{"id":"https://openalex.org/W2067841194","doi":"https://doi.org/10.1305/ndjfl/1093888026","title":"The continuum hypothesis is independent of second-order ZF.","display_name":"The continuum hypothesis is independent of second-order ZF.","publication_year":1977,"publication_date":"1977-07-01","ids":{"openalex":"https://openalex.org/W2067841194","doi":"https://doi.org/10.1305/ndjfl/1093888026","mag":"2067841194"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093888026","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093888026","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/The-continuum-hypothesis-is-independent-of-second-order-ZF/10.1305/ndjfl/1093888026.pdf","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":true,"oa_status":"bronze","oa_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/The-continuum-hypothesis-is-independent-of-second-order-ZF/10.1305/ndjfl/1093888026.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5078532689","display_name":"T. S. Weston","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Thomas S. Weston","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5078532689"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.36376365,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"18","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10892","display_name":"Drilling and Well Engineering","score":0.3476000130176544,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10892","display_name":"Drilling and Well Engineering","score":0.3476000130176544,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.3443000018596649,"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/T13885","display_name":"Geophysics and Sensor Technology","score":0.30329999327659607,"subfield":{"id":"https://openalex.org/subfields/2212","display_name":"Ocean Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5098447203636169},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5042029619216919},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.32588309049606323},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.041267603635787964}],"concepts":[{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5098447203636169},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5042029619216919},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.32588309049606323},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.041267603635787964},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1305/ndjfl/1093888026","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093888026","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/The-continuum-hypothesis-is-independent-of-second-order-ZF/10.1305/ndjfl/1093888026.pdf","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/1093888026","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093888026","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":{"id":"doi:10.1305/ndjfl/1093888026","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093888026","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-18/issue-3/The-continuum-hypothesis-is-independent-of-second-order-ZF/10.1305/ndjfl/1093888026.pdf","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"},"sustainable_development_goals":[{"display_name":"Reduced inequalities","id":"https://metadata.un.org/sdg/10","score":0.5}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2067841194.pdf","grobid_xml":"https://content.openalex.org/works/W2067841194.grobid-xml"},"referenced_works_count":14,"referenced_works":["https://openalex.org/W1504706885","https://openalex.org/W1513919859","https://openalex.org/W1556162816","https://openalex.org/W1577927442","https://openalex.org/W1970352212","https://openalex.org/W2020181945","https://openalex.org/W2041833446","https://openalex.org/W2065038407","https://openalex.org/W4230318158","https://openalex.org/W4240398628","https://openalex.org/W6605764590","https://openalex.org/W6630901844","https://openalex.org/W6634604590","https://openalex.org/W6812759502"],"related_works":["https://openalex.org/W1979597421","https://openalex.org/W2007980826","https://openalex.org/W4245490552","https://openalex.org/W4225152035","https://openalex.org/W2061531152","https://openalex.org/W3002753104","https://openalex.org/W2077600819","https://openalex.org/W1587224694","https://openalex.org/W2911598644","https://openalex.org/W2042127053"],"abstract_inverted_index":{"In":[0],"the":[1,32,35,42,53,66,80,109,114,155,161,175,190,237,265],"last":[2],"few":[3],"years,":[4],"several":[5],"authors":[6],"have":[7,29,63,278],"discussed":[8,65],"formulations":[9],"of":[10,16,26,34,37,84,102,130,132,177,189,202],"set":[11,59,88,115,195,209],"theory":[12,60,89,116,157,284],"with":[13,285],"underlying":[14],"logic":[15,39],"order":[17],"greater":[18],"than":[19],"one":[20],"([4],":[21,45],"[5],":[22],"[\u03b2],":[23],"[8],":[24,46],"[9]).Some":[25],"these":[27],"writers":[28],"also":[30,141],"considered":[31],"question":[33],"utility":[36],"higher":[38],"in":[40,57,79,113,150,236],"deciding":[41],"continuum":[43],"hypothesis":[44],"[9]).Kreisel,":[47],"for":[48,172,300],"example,":[49],"has":[50,140,200],"claimed":[51],"that":[52,78,108,154,169,198,268],"CH":[54,110,133,178],"is":[55,98,111,153,159,185,235,281,297],"\"decided\"":[56],"second-order":[58,86,283],"([8]).(Elsewhere,":[61],"I":[62],"critically":[64],"model-theoretic":[67],"result":[68,148],"to":[69],"which":[70,210,221],"Kreisel":[71],"refers.See":[72],"[12].)This":[73],"note":[74],"presents":[75],"a":[76,99,282],"proof":[77,97,106,129],"usual":[81],"(proof-theoretic)":[82],"sense":[83],"\"decided\",":[85],"Zermelo-Frankel":[87],"(ZF":[90],"2":[91,166,181],")":[92],"does":[93,259],"not":[94,260],"decide":[95],"CH.The":[96],"straight-forward":[100],"adaptation":[101],"L.":[103],"Tharp's":[104,170],"unpublished":[105],"([10])":[107],"undecided":[112],"variously":[117],"called":[118],"VBI":[119,135,158,173,184,234],"(for":[120],"von":[121,193],"Neuman-Bernays-Impredicative),":[122],"Kelley-Morse,":[123],"([3],":[124],"[7])":[125],"or":[126],"NQ":[127],"([11]).A":[128],"independence":[131,176],"from":[134,179],"using":[136],"an":[137,186],"inaccessible":[138],"cardinal":[139],"been":[142],"published":[143],"by":[144],"R.":[145],"Chuaqui":[146],"([l]).The":[147],"presented":[149],"section":[151],"3":[152],"first-order":[156],"\"almost":[160],"same":[162],"theory\"":[163],"as":[164,214,216,228],"ZF":[165,180],",":[167],"so":[168],"results":[171],"show":[174],".":[182,241,242,257,270],"1":[183],"impredicative":[187],"extension":[188],"more":[191],"familiar":[192],"Neuman-Bernays":[194],"theory,":[196],"VB.Recall":[197],"VB":[199,232],"axioms":[201],"Nullset,":[203],"Pairing,":[204],"Sumset,":[205],"Infinity,":[206],"and":[207,219,233],"Power":[208],"concern":[211],"only":[212,294],"sets,":[213],"well":[215],"Foundation,":[217],"Replacement,":[218],"Extensionality":[220],"quantify":[222],"over":[223],"classes":[224],"(upper":[225,289],"case":[226,290],"variables)":[227],"well.The":[229],"difference":[230],"between":[231],"Comprehension":[238],"Schema:(1)":[239],"(Xi)":[240],".(X":[243],"n":[244,253],")(3Y)(xe":[245],"Y":[246],"=":[247],"\u03c6(x,":[248,256,269],"X":[249,252],"l9":[250],"...,":[251],"))":[254],"where":[255],"..)":[258,271],"contain":[261,272],"Yfree.If":[262],"we":[263,277],"impose":[264],"additional":[266],"limitation":[267],"no":[273],"bound":[274],"class":[275],"variables,":[276],"VB.":[279],"ZF2":[280],"one-place":[286],"predicate":[287],"variables":[288],"letters).Like":[291],"VBI,":[292],"its":[293],"non-logical":[295],"constant":[296],"*e\\":[298],"Equality":[299]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2018,"cited_by_count":1}],"updated_date":"2026-08-05T07:39:15.569665","created_date":"2025-10-10T00:00:00"}