{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T11:01:03Z","timestamp":1698490863967},"reference-count":11,"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":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"Let T<\/jats:italic>1<\/jats:sub> be the complete first-order theory of the additive group of the integers with 1 as distinguished element (in symbols, T<\/jats:italic>1<\/jats:sub> = Th(Z, +, 1)). In this paper we prove that all models of T<\/jats:italic>1<\/jats:sub> are \u21350<\/jats:sub>-homogeneous (\u00a72), classify them (and lists of elements in them) up to isomorphism or L\u221e\u03ba<\/jats:italic><\/jats:sub>-equivalence (\u00a7\u00a73 and 4) and show that they may be as complex as arbitrary sets of real numbers from the point of view of admissible set theory (\u00a75). The results of \u00a7\u00a72 and 5 together show that while the Scott heights of all models of T<\/jats:italic>1<\/jats:sub> are \u2264 \u03c9<\/jats:italic> (by \u21350<\/jats:sub>-homogeneity) their HYP-heights form an unbounded subset of the cardinal .<\/jats:p>In addition to providing this unusual example of the relation between Scott heights and HYP-heights, the theory T<\/jats:italic>1<\/jats:sub> has served (using the homogeneity results of \u00a72) as an example for certain combinations of properties that people had looked for in stability theory (see end of \u00a74). In \u00a76 it is shown that not all models of T<\/jats:italic> = Th(Z, +) are \u21350<\/jats:sub>-homogeneous, so that the availability of the constant for 1 is essential for the result of \u00a72.<\/jats:p>The two main results of this paper (2.2 and essentially Theorem 5.3) were obtained in the summer of 1979. Later we learnt from Victor Harnik and Julia Knight that T1<\/jats:sub> is of some interest for stability theory, and were encouraged to write up our proofs.<\/jats:p>During 1982\/3 we improved the proofs and added some results.<\/jats:p>","DOI":"10.2307\/2274950","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:33:48Z","timestamp":1146954828000},"page":"1-20","source":"Crossref","is-referenced-by-count":5,"title":["On models of the elementary theory of (Z, +, 1)"],"prefix":"10.1017","volume":"55","author":[{"given":"Mark","family":"Nadel","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Stavi","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026396_ref010","first-page":"833","volume":"47","author":"Knight","year":"1983","journal-title":"Models of arithmetic and closed ideals"},{"key":"S0022481200026396_ref006","volume-title":"A mathematical introduction to logic","author":"Enderton","year":"1972"},{"key":"S0022481200026396_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-11035-5"},{"key":"S0022481200026396_ref008","volume-title":"Infinite abelian groups","volume":"1","author":"Fuchs","year":"1970"},{"key":"S0022481200026396_ref007","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-1-57-103"},{"key":"S0022481200026396_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90013-7"},{"key":"S0022481200026396_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF02765017"},{"key":"S0022481200026396_ref004","volume-title":"Model theory","author":"Chang","year":"1973"},{"key":"S0022481200026396_ref011","first-page":"33","volume":"42","author":"Nadel","year":"1977","journal-title":"The pure part of HYP"},{"key":"S0022481200026396_ref003","first-page":"531","volume":"41","author":"Barwise","year":"1976","journal-title":"An introduction to recursively saturated and resplendent models"},{"key":"S0022481200026396_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02945115"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026396","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T21:52:13Z","timestamp":1558216333000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026396\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":11,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026396"],"URL":"https:\/\/doi.org\/10.2307\/2274950","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}