{"id":"https://openalex.org/W1998072591","doi":"https://doi.org/10.1017/s0022481200031169","title":"On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. Part A","display_name":"On the number of nonisomorphic models of an infinitary theory which has the infinitary order property. Part A","publication_year":1986,"publication_date":"1986-06-01","ids":{"openalex":"https://openalex.org/W1998072591","doi":"https://doi.org/10.1017/s0022481200031169","mag":"1998072591"},"language":"en","primary_location":{"id":"doi:10.1017/s0022481200031169","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200031169","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/A5027331051","display_name":"Rami Grossberg","orcid":"https://orcid.org/0000-0002-8014-4607"},"institutions":[{"id":"https://openalex.org/I52357470","display_name":"The Ohio State University","ror":"https://ror.org/00rs6vg23","country_code":"US","type":"education","lineage":["https://openalex.org/I52357470"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Rami Grossberg","raw_affiliation_strings":["Department of Mathematics, Ohio State University, Columbus, Ohio 43210"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Ohio State University, Columbus, Ohio 43210","institution_ids":["https://openalex.org/I52357470"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024923217","display_name":"Saharon Shelah","orcid":"https://orcid.org/0000-0003-0462-3152"},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]}],"countries":["IL"],"is_corresponding":false,"raw_author_name":"Saharon Shelah","raw_affiliation_strings":["Institute of Mathematics, The Hebrew University, Jerusalem, Israel","Institute of Mathematics, The Hebrew University, Jerusalem, Israel#TAB#"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Mathematics, The Hebrew University, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]},{"raw_affiliation_string":"Institute of Mathematics, The Hebrew University, Jerusalem, Israel#TAB#","institution_ids":["https://openalex.org/I197251160"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5027331051"],"corresponding_institution_ids":["https://openalex.org/I52357470"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.4932,"has_fulltext":false,"cited_by_count":21,"citation_normalized_percentile":{"value":0.59132064,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"51","issue":"2","first_page":"302","last_page":"322"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9998000264167786,"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":0.9998000264167786,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.998199999332428,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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":0.9947999715805054,"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/cardinality","display_name":"Cardinality (data modeling)","score":0.8774325847625732},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7046619057655334},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.6094138622283936},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5439302325248718},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.5288625359535217},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5163487195968628},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4919993579387665},{"id":"https://openalex.org/keywords/cardinal-number","display_name":"Cardinal number (linguistics)","score":0.4717353582382202},{"id":"https://openalex.org/keywords/model-theory","display_name":"Model theory","score":0.44893205165863037},{"id":"https://openalex.org/keywords/isomorphism-theorem","display_name":"Isomorphism theorem","score":0.41791465878486633},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10030046105384827}],"concepts":[{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.8774325847625732},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7046619057655334},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.6094138622283936},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5439302325248718},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5288625359535217},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5163487195968628},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4919993579387665},{"id":"https://openalex.org/C2778408900","wikidata":"https://www.wikidata.org/wiki/Q1329258","display_name":"Cardinal