{"id":"https://openalex.org/W2158319126","doi":"https://doi.org/10.2307/2274262","title":"Typical ambiguity and the axiom of choice","display_name":"Typical ambiguity and the axiom of choice","publication_year":1984,"publication_date":"1984-12-01","ids":{"openalex":"https://openalex.org/W2158319126","doi":"https://doi.org/10.2307/2274262","mag":"2158319126"},"language":"en","primary_location":{"id":"doi:10.2307/2274262","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274262","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/A5000567568","display_name":"Marcel Crabb\u00e9","orcid":null},"institutions":[{"id":"https://openalex.org/I95674353","display_name":"UCLouvain","ror":"https://ror.org/02495e989","country_code":"BE","type":"education","lineage":["https://openalex.org/I95674353"]}],"countries":["BE"],"is_corresponding":true,"raw_author_name":"Marcel Crabb\u00e9","raw_affiliation_strings":["Universit\u00e9 de Louvain, 1348 Louvain-la-Neuve, Belgique"],"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Louvain, 1348 Louvain-la-Neuve, Belgique","institution_ids":["https://openalex.org/I95674353"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5000567568"],"corresponding_institution_ids":["https://openalex.org/I95674353"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.27248851,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"49","issue":"4","first_page":"1074","last_page":"1078"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9901999831199646,"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":0.9901999831199646,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.988099992275238,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9879999756813049,"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/axiom-of-choice","display_name":"Axiom of choice","score":0.8133578300476074},{"id":"https://openalex.org/keywords/ambiguity","display_name":"Ambiguity","score":0.8041749000549316},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.8034929037094116},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7681781053543091},{"id":"https://openalex.org/keywords/zermelo\u2013fraenkel-set-theory","display_name":"Zermelo\u2013Fraenkel set theory","score":0.7578269243240356},{"id":"https://openalex.org/keywords/urelement","display_name":"Urelement","score":0.5412226915359497},{"id":"https://openalex.org/keywords/conjecture","display_name":"Conjecture","score":0.5196360349655151},{"id":"https://openalex.org/keywords/proposition","display_name":"Proposition","score":0.4907304048538208},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.47436270117759705},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4653283953666687},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4157566428184509},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.4065651297569275},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.3766433000564575},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.33631253242492676},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14154183864593506},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.12839967012405396},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.08707422018051147},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.07146784663200378}],"concepts":[{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.8133578300476074},{"id":"https://openalex.org/C2780522230","wikidata":"https://www.wikidata.org/wiki/Q1140419","display_name":"Ambiguity","level":2,"score":0.8041749000549316},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.8034929037094116},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7681781053543091},{"id":"https://openalex.org/C51460","wikidata":"https://www.wikidata.org/wiki/Q191849","display_name":"Zermelo\u2013Fraenkel set theory","level":5,"score":0.7578269243240356},{"id":"https://openalex.org/C78550038","wikidata":"https://www.wikidata.org/wiki/Q1896657","display_name":"Urelement","level":5,"score":0.5412226915359497},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.5196360349655151},{"id":"https://openalex.org/C2777152325","wikidata":"https://www.wikidata.org/wiki/Q108163","display_name":"Proposition","level":2,"score":0.4907304048538208},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.47436270117759705},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4653283953666687},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4157566428184509},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.4065651297569275},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3766433000564575},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.33631253242492676},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14154183864593506},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.12839967012405396},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.08707422018051147},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.07146784663200378},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2307/2274262","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274262","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/1183741689","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741689","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:dial.uclouvain.be:boreal:55501","is_oa":false,"landing_page_url":"http://hdl.handle.net/2078.1/55501","pdf_url":null,"source":{"id":"https://openalex.org/S4306401974","display_name":"DIAL (Catholic University of Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Journal of Symbolic Logic, Vol. 49, no. 4, p. 1074-1078 (1984)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Peace, Justice and strong institutions","score":0.44999998807907104,"id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":5,"referenced_works":["https://openalex.org/W1486610321","https://openalex.org/W1976151458","https://openalex.org/W1978602663","https://openalex.org/W2061498086","https://openalex.org/W4243558221"],"related_works":["https://openalex.org/W1574825745","https://openalex.org/W2016661762","https://openalex.org/W3197691547","https://openalex.org/W2079743999","https://openalex.org/W1511476185","https://openalex.org/W102647493","https://openalex.org/W2741238752","https://openalex.org/W2105527873","https://openalex.org/W2017481539","https://openalex.org/W1538212383"],"abstract_inverted_index":{"E.":[0],"Specker":[1,27,230],"has":[2,105],"proved":[3],"that":[4,198,208,210],"the":[5,46,141,144,154,175,204],"axiom":[6,214],"of":[7,26,40,45,70,76,84,138,143,156,160,174,182,229],"choice":[8],"(AC)":[9],"is":[10,17,135,150,184,200,215],"false":[11],"in":[12,32,68,79,98,115,122,163,177,187,201],"NF":[13],"[6].":[14],"Since":[15],"AC":[16,77,183],"stratified,":[18],"one":[19,61,96,166],"can,":[20],"according":[21],"to":[22,62,107,140,167,226],"another":[23],"famous":[24],"result":[25],"[7],":[28],"prove":[29],"directly":[30,78],"\u00acAC":[31],"type":[33,64],"theory":[34],"(TT)":[35],"plus":[36,81,189,219],"some":[37,82],"finite":[38],"set":[39,124],"ambiguity":[41,192,213],"axioms,":[42],"i.e.":[43],"sentences":[44,170],"form":[47],"\u03c6":[48,50,54,58],"\u2194":[49],"+":[51,55],",":[52],"where":[53],"results":[56],"from":[57],"by":[59],"adding":[60],"its":[63],"indices.":[65],"We":[66],"shall":[67],"\u00a72":[69],"this":[71,139,199],"paper":[72],"give":[73],"a":[74,202,223,227],"disproof":[75,181],"TT":[80,188,218],"axioms":[83],"ambiguity.":[85,111],"The":[86,94,132,180],"argument":[87],"will":[88],"be":[89],"split":[90],"into":[91],"two":[92,190],"parts.":[93],"first":[95],"(contained":[97],"Proposition":[99],"2)":[100],"concerns":[101],"cardinal":[102],"arithmetic":[103],"and":[104],"nothing":[106],"do":[108],"with":[109,217],"typical":[110],"Though":[112],"carried":[113],"out":[114],"TT,":[116],"it":[117],"could":[118],"have":[119],"been":[120],"done":[121],"other":[123],"theories":[125],"such":[126],"as":[127,171],"Zermelo's":[128],"Z":[129],"or":[130],"ZF.":[131],"second":[133],"part":[134],"an":[136,157],"application":[137],"cardinals":[142],"universes":[145],"at":[146],"different":[147],"types.":[148],"This":[149],"made":[151],"possible":[152,206],"through":[153],"introduction":[155],"appropriate":[158],"definition":[159],"2":[161],"\u03b1":[162],"\u00a71":[164],"enabling":[165],"express":[168],"shifting":[169],"\u201ctyped":[172],"properties\u201d":[173],"universe,":[176],"Boffa's":[178],"sense.":[179],"then":[185],"completed":[186],"extra":[191],"axioms.":[193],"In":[194],"\u00a73,":[195],"we":[196],"show":[197],"sense":[203],"best":[205],"result:":[207],"means":[209],"every":[211],"single":[212],"consistent":[216],"AC,":[220],"thus":[221],"giving":[222],"positive":[224],"solution":[225],"conjecture":[228],"[7,":[231],"p.":[232],"119].":[233]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}