{"id":"https://openalex.org/W2011606100","doi":"https://doi.org/10.2307/2272543","title":"Describing ordinals using functionals of transfinite type","display_name":"Describing ordinals using functionals of transfinite type","publication_year":1972,"publication_date":"1972-03-01","ids":{"openalex":"https://openalex.org/W2011606100","doi":"https://doi.org/10.2307/2272543","mag":"2011606100"},"language":"en","primary_location":{"id":"doi:10.2307/2272543","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272543","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/A5051943015","display_name":"Peter Aczel","orcid":null},"institutions":[{"id":"https://openalex.org/I28407311","display_name":"University of Manchester","ror":"https://ror.org/027m9bs27","country_code":"GB","type":"education","lineage":["https://openalex.org/I28407311"]},{"id":"https://openalex.org/I87253917","display_name":"Manchester University","ror":"https://ror.org/02aswte26","country_code":"US","type":"education","lineage":["https://openalex.org/I87253917"]}],"countries":["GB","US"],"is_corresponding":true,"raw_author_name":"Peter Aczel","raw_affiliation_strings":["Manchester University, Manchester, England"],"affiliations":[{"raw_affiliation_string":"Manchester University, Manchester, England","institution_ids":["https://openalex.org/I28407311","https://openalex.org/I87253917"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5051943015"],"corresponding_institution_ids":["https://openalex.org/I28407311","https://openalex.org/I87253917"],"apc_list":null,"apc_paid":null,"fwci":0.5357,"has_fulltext":false,"cited_by_count":18,"citation_normalized_percentile":{"value":0.60182482,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"37","issue":"1","first_page":"35","last_page":"47"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9670000076293945,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9670000076293945,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9452000260353088,"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/T14351","display_name":"Statistical and Computational Modeling","score":0.934499979019165,"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/transfinite-number","display_name":"Transfinite number","score":0.934776782989502},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8248550891876221},{"id":"https://openalex.org/keywords/cofinality","display_name":"Cofinality","score":0.6625506281852722},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.6278297901153564},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5532255172729492},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.4989454746246338},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44900402426719666},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4223648011684418},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.41272294521331787},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3644607365131378},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.14821049571037292},{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.08948501944541931},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.08129003643989563}],"concepts":[{"id":"https://openalex.org/C147358099","wikidata":"https://www.wikidata.org/wiki/Q1069891","display_name":"Transfinite number","level":2,"score":0.934776782989502},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8248550891876221},{"id":"https://openalex.org/C148070024","wikidata":"https://www.wikidata.org/wiki/Q1283623","display_name":"Cofinality","level":4,"score":0.6625506281852722},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6278297901153564},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5532255172729492},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4989454746246338},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44900402426719666},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4223648011684418},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41272294521331787},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3644607365131378},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.14821049571037292},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.08948501944541931},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.08129003643989563},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.2307/2272543","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272543","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/1183738076","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183738076","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"},{"id":"pmh:oai:pure.atira.dk:openaire_cris_publications/b8a8ee1a-5966-43ef-86fa-5e497a76a72c","is_oa":false,"landing_page_url":"https://research.manchester.ac.uk/en/publications/b8a8ee1a-5966-43ef-86fa-5e497a76a72c","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Aczel, P 