{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:26:28Z","timestamp":1771471588382,"version":"3.50.1"},"reference-count":20,"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":13068,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,6]]},"abstract":"<jats:p>One of the most significant by-products of the study of admissible sets with urelements is the emphasis it has given to recursively saturated models. As suggested in [Schlipf, 1977], countable recursively saturated models (for finite languages) possess many of the desirable properties of saturated and special models. The notion of resplendency was introduced to isolate some of these desirable properties. In \u00a7\u00a71 and 2 of this paper we study these parallels, showing how they can be exploited to give new proofs of some traditional model theoretic theorems. This yields both pedagogical and philosophical advantages: pedagogical since countable recursively saturated models are easier to build and manipulate than saturated and special models; philosophical since it shows that uncountable models \u2014 which the downward Lowenheim\u2013Skolem theorem tells us are in some sense not basic in the study of countable theories \u2014 are not needed in model theoretic proofs of these theorems. In \u00a73 we apply our local results to get results about resplendent models of ZF set theory and PA (Peano arithmetic). In \u00a74 we shall examine certain analogous results for admissible languages, most similar to, and seemingly generally slightly weaker than, already known results. (The Chang\u2013Makkai sort of result, however, is new.)<\/jats:p><jats:p>Although this paper is an outgrowth of work with admissible sets with urelements, I have tried to keep it as accessible as possible to those with a background only in finitary model theory. Thus \u00a7\u00a71,2, and 3 should not involve any work with admissible sets. \u00a74, however, is concerned with some admissible analogues to results in \u00a72 and necessarily uses certain technical results of \u00a711 5 of [Schlipf, 1977].<\/jats:p>","DOI":"10.2307\/2272817","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:47:35Z","timestamp":1146952055000},"page":"183-206","source":"Crossref","is-referenced-by-count":32,"title":["Toward model theory through recursive saturation"],"prefix":"10.1017","volume":"43","author":[{"given":"John Stewart","family":"Schlipf","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049537_ref020","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-01380-9","volume-title":"Modern algebra","author":"van der Waerden","year":"1964"},{"key":"S0022481200049537_ref019","first-page":"376","volume-title":"The theory of models","author":"Svenonius","year":"1965"},{"key":"S0022481200049537_ref015","doi-asserted-by":"publisher","DOI":"10.1016\/S0003-4843(70)80006-7"},{"key":"S0022481200049537_ref008","volume-title":"Model theory for infinitary logic","author":"Keisler","year":"1971"},{"key":"S0022481200049537_ref012","unstructured":"Nadel M. [1971], Model theory in admissible sets, Ph.D. Thesis, University of Wisconsin."},{"key":"S0022481200049537_ref017","first-page":"282","volume":"41","author":"Schlipf","year":"1976","journal-title":"Recursively saturated models and the Chang-Makkai Theorem"},{"key":"S0022481200049537_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90003-3"},{"key":"S0022481200049537_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1973-13258-6"},{"key":"S0022481200049537_ref009","first-page":"27","volume-title":"Two papers on the predicate calculus, Memoirs of the American Mathematical Society","author":"Kleene","year":"1952"},{"key":"S0022481200049537_ref007","first-page":"158","volume-title":"The theory of models","author":"Keisler","year":"1965"},{"key":"S0022481200049537_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0066789"},{"key":"S0022481200049537_ref016","unstructured":"Schlipf J. [1975], Some hyperelementary aspects of model theory, Ph.D. Thesis, University of Wisconsin."},{"key":"S0022481200049537_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200049537_ref005","first-page":"934","article-title":"On \u03a311-relations on special models","volume":"15","author":"Chang","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200049537_ref014","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(77)90009-2"},{"key":"S0022481200049537_ref003","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200049537_ref013","unstructured":"Ressayre J. P. [1972], Mod\u00e9les bool\u00e9ens et langages du 1er order, Doctoral Thesis, University of Paris."},{"key":"S0022481200049537_ref018","volume-title":"Annals of Mathematical Logic","author":"Schlipf","year":"1977"},{"key":"S0022481200049537_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0080973"},{"key":"S0022481200049537_ref004","volume-title":"Model theory","author":"Chang","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049537","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T19:58:05Z","timestamp":1558987085000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049537\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,6]]},"references-count":20,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1978,6]]}},"alternative-id":["S0022481200049537"],"URL":"https:\/\/doi.org\/10.2307\/2272817","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,6]]}}}