{"id":"https://openalex.org/W2058981710","doi":"https://doi.org/10.2307/2273386","title":"On the consistency of an impredicative subsystem of Quine's NF","display_name":"On the consistency of an impredicative subsystem of Quine's NF","publication_year":1982,"publication_date":"1982-03-01","ids":{"openalex":"https://openalex.org/W2058981710","doi":"https://doi.org/10.2307/2273386","mag":"2058981710"},"language":"en","primary_location":{"id":"doi:10.2307/2273386","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273386","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":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2078.1/57616","any_repository_has_fulltext":true},"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 (U.C.L.), Institut Sup\u00e9rieur de Philosophie, Louvain-La-Neuve, Belgium","Universit\u00e9 de Louvain (U.C.L.)"],"affiliations":[{"raw_affiliation_string":"Universit\u00e9 de Louvain (U.C.L.), Institut Sup\u00e9rieur de Philosophie, Louvain-La-Neuve, Belgium","institution_ids":["https://openalex.org/I95674353"]},{"raw_affiliation_string":"Universit\u00e9 de Louvain (U.C.L.)","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":21,"citation_normalized_percentile":{"value":0.15530533,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"47","issue":"1","first_page":"131","last_page":"136"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.984499990940094,"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"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.984499990940094,"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"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.9505000114440918,"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/T12170","display_name":"History and Theory of Mathematics","score":0.9480000138282776,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/predicative-expression","display_name":"Predicative expression","score":0.9127205610275269},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.7818840742111206},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.7031194567680359},{"id":"https://openalex.org/keywords/quine","display_name":"Quine","score":0.6577196717262268},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5418593883514404},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5192742347717285},{"id":"https://openalex.org/keywords/fragment","display_name":"Fragment (logic)","score":0.5148047804832458},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.4409518539905548},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.42921558022499084},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.33449602127075195},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.31928038597106934},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.27304327487945557},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.16086167097091675},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.11272889375686646}],"concepts":[{"id":"https://openalex.org/C129131279","wikidata":"https://www.wikidata.org/wiki/Q1931259","display_name":"Predicative expression","level":2,"score":0.9127205610275269},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.7818840742111206},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.7031194567680359},{"id":"https://openalex.org/C9270016","wikidata":"https://www.wikidata.org/wiki/Q1047886","display_name":"Quine","level":2,"score":0.6577196717262268},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5418593883514404},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5192742347717285},{"id":"https://openalex.org/C2776235265","wikidata":"https://www.wikidata.org/wiki/Q18392052","display_name":"Fragment (logic)","level":2,"score":0.5148047804832458},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.4409518539905548},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.42921558022499084},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.33449602127075195},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.31928038597106934},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.27304327487945557},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.16086167097091675},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.11272889375686646},{"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:10.2307/2273386","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273386","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/1183740945","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183740945","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:57616","is_oa":true,"landing_page_url":"http://hdl.handle.net/2078.1/57616","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Journal of Symbolic Logic, Vol. 47, no. 1, p. 131-136 (1982)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:dial.uclouvain.be:boreal:57616","is_oa":true,"landing_page_url":"http://hdl.handle.net/2078.1/57616","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":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Journal of Symbolic Logic, Vol. 47, no. 1, p. 131-136 (1982)","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[{"score":0.5099999904632568,"id":"https://metadata.un.org/sdg/4","display_name":"Quality Education"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W42912522","https://openalex.org/W963800692","https://openalex.org/W1708393443","https://openalex.org/W1976151458","https://openalex.org/W1992693426","https://openalex.org/W2038831710","https://openalex.org/W2066678816","https://openalex.org/W2461193753","https://openalex.org/W3039529476","https://openalex.org/W4237554060","https://openalex.org/W4245796794"],"related_works":["https://openalex.org/W2166103926","https://openalex.org/W4390350639","https://openalex.org/W2393468107","https://openalex.org/W3194519741","https://openalex.org/W4297823556","https://openalex.org/W3037732692","https://openalex.org/W3044806169","https://openalex.org/W1744069003","https://openalex.org/W2354474640","https://openalex.org/W2373796170"],"abstract_inverted_index":{"NFP":[0,44,50,79,208],"is":[1,37,51,58,92,112,118,172],"the":[2,68,72,97,103,127,158,174,184,187,203],"predicative":[3],"fragment":[4,41],"of":[5,32,105,157,160,164,186,190,205],"NF.":[6,46,61,115],"In":[7,123,148],"this":[8],"system":[9],"we":[10,75,100,125,150],"do":[11],"not":[12,173],"allow":[13],"a":[14,33,38,132,138,153,181],"set":[15],"to":[16,178,202],"exist":[17],"if":[18],"it":[19,191],"cannot":[20],"be":[21,65,81,143,169,194,200],"defined":[22],"without":[23],"using":[24],"quantifiers":[25],"ranging":[26],"over":[27],"its":[28],"type":[29],"or":[30],"parameters":[31],"higher":[34],"type.":[35],"NFI":[36,57,106,161],"less":[39],"restrictive":[40],"located":[42],"between":[43],"and":[45,87,130,209],"We":[47],"show":[48,77],"that":[49,78,88,140,192],"really":[52],"weaker":[53,59],"than":[54,60],"NFI;":[55,95],"similarly,":[56],"This":[62],"result":[63],"will":[64,76,101,142,151,168,193,199],"obtained":[66],"in":[67,84,94,107,114,120,136,180,196,207],"following":[69],"manner:":[70],"on":[71,96],"one":[73],"hand,":[74,99],"can":[80],"proved":[82],"consistent":[83],"elementary":[85],"arithmetic":[86,91],"second":[89],"order":[90,109],"interpretable":[93],"other":[98],"prove":[102],"consistency":[104,159],"third":[108],"arithmetic,":[110],"which":[111],"contained":[113],"The":[116,166],"paper":[117],"divided":[119],"four":[121],"sections.":[122],"\u00a71,":[124],"define":[126],"concepts":[128],"needed":[129],"collect":[131],"few":[133],"results":[134],"together":[135],"such":[137],"way":[139],"they":[141],"ready":[144],"for":[145],"later":[146],"use.":[147],"\u00a72,":[149],"present":[152],"model-theoretic":[154],"(quick)":[155],"proof":[156,167],"(and":[162],"thus":[163],"NFP).":[165],"chosen":[170],"(it":[171],"quickest!)":[175],"so":[176],"as":[177],"motivate":[179],"natural":[182],"manner":[183],"details":[185],"proof-theoretical":[188],"version":[189],"presented":[195],"\u00a73.":[197],"\u00a74":[198],"devoted":[201],"axiom":[204],"infinity":[206],"NFI.":[210]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}