1972, 'Describing Ordinals Using Functionals of Transfinite Type', The Journal of Symbolic Logic, vol. 37, no. 1, pp. 35-47. < http://dblp.uni-trier.de/db/journals/jsyml/jsyml37.html#Aczel72 >","raw_type":"article"},{"id":"pmh:oai:pure.atira.dk:publications/b8a8ee1a-5966-43ef-86fa-5e497a76a72c","is_oa":false,"landing_page_url":"https://www.research.manchester.ac.uk/portal/en/publications/describing-ordinals-using-functionals-of-transfinite-type(b8a8ee1a-5966-43ef-86fa-5e497a76a72c).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400662","display_name":"Research Explorer (The University of Manchester)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I28407311","host_organization_name":"University of Manchester","host_organization_lineage":["https://openalex.org/I28407311"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Aczel, P 1972, 'Describing Ordinals Using Functionals of Transfinite Type', The Journal of Symbolic Logic, vol. 37, no. 1, pp. 35-47. < http://dblp.uni-trier.de/db/journals/jsyml/jsyml37.html#Aczel72 >","raw_type":"article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth","score":0.4699999988079071}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":4,"referenced_works":["https://openalex.org/W132036571","https://openalex.org/W1434851050","https://openalex.org/W2066527932","https://openalex.org/W2081002360"],"related_works":["https://openalex.org/W2083332186","https://openalex.org/W2950392144","https://openalex.org/W4302039485","https://openalex.org/W2103474290","https://openalex.org/W2168631676","https://openalex.org/W1510434119","https://openalex.org/W2806221637","https://openalex.org/W2973517705","https://openalex.org/W2002854980","https://openalex.org/W2011606100"],"abstract_inverted_index":{"Bachmann,":[0],"in":[1,78,195,203],"[2]":[2],"shows":[3],"how":[4],"certain":[5],"ordinals":[6,35,128],"&lt;":[7,36,53,101,220,228,230,259],"\u03a9":[8,10,12,15,37,54,72,88,90,102,146,155,158,173,221,231,269],"(":[9,109],"=":[11,112],"1":[13,114],"where":[14,257],"\u03be":[16,21,262],"is":[17,45,134,181,248,263],"the":[18,59,127,162,165,201,254],"(1":[19],"+":[20,115],")th":[22],"infinite":[23],"initial":[24,166],"ordinal)":[25],"may":[26],"be":[27],"described":[28],"from":[29,193],"below":[30],"using":[31],"suitable":[32,124],"descriptions":[33],"of":[34,42,67,82,120,126,149,164,175,187,267],"2":[38,103,147,156,159,222,232],".":[39,73,117,132,176,182,184,223,245],"The":[40,74,118,177,246],"aim":[41],"this":[43,240],"paper":[44],"to":[46,50,161,238,253],"consider":[47],"another":[48],"approach":[49,63],"describing":[51],"ordinal":[52,178],"and":[55,93,199,251,261,271],"compare":[56],"it":[57,234],"with":[58,106],"Bachmann":[60,75,224],"method.":[61],"Our":[62,185],"will":[64,190],"use":[65],"functionals":[66],"transfinite":[68],"type":[69],"based":[70],"on":[71,122],"method":[76],"consists":[77],"denning":[79],"a":[80,123,138,264],"hierarchy":[81,139],"normal":[83,150,208],"functions":[84,151,209],"\u03d5":[85,107],"\u03b4":[86,97,142,144,153,169,171,219],":":[87,154],"\u2192":[89,157],"(i.e.":[91],"continuous":[92],"strictly":[94],"increasing)":[95],"for":[96,217,226,250],"\u2264":[98,129,145,172],"\u03b7":[99,130,179],"0":[100,108,131,180,214,227],",":[104,270],"starting":[105],"\u03bb":[110,116,229],")":[111],"\u03c9":[113,260],"definition":[119,163],"depends":[121],"description":[125,186],"This":[133],"obtained":[135],"by":[136],"defining":[137],"\u3008":[140],"F":[141,152],"\u2223":[143,170],"\u3009":[148,174],"analogously":[160],"segment":[167],"\u3008\u03d5":[168],"Note":[183],"Bachmann's":[188,196,207],"hierarchies":[189,202],"differ":[191],"slightly":[192],"those":[194],"paper.":[197],"Let":[198],"denote":[200],"[2].":[204],"Then":[205],"as":[206],"are":[210],"not":[211],"defined":[212],"at":[213],"we":[215,243],"let":[216,244],"\u03bb,":[218],"defines":[225],"but":[233],"seems":[235],"more":[236],"natural":[237],"omit":[239],"so":[241],"that":[242],"situation":[247],"analogous":[249],"leads":[252],"following":[255],"definitions:":[256],"n":[258],"limit":[265],"number":[266],"cofinality":[268]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":2}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
