{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T08:17:06Z","timestamp":1775463426232,"version":"3.50.1"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":15167,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,9]]},"abstract":"<jats:p>The central result of this paper was proved in order to settle a problem arising from B. H. Neumann's paper [10].<\/jats:p><jats:p>In [10] Neumann proved that if a finitely generated group <jats:italic>H<\/jats:italic> is recursively absolutely presentable then <jats:italic>H<\/jats:italic> is embeddable in all nontrivial algebraically-closed groups. Harry Simmons [14] clarified this by showing that a finitely generated group <jats:italic>H<\/jats:italic> is recursively absolutely presentable if and only if <jats:italic>H<\/jats:italic> can be recursively presented with solvable word-problem. Therefore, if a finitely generated group <jats:italic>H<\/jats:italic> can be recursively presented with solvable word-problem then <jats:italic>H<\/jats:italic> is embeddable in all nontrivial algebraically-closed groups.<\/jats:p><jats:p>The problem arises of characterizing those finitely generated groups which are embeddable in all nontrivial algebraically-closed groups. In this paper we prove, by a forcing argument, that if a finitely generated group <jats:italic>H<\/jats:italic> is embeddable in all non-trivial algebraically-closed groups then <jats:italic>H<\/jats:italic> can be recursively presented with solvable word-problem. Thus Neumann's result is sharp.<\/jats:p><jats:p>Our results are obtained by the method of forcing in model-theory, as developed in [1], [12]. Our method of proof has nothing to do with group-theory. We prove general results, Theorems 1 and 2 below, about constructing generic structures without certain isomorphism-types of finitely generated substructures. The formulation of these results requires the notion of Turing degree. As an application of the central result we prove Theorem 3 which gives information about the number of countable <jats:italic>K<\/jats:italic>-generic structures.<\/jats:p><jats:p>We gratefully acknowledge many helpful conversations with Harry Simmons.<\/jats:p>","DOI":"10.2307\/2272737","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:17:55Z","timestamp":1146935875000},"page":"512-520","source":"Crossref","is-referenced-by-count":28,"title":["Omitting quantifier-free types in generic structures"],"prefix":"10.1017","volume":"37","author":[{"given":"Angus","family":"Macintyre","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200079068_ref015","volume":"37","author":"Simmons","year":"1972","journal-title":"Existentially closed structures"},{"key":"S0022481200079068_ref014","article-title":"The word-problem for absolute presentations","author":"Simmons","journal-title":"Journal of the London Mathematical Society"},{"key":"S0022481200079068_ref008","first-page":"14","volume":"35","author":"Morley","year":"1970","journal-title":"The number of countable models"},{"key":"S0022481200079068_ref013","unstructured":"Shelah S. , A note on model complete models and generic models (to appear)."},{"key":"S0022481200079068_ref010","volume-title":"Proceedings of a Conference on Decision Problems in Group Theory","author":"Neumann","year":"1971"},{"key":"S0022481200079068_ref007","first-page":"265","volume-title":"Theory of models","author":"Morley","year":"1965"},{"key":"S0022481200079068_ref012","first-page":"69","article-title":"Forcing in model-theory","volume":"5","author":"Robinson","year":"1969","journal-title":"Proceedings of the Colloquium on Model Theory"},{"key":"S0022481200079068_ref004","volume-title":"Annals of Mathematics","author":"Macintyre"},{"key":"S0022481200079068_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90008-2"},{"key":"S0022481200079068_ref006","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1965-0175782-0"},{"key":"S0022481200079068_ref009","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-27.2.247"},{"key":"S0022481200079068_ref002","first-page":"30","volume":"18","author":"Craig","year":"1953","journal-title":"On axiomatizability within a system"},{"key":"S0022481200079068_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90016-7"},{"key":"S0022481200079068_ref005","unstructured":"Miller C. F. III , The word problem in quotients of a group (to appear)."},{"key":"S0022481200079068_ref011","first-page":"185","volume-title":"Studies in Pure Mathematics","author":"Neumann","year":"1971"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200079068","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T16:58:05Z","timestamp":1559235485000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200079068\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,9]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1972,9]]}},"alternative-id":["S0022481200079068"],"URL":"https:\/\/doi.org\/10.2307\/2272737","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,9]]}}}