number (linguistics)","level":2,"score":0.4717353582382202},{"id":"https://openalex.org/C47030870","wikidata":"https://www.wikidata.org/wiki/Q467606","display_name":"Model theory","level":2,"score":0.44893205165863037},{"id":"https://openalex.org/C26231708","wikidata":"https://www.wikidata.org/wiki/Q1065966","display_name":"Isomorphism theorem","level":2,"score":0.41791465878486633},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10030046105384827},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0022481200031169","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200031169","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/1183742095","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183742095","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":19,"referenced_works":["https://openalex.org/W65053296","https://openalex.org/W128909481","https://openalex.org/W1537266584","https://openalex.org/W1596591404","https://openalex.org/W1966384406","https://openalex.org/W1994831517","https://openalex.org/W2005659848","https://openalex.org/W2017769265","https://openalex.org/W2020913456","https://openalex.org/W2047282968","https://openalex.org/W2069110520","https://openalex.org/W2073020252","https://openalex.org/W2081441842","https://openalex.org/W2088386286","https://openalex.org/W2095579545","https://openalex.org/W2173438417","https://openalex.org/W2799061192","https://openalex.org/W6632234802","https://openalex.org/W6648983780"],"related_works":["https://openalex.org/W4321499188","https://openalex.org/W2331093826","https://openalex.org/W2170102489","https://openalex.org/W4388419030","https://openalex.org/W4225837288","https://openalex.org/W1973109584","https://openalex.org/W2941284322","https://openalex.org/W2168299207","https://openalex.org/W4233846194","https://openalex.org/W1998072591"],"abstract_inverted_index":{"Abstract":[0],"Let":[1,24,76],"\u03ba":[2,21,58,142],"and":[3,61,86,153,172,248],"\u03bb":[4,10,12,144],"be":[5,26],"infinite":[6],"cardinals":[7],"such":[8,181],"that":[9,120,125,182,229],"\u2264":[11,22,84,136],"(we":[13],"have":[14,138,146],"new":[15,159],"information":[16],"for":[17,100,194,204],"the":[18,101,116,221,225,232,239,244,254],"case":[19],"when":[20],"\u03bb).":[23],"T":[25,77,109,167,183,189,200,209,215],"a":[27,50,158],"theory":[28,51],"in":[29,45,52,150,231],"L\u03ba":[30,53,65,79,169,178,185,217],"+,":[31,54,66,80,170,179,186,218],"\u03c9":[32,55,81,180],"of":[33,56,73,82,103,106,108,110,175,224,253],"cardinality":[34,57,83,111],"at":[35,59],"most":[36],"\u03ba,":[37,85],"let":[38],".":[39,87,121,131,139],"Now":[40],"define":[41],"Our":[42],"main":[43],"concept":[44,69],"this":[46,154],"paper":[47],"is":[48,49,70,155,190,201,210,216,235],"most,":[60],"\u03c6(x,":[62],"y)":[63],"\u03f5":[64],"\u03c9}.":[67],"This":[68],"interesting":[71],"because":[72],"Theorem":[74,129,132,151,163],"1.":[75],"\u2286":[78,168,177],"If":[88],"then":[89,199],"(\u2200\u03c7":[90],">":[91],"\u03ba)I(\u03c7,":[92],"T)":[93,98,242],"=":[94,197],"2\u03c7":[95],"(where":[96],"I(\u03c7,":[97,241],"stands":[99],"number":[102],"isomorphism":[104],"types":[105],"models":[107],"\u03c7).":[112],"Many":[113],"years":[114],"ago":[115],"second":[117,222],"author":[118],"proved":[119,156],"Here":[122],"we":[123,145,227,249],"continue":[124],"work":[126],"by":[127,243],"proving":[128],"2.":[130],"3.":[133],"For":[134,140,165],"every\u03ba":[135],"\u03bbwe":[137],"some":[141],"or":[143],"better":[147],"bounds":[148],"than":[149],"3,":[152],"via":[157],"two":[160],"cardinal":[161],"theorem.":[162],"4.":[164],"every":[166,205],"\u03c9,":[171,187],"any":[173],"set":[174],"formulas":[176],"\u2287":[184],"if":[188],"(":[191,211],",":[192,212],"\u03bc)-unstable":[193],"\u03bc":[195,198],"satisfying\u03bc\u03bc*(\u03bb,\u03ba)":[196],"-unstable":[202],"(i.e.":[203],"\u03c7":[206],"\u2265":[207],"\u03bb,":[208],"\u03c7)-unstable).":[213],"Moreover,":[214],"\u03c9-unstable.":[219],"In":[220],"part":[223],"paper,":[226],"show":[228],"always":[230],"applications":[233],"it":[234],"possible":[236],"to":[237,256],"replace":[238],"function":[240,245],"IE(\u03c7,":[246],"T),":[247],"give":[250],"an":[251],"application":[252],"theorems":[255],"Boolean":[257],"powers.":[258]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}