{"id":"https://openalex.org/W2042415135","doi":"https://doi.org/10.2307/2273190","title":"Theories with exactly three countable models and theories with algebraic prime models","display_name":"Theories with exactly three countable models and theories with algebraic prime models","publication_year":1980,"publication_date":"1980-06-01","ids":{"openalex":"https://openalex.org/W2042415135","doi":"https://doi.org/10.2307/2273190","mag":"2042415135"},"language":"en","primary_location":{"id":"doi:10.2307/2273190","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273190","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/I98640878","display_name":"Bedford College","ror":"https://ror.org/01aa0nz04","country_code":"GB","type":"education","lineage":["https://openalex.org/I4405252469","https://openalex.org/I98640878"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Anand Pillay","raw_affiliation_strings":["Bedford College, London NW1 4NS, England","#N#        Bedford College"],"affiliations":[{"raw_affiliation_string":"Bedford College, London NW1 4NS, England","institution_ids":["https://openalex.org/I98640878"]},{"raw_affiliation_string":"#N#        Bedford College","institution_ids":["https://openalex.org/I98640878"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110147319"],"corresponding_institution_ids":["https://openalex.org/I98640878"],"apc_list":null,"apc_paid":null,"fwci":1.4015,"has_fulltext":false,"cited_by_count":7,"citation_normalized_percentile":{"value":0.80253623,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"45","issue":"2","first_page":"302","last_page":"310"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9987000226974487,"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":0.9987000226974487,"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":0.9751999974250793,"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"}},{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.949400007724762,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.8378826975822449},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8054466247558594},{"id":"https://openalex.org/keywords/model-theory","display_name":"Model theory","score":0.599686861038208},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5994384288787842},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5680257678031921},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5148400664329529},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5127342939376831},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.462529718875885}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.8378826975822449},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8054466247558594},{"id":"https://openalex.org/C47030870","wikidata":"https://www.wikidata.org/wiki/Q467606","display_name":"Model theory","level":2,"score":0.599686861038208},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5994384288787842},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5680257678031921},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5148400664329529},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5127342939376831},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.462529718875885},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273190","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273190","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/1183740561","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183740561","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":4,"referenced_works":["https://openalex.org/W2023328148","https://openalex.org/W2097507397","https://openalex.org/W2167651941","https://openalex.org/W2799061192"],"related_works":["https://openalex.org/W4300828264","https://openalex.org/W2216555165","https://openalex.org/W3105928773","https://openalex.org/W2106480630","https://openalex.org/W177522066","https://openalex.org/W1991893284","https://openalex.org/W1981494362","https://openalex.org/W2962677089","https://openalex.org/W2101752691","https://openalex.org/W1994483064"],"abstract_inverted_index":{"We":[0,88,173],"prove":[1,180],"first":[2,162],"that":[3,40,90,113,148,167],"if":[4,41,114,149],"T":[5,20,26,42,52,62,74,115],"is":[6,27,43,63,116,145,160,199,208],"a":[7,35,94,117,122,209,240],"countable":[8,17,118],"complete":[9],"theory":[10,119],"with":[11,75],"n(T)":[12,58,76,141,150,154,168],",":[13,21],"the":[14,30,45,49,85,101,109,127,161,181,184,188,192,195,202,225,228],"number":[15],"of":[16,19,33,55,72,104,129,136,177,191,212,227],"models":[18],"equal":[22],"to":[23,29,179,218,231],"three,":[24],"then":[25,61,140,153],"similar":[28],"Ehrenfeucht":[31,50,86],"example":[32],"such":[34,91],"theory.":[36],"Woodrow":[37],"[4]":[38],"showed":[39],"in":[44,108,124,194,201],"same":[46],"language":[47],"as":[48,207],"example,":[51],"has":[53,121],"elimination":[54],"quantifiers,":[56],"and":[57,78,230],"=":[59],"3":[60],"very":[64],"much":[65],"like":[66],"this":[67],"example.":[68,87],"All":[69],"known":[70,146],"examples":[71],"theories":[73,92],"finite":[77,189],"greater":[79],"than":[80],"one":[81],"are":[82,93,134,216],"based":[83],"on":[84,187],"feel":[89],"pathological":[95],"case.":[96],"Our":[97],"second":[98],"theorem":[99,107],"strengthens":[100],"main":[102],"result":[103,159,182],"[2].":[105],"The":[106],"present":[110],"paper":[111],"says":[112],"which":[120,125],"model":[123],"all":[126],"elements":[128,193],"some":[130],"infinite":[131],"definable":[132],"set":[133],"algebraic":[135],"uniformly":[137],"bounded":[138],"degree,":[139],"\u2265":[142,169],"4.":[143],"It":[144],"[3]":[147],"&gt;":[151,155],"1,":[152],"3,":[156],"so":[157],"our":[158],"nontrivial":[163],"step":[164],"towards":[165],"proving":[166],"\u2135":[170],"0":[171],".":[172],"would":[174],"also":[175],"like,":[176],"course,":[178],"without":[183],"uniform":[185],"bound":[186],"degrees":[190],"subset.":[196],"Theorem":[197,213],"2.1":[198],"included":[200],"author's":[203],"Ph.":[204,241],"D.":[205,242],"thesis,":[206],"weaker":[210],"version":[211],"3.7.":[214],"Thanks":[215],"due":[217],"Harry":[219],"Simmons":[220],"for":[221,234],"his":[222,235],"suggestions":[223],"concerning":[224],"presentation":[226],"material,":[229],"Wilfrid":[232],"Hodges":[233],"advice":[236],"while":[237],"I":[238],"was":[239],"student.":[243]},"counts_by_year":[{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
