{"id":"https://openalex.org/W1964785514","doi":"https://doi.org/10.2307/2273329","title":"A note on finitely generated models","display_name":"A note on finitely generated models","publication_year":1983,"publication_date":"1983-03-01","ids":{"openalex":"https://openalex.org/W1964785514","doi":"https://doi.org/10.2307/2273329","mag":"1964785514"},"language":"en","primary_location":{"id":"doi:10.2307/2273329","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273329","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5110147319","display_name":"Anand Pillay","orcid":"https://orcid.org/0000-0002-2471-4097"},"institutions":[{"id":"https://openalex.org/I204730241","display_name":"Universit\u00e9 Paris Cit\u00e9","ror":"https://ror.org/05f82e368","country_code":"FR","type":"education","lineage":["https://openalex.org/I204730241"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Anand Pillay","raw_affiliation_strings":["Universit\u00e9 De Paris VII, 75221 Paris, France","Universite de Paris VII"],"affiliations":[{"raw_affiliation_string":"Universit\u00e9 De Paris VII, 75221 Paris, France","institution_ids":["https://openalex.org/I204730241"]},{"raw_affiliation_string":"Universite de Paris VII","institution_ids":["https://openalex.org/I204730241"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110147319"],"corresponding_institution_ids":["https://openalex.org/I204730241"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.0759987,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"48","issue":"1","first_page":"163","last_page":"166"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9287999868392944,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.8106203675270081},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8086581826210022},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5783470273017883},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.5190887451171875},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.49208396673202515},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4876495599746704},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4855074882507324},{"id":"https://openalex.org/keywords/stallings-theorem-about-ends-of-groups","display_name":"Stallings theorem about ends of groups","score":0.47163888812065125},{"id":"https://openalex.org/keywords/finitely-generated-abelian-group","display_name":"Finitely-generated abelian group","score":0.4446724057197571},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.4197169542312622},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.389621376991272},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10503539443016052}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.8106203675270081},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8086581826210022},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5783470273017883},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.5190887451171875},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.49208396673202515},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4876495599746704},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4855074882507324},{"id":"https://openalex.org/C58249144","wikidata":"https://www.wikidata.org/wiki/Q7597316","display_name":"Stallings theorem about ends of groups","level":3,"score":0.47163888812065125},{"id":"https://openalex.org/C162860070","wikidata":"https://www.wikidata.org/wiki/Q142140","display_name":"Finitely-generated abelian group","level":2,"score":0.4446724057197571},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.4197169542312622},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.389621376991272},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10503539443016052},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273329","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273329","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183741199","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741199","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W1523486281","https://openalex.org/W2005659848","https://openalex.org/W2069666741"],"related_works":["https://openalex.org/W3105973910","https://openalex.org/W2952241151","https://openalex.org/W1590360997","https://openalex.org/W2101435336","https://openalex.org/W1981717402","https://openalex.org/W2087413461","https://openalex.org/W2089823147","https://openalex.org/W2012835417","https://openalex.org/W2321123445","https://openalex.org/W2014013251"],"abstract_inverted_index":{"A":[0,49,62,76,94,102,114,153,241,251,262],"model":[1,42,128,159],"M":[2,32,51,54,69,72,84,90,98,109,134,137],"(of":[3],"a":[4,20,27,40,245,271],"countable":[5,158],"first":[6],"order":[7],"language)":[8],"is":[9,17,26,39,55,65,73,95,110,126,138,143,150,155,165,181,210,268],"said":[10,56],"to":[11,57,97],"be":[12,58,242,259],"finitely":[13,59,139,144,151,173],"generated":[14,60,100,145,174],"if":[15,24,48,63,82,124,183],"it":[16],"prime":[18,41,74,92],"over":[19,61,75,93,146,244,261,270],"finite":[21,28,66,215,229,246,254,272],"set,":[22,247],"namely":[23],"there":[25,64],"tuple":[29],"\u0101":[30,37,67,78,127,249,264],"in":[31,68,103,195,205,222],"such":[33,70],"that":[34,71,81,107,123,156,178,207],"(":[35],"M,":[36,147],")":[38,85,252],"of":[43,113,117,129,160,168,172],"its":[44],"own":[45],"theory.":[46],"Similarly,":[47],"\u2282":[50],",":[52],"then":[53,89,148],"\u22c3":[77,263],".":[79,218,265],"(Note":[80],"Th(":[83],"has":[86,186],"Skolem":[87,187],"functions,":[88],"being":[91,99],"equivalent":[96],"by":[101],"the":[104,111,118,166,184,193,200,208,220,225,232],"usual":[105],"sense,":[106],"is,":[108],"closure":[112],"under":[115],"functions":[116],"language.)":[119],"We":[120],"show":[121,199],"here":[122,191],"N":[125,142,149],"an":[130,161,169],"\u03c9":[131,162,238],"-stable":[132,163],"theory,":[133],"\u227a":[135],"N,":[136],"generated,":[140],"and":[141,257],"generated.":[152],"corollary":[154],"any":[157],"theory":[164,185,209],"union":[167],"elementary":[170],"chain":[171],"models.":[175],"Note":[176],"again":[177],"all":[179],"this":[180],"trivial":[182],"functions.":[188],"The":[189],"result":[190],"strengthens":[192],"results":[194],"[3],":[196],"where":[197],"we":[198],"same":[201],"thing":[202],"but":[203],"assuming":[204],"addition":[206],"either":[211],"nonmultidimensional":[212],"or":[213],"with":[214],"\u03b1":[216,227],"T":[217,228],"However":[219],"proof":[221],"[3]":[223],"for":[224],"case":[226],"actually":[230],"shows":[231],"following":[233],"which":[234],"does":[235],"not":[236],"assume":[237],"-stability):":[239],"Let":[240],"atomic":[243,260,269],"tp(":[248],"/":[250],"have":[253],"Cantor-Bendixson":[255],"rank,":[256],"B":[258,267],"Then":[266],"set.":[273]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}