{"id":"https://openalex.org/W2168707295","doi":"https://doi.org/10.1305/ndjfl/1093870217","title":"On the Borel classification of the isomorphism class of a countable model.","display_name":"On the Borel classification of the isomorphism class of a countable model.","publication_year":1983,"publication_date":"1983-01-01","ids":{"openalex":"https://openalex.org/W2168707295","doi":"https://doi.org/10.1305/ndjfl/1093870217","mag":"2168707295"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093870217","is_oa":false,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093870217","pdf_url":null,"source":{"id":"https://openalex.org/S78544662","display_name":"Notre Dame Journal of Formal Logic","issn_l":"0029-4527","issn":["0029-4527","1939-0726"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315698","host_organization_name":"Duke University Press","host_organization_lineage":["https://openalex.org/P4310315698","https://openalex.org/P4310315572"],"host_organization_lineage_names":["Duke University Press","Duke University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Notre Dame Journal of Formal 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/A5062397219","display_name":"Arnold W. Miller","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Arnold W. Miller","raw_affiliation_strings":["Research"],"affiliations":[{"raw_affiliation_string":"Research","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5062397219"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.6548,"has_fulltext":false,"cited_by_count":20,"citation_normalized_percentile":{"value":0.69303465,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"24","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9948999881744385,"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":0.9948999881744385,"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":0.989799976348877,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9783999919891357,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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":0.8264040946960449},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.758013129234314},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.5985351800918579},{"id":"https://openalex.org/keywords/borel-set","display_name":"Borel set","score":0.5587413311004639},{"id":"https://openalex.org/keywords/quantifier","display_name":"Quantifier (linguistics)","score":0.5190913677215576},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5067387223243713},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5046347379684448},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.5021183490753174},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.49763038754463196},{"id":"https://openalex.org/keywords/sentence","display_name":"Sentence","score":0.45936909317970276},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3675670921802521},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.18326735496520996},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.12450182437896729},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.09108197689056396},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08785685896873474}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.8264040946960449},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.758013129234314},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.5985351800918579},{"id":"https://openalex.org/C39952206","wikidata":"https://www.wikidata.org/wiki/Q21086747","display_name":"Borel set","level":2,"score":0.5587413311004639},{"id":"https://openalex.org/C2778523021","wikidata":"https://www.wikidata.org/wiki/Q5532504","display_name":"Quantifier (linguistics)","level":2,"score":0.5190913677215576},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5067387223243713},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5046347379684448},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.5021183490753174},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.49763038754463196},{"id":"https://openalex.org/C2777530160","wikidata":"https://www.wikidata.org/wiki/Q41796","display_name":"Sentence","level":2,"score":0.45936909317970276},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3675670921802521},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.18326735496520996},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.12450182437896729},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.09108197689056396},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08785685896873474},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"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/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1305/ndjfl/1093870217","is_oa":false,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093870217","pdf_url":null,"source":{"id":"https://openalex.org/S78544662","display_name":"Notre Dame Journal of Formal Logic","issn_l":"0029-4527","issn":["0029-4527","1939-0726"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315698","host_organization_name":"Duke University Press","host_organization_lineage":["https://openalex.org/P4310315698","https://openalex.org/P4310315572"],"host_organization_lineage_names":["Duke University Press","Duke University"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Notre Dame Journal of Formal Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.ndjfl/1093870217","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093870217","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W40029885","https://openalex.org/W775270089","https://openalex.org/W999833448","https://openalex.org/W1180449143","https://openalex.org/W1555138513","https://openalex.org/W2163355682","https://openalex.org/W2200966179","https://openalex.org/W2301796307","https://openalex.org/W2406763711","https://openalex.org/W4293117209"],"related_works":["https://openalex.org/W4243638815","https://openalex.org/W2091162291","https://openalex.org/W2007535792","https://openalex.org/W3149940161","https://openalex.org/W3105729805","https://openalex.org/W2094635717","https://openalex.org/W4214646115","https://openalex.org/W1482930854","https://openalex.org/W4296980826","https://openalex.org/W1482321664"],"abstract_inverted_index":{"Introduction":[0],"For":[1,26,128],"p,":[2],"a":[3,53,69,97,112,130],"countable":[4],"similarity":[5,15],"type,":[6],"let":[7,32],"Xp":[8,41,84],"be":[9,34,123],"the":[10,35,79,124,133,139],"space":[11],"of":[12,14,30,37,40,56,72,81,83,88,100,115,117,138],"structures":[13],"type":[16],"p":[17],"whose":[18],"universe":[19],"is":[20,52,68,77,111],"\u03c9":[21],"(see":[22,90],"[13],":[23],"Section":[24],"3).":[25],"any":[27,64,129],"element":[28],"&":[29],"Xp,":[31],"\\d/":[33],"set":[36,80],"all":[38],"elements":[39,82],"which":[42,85],"are":[43,86,136],"isomorphic":[44],"to":[45],"CO.":[46],"Scott":[47,101],"[10]":[48],"showed":[49,61],"that":[50,62,75],"\\\u03b2/":[51],"Borel":[54],"subset":[55],"Xp.":[57],"In":[58,103],"fact,":[59],"he":[60],"for":[63,96],"such":[65,74],"(2/":[66],"there":[67],"sentence":[70],"\u03b8":[71,89],"L\u03c9\u03c7\u03c9":[73],"\\C0":[76],"exactly":[78],"models":[87],"[":[91],"1":[92,132],"],":[93],"Ch.":[94],"VII,":[95],"good":[98],"write-up":[99],"sentences).":[102],"[13]":[104],"Vaught":[105],"considerably":[106],"strengthened":[107],"Scott's":[108],"result.":[109],"There":[110],"natural":[113],"hierarchy":[114],"formulas":[116,135],"L\u03c9\u03c7\u03c9.":[118],"Let":[119],"\u03a0Q":[120],"=":[121],"\u03a3Q":[122],"quantifier-free":[125],"first-order":[126],"formulas.":[127],">":[131],"\u03a0\u00b0":[134],"those":[137],"form:":[140]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}