d). The superscript \u201c<jats:italic>L<\/jats:italic>(\u211d)\u201d indicates, here and below, that the notion in question is to be interpreted in L(R).) HOD<jats:sup><jats:italic>L<\/jats:italic>(\u211d)<\/jats:sup> is reasonably close to the full <jats:italic>L<\/jats:italic>(\u211d), in ways we shall make precise in \u00a7 1. The most important of the questions we shall answer concern HOD<jats:sup><jats:italic>L<\/jats:italic>(\u211d)<\/jats:sup>: what is its first order theory, and in particular, does it satisfy GCH?<\/jats:p><jats:p>These questions first drew attention in the 70's and early 80's. (See [4, p. 223]; also [12, p. 573] for variants involving finer notions of definability.)<\/jats:p>","DOI":"10.2307\/420947","type":"journal-article","created":{"date-parts":[[2006,4,18]],"date-time":"2006-04-18T16:43:38Z","timestamp":1145378618000},"page":"75-84","source":"Crossref","is-referenced-by-count":30,"title":["HOD<sup><i>L<\/i>(\u211d)<\/sup> is a Core Model Below \u0398"],"prefix":"10.1017","volume":"1","author":[{"given":"John R.","family":"Steel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,1,15]]},"reference":[{"key":"S1079898600008301_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21903-4"},{"key":"S1079898600008301_ref005","volume-title":"Set theory","author":"Jech","year":"1978"},{"key":"S1079898600008301_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90037-E"},{"key":"S1079898600008301_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(80)90010-8"},{"key":"S1079898600008301_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1994-1224594-7"},{"key":"S1079898600008301_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090242"},{"key":"S1079898600008301_ref015","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.85.18.6587"},{"key":"S1079898600008301_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1989-0955605-X"},{"key":"S1079898600008301_ref004","first-page":"221","volume-title":"Cabal seminar 81\u201385","author":"Delfino","year":"1988"},{"key":"S1079898600008301_ref014","unstructured":"Woodin H. , Omega Woodin cardinals from AD, unpublished lecture notes taken by E. Schimmerling."},{"key":"S1079898600008301_ref012","volume-title":"Descriptive set theory","author":"Moschovakis","year":"1980"},{"key":"S1079898600008301_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069296"},{"key":"S1079898600008301_ref006","unstructured":"Jensen R. B. , Inner models and large cardinals, forthcoming in This Journal."},{"key":"S1079898600008301_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/031\/763889"},{"key":"S1079898600008301_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0071697"}],"container-title":["Bulletin of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1079898600008301","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,14]],"date-time":"2019-05-14T19:16:57Z","timestamp":1557861417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1079898600008301\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1995,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1995,3]]}},"alternative-id":["S1079898600008301"],"URL":"https:\/\/doi.org\/10.2307\/420947","relation":{},"ISSN":["1079-8986","1943-5894"],"issn-type":[{"value":"1079-8986","type":"print"},{"value":"1943-5894","type":"electronic"}],"subject":[],"published":{"date-parts":[[1995,3]]}}}