{"id":"https://openalex.org/W1965610596","doi":"https://doi.org/10.2307/2271902","title":"Extending G\u00f6del's negative interpretation to ZF","display_name":"Extending G\u00f6del's negative interpretation to ZF","publication_year":1975,"publication_date":"1975-06-01","ids":{"openalex":"https://openalex.org/W1965610596","doi":"https://doi.org/10.2307/2271902","mag":"1965610596"},"language":"en","primary_location":{"id":"doi:10.2307/2271902","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2271902","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/A5084426243","display_name":"William C. Powell","orcid":null},"institutions":[{"id":"https://openalex.org/I63190737","display_name":"University at Buffalo, State University of New York","ror":"https://ror.org/01y64my43","country_code":"US","type":"education","lineage":["https://openalex.org/I63190737"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"William C. Powell","raw_affiliation_strings":["Suny at Buffalo, Amherst, New York 14226"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Suny at Buffalo, Amherst, New York 14226","institution_ids":["https://openalex.org/I63190737"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5084426243"],"corresponding_institution_ids":["https://openalex.org/I63190737"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.5324,"has_fulltext":false,"cited_by_count":31,"citation_normalized_percentile":{"value":0.57835821,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"40","issue":"2","first_page":"221","last_page":"229"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9936000108718872,"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"}},"topics":[{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9936000108718872,"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":0.9911999702453613,"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/T10258","display_name":"Philosophy and Theoretical Science","score":0.9779999852180481,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7862686514854431},{"id":"https://openalex.org/keywords/peano-axioms","display_name":"Peano axioms","score":0.7246168851852417},{"id":"https://openalex.org/keywords/interpretation","display_name":"Interpretation (philosophy)","score":0.7172809839248657},{"id":"https://openalex.org/keywords/continuum-hypothesis","display_name":"Continuum hypothesis","score":0.5893247127532959},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5679635405540466},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.513934314250946},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.4336512088775635},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41284751892089844},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34942305088043213},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3455970883369446},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3424926996231079},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.255156934261322},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12004035711288452},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.10756245255470276},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.07637423276901245},{"id":"https://openalex.org/keywords/linguistics","display_name":"Linguistics","score":0.06519085168838501},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.063254714012146},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06286898255348206}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7862686514854431},{"id":"https://openalex.org/C97489613","wikidata":"https://www.wikidata.org/wiki/Q842755","display_name":"Peano axioms","level":2,"score":0.7246168851852417},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.7172809839248657},{"id":"https://openalex.org/C96488702","wikidata":"https://www.wikidata.org/wiki/Q208416","display_name":"Continuum hypothesis","level":2,"score":0.5893247127532959},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5679635405540466},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.513934314250946},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.4336512088775635},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41284751892089844},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34942305088043213},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3455970883369446},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3424926996231079},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.255156934261322},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12004035711288452},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.10756245255470276},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.07637423276901245},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.06519085168838501},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.063254714012146},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06286898255348206},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"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/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2271902","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2271902","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/1183739383","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183739383","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":[{"score":0.6000000238418579,"display_name":"Peace, Justice and strong institutions","id":"https://metadata.un.org/sdg/16"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W60489380","https://openalex.org/W141536115","https://openalex.org/W158226905","https://openalex.org/W567983113","https://openalex.org/W1220155927","https://openalex.org/W1978270323","https://openalex.org/W2024315465","https://openalex.org/W2148456777","https://openalex.org/W2164241438","https://openalex.org/W2169772316","https://openalex.org/W2243154890","https://openalex.org/W4252318916"],"related_works":["https://openalex.org/W4323354904","https://openalex.org/W2993442435","https://openalex.org/W2375826828","https://openalex.org/W4255069280","https://openalex.org/W1965300620","https://openalex.org/W2505434214","https://openalex.org/W3047205667","https://openalex.org/W3101572115","https://openalex.org/W2613063579","https://openalex.org/W3027428265"],"abstract_inverted_index":{"In":[0,9,28],"[5]":[1],"G\u00f6del":[2],"interpreted":[3],"Peano":[4],"arithmetic":[5],"in":[6,67,73,103,125,141,161],"Heyting":[7],"arithmetic.":[8,27],"[8,":[10],"p.":[11,15,32],"153],":[12],"and":[13,37,92,158],"[7,":[14],"344,":[16],"(iii)],":[17],"Kreisel":[18],"observed":[19],"that":[20,49,117,121,164],"G\u00f6del's":[21,50],"interpretation":[22,42,52],"extended":[23,40,55],"to":[24,43,56,217],"second":[25],"order":[26],"[11]":[29],"(see":[30],"[4,":[31],"92]":[33],"for":[34],"a":[35,62,174,218,230],"correction)":[36],"[10]":[38],"Myhill":[39],"the":[41,68,74,77,84,87,105,126,142,162,186],"type":[44],"theory.":[45,59,180],"We":[46,60],"will":[47,228],"show":[48,116],"negative":[51,127],"can":[53,212],"be":[54,193],"Zermelo-Fraenkel":[57,108,178],"set":[58,63,109,179],"consider":[61],"theory":[64,89,110,220],"T":[65,216],"formulated":[66],"minimal":[69],"predicate":[70],"calculus,":[71],"which":[72,104],"presence":[75],"of":[76,80,90,107,155,177,232],"full":[78],"law":[79],"excluded":[81],"middle":[82],"is":[83,122,173,184],"same":[85],"as":[86],"classical":[88],"Zermelo":[91],"Fraenkel.":[93],"Then,":[94],"following":[95],"Myhill,":[96],"we":[97,115,211,227],"define":[98],"an":[99],"inner":[100],"model":[101,176],"S":[102,183,196,233],"axioms":[106],"are":[111],"true.":[112],"More":[113],"generally":[114],"any":[118],"class":[119,143,182,219],"X":[120,167],"(i)":[123],"transitive":[124],"sense,":[128],"\u2200x":[129],"\u2208":[130,132,137,150,153,171,207],"X\u2200y":[131],"x":[133,136,170,206],"\u00ac":[134,135,148],"X,":[138],"(ii)":[139],"contained":[140],"St":[144],"=":[145,197],"{x:":[146],"\u2200u(\u00ac":[147],"u":[149,152],"x\u2192":[151],"x)}":[154],"stable":[156],"sets,":[157],"(iii)":[159],"closed":[160],"sense":[163],"\u2200x(x":[165,199],"\u2286":[166,200],"\u223c":[168,169,201,202,204,205],"X),":[172],"standard":[175],"The":[181],"simply":[185],"\u2286-least":[187],"such":[188],"class,":[189],"and,":[190],"hence,":[191],"could":[192],"defined":[194],"by":[195],"\u22c2{X:":[198],"X\u2192":[203],"X)}.":[208],"However,":[209],"since":[210],"only":[213],"conservatively":[214],"extend":[215],"with":[221,225],"\u039401-comprehension,":[222],"but":[223],"not":[224],"\u039411-comprehension,":[226],"give":[229],"\u039401-definition":[231],"within":[234],"T.":[235]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}