{"id":"https://openalex.org/W2052137788","doi":"https://doi.org/10.1305/ndjfl/1093956974","title":"Certain formulas equivalent to the axiom of choice.","display_name":"Certain formulas equivalent to the axiom of choice.","publication_year":1961,"publication_date":"1961-01-01","ids":{"openalex":"https://openalex.org/W2052137788","doi":"https://doi.org/10.1305/ndjfl/1093956974","mag":"2052137788"},"language":"en","primary_location":{"id":"doi:10.1305/ndjfl/1093956974","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093956974","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-2/issue-4/Certain-formulas-equivalent-to-the-axiom-of-choice/10.1305/ndjfl/1093956974.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-2/issue-4/Certain-formulas-equivalent-to-the-axiom-of-choice/10.1305/ndjfl/1093956974.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028676996","display_name":"Boles\u0142aw Soboci\u0144ski","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Boles\u0142aw Soboci\u0144ski","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5028676996"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.11179557,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"2","issue":"4","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12137","display_name":"Economic theories and models","score":0.11550000309944153,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T12137","display_name":"Economic theories and models","score":0.11550000309944153,"subfield":{"id":"https://openalex.org/subfields/2002","display_name":"Economics and Econometrics"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.7551827430725098},{"id":"https://openalex.org/keywords/axiom-independence","display_name":"Axiom independence","score":0.6940664052963257},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5485039353370667},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5406131744384766},{"id":"https://openalex.org/keywords/zermelo\u2013fraenkel-set-theory","display_name":"Zermelo\u2013Fraenkel set theory","score":0.5228822231292725},{"id":"https://openalex.org/keywords/mathematical-economics","display_name":"Mathematical economics","score":0.5150696635246277},{"id":"https://openalex.org/keywords/constructive-set-theory","display_name":"Constructive set theory","score":0.4772615134716034},{"id":"https://openalex.org/keywords/urelement","display_name":"Urelement","score":0.4411011338233948},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3226770758628845},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.20453396439552307},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.088198721408844},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.08528760075569153},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.05638357996940613},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.038784950971603394},{"id":"https://openalex.org/keywords/programming-language","display_name":"Programming language","score":0.03385442495346069}],"concepts":[{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.7551827430725098},{"id":"https://openalex.org/C170088968","wikidata":"https://www.wikidata.org/wiki/Q4830547","display_name":"Axiom independence","level":3,"score":0.6940664052963257},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5485039353370667},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5406131744384766},{"id":"https://openalex.org/C51460","wikidata":"https://www.wikidata.org/wiki/Q191849","display_name":"Zermelo\u2013Fraenkel set theory","level":5,"score":0.5228822231292725},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.5150696635246277},{"id":"https://openalex.org/C556429856","wikidata":"https://www.wikidata.org/wiki/Q5164501","display_name":"Constructive set theory","level":5,"score":0.4772615134716034},{"id":"https://openalex.org/C78550038","wikidata":"https://www.wikidata.org/wiki/Q1896657","display_name":"Urelement","level":5,"score":0.4411011338233948},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3226770758628845},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.20453396439552307},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.088198721408844},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.08528760075569153},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.05638357996940613},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.038784950971603394},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.03385442495346069},{"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:10.1305/ndjfl/1093956974","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093956974","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-2/issue-4/Certain-formulas-equivalent-to-the-axiom-of-choice/10.1305/ndjfl/1093956974.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/1093956974","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.ndjfl/1093956974","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/1093956974","is_oa":true,"landing_page_url":"https://doi.org/10.1305/ndjfl/1093956974","pdf_url":"https://projecteuclid.org/journals/notre-dame-journal-of-formal-logic/volume-2/issue-4/Certain-formulas-equivalent-to-the-axiom-of-choice/10.1305/ndjfl/1093956974.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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2052137788.pdf","grobid_xml":"https://content.openalex.org/works/W2052137788.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W854537603","https://openalex.org/W1041856972","https://openalex.org/W1565367437","https://openalex.org/W2150810833","https://openalex.org/W2326782760"],"related_works":["https://openalex.org/W288388583","https://openalex.org/W1995120876","https://openalex.org/W2350176099","https://openalex.org/W2491940682","https://openalex.org/W2796622564","https://openalex.org/W2076278624","https://openalex.org/W2980408391","https://openalex.org/W423103167","https://openalex.org/W207680367","https://openalex.org/W1504849543"],"abstract_inverted_index":{"In":[0],"this":[1],"note":[2],"it":[3,168,253],"will":[4],"be":[5,170],"shown":[6],"that":[7,46,86,173,235,278],"each":[8],"of":[9,164,228,231,243,246,262,267,285],"the":[10,33,73,162,174,183,203,226,229,236,244,254,283],"following":[11,184],"four":[12],"set-theoretical":[13],"formulas:":[14],"21.For":[15],"any":[16,56,118,187],"cardinal":[17,57,98,119,135,188,249],"numbers":[18,58,99,120,136,189],"m":[19,40,53,59,80,93,100,126,140,190,210],"and":[20,29,41,44,60,69,81,84,178,191,199,211,214,272],"n":[21,42,61,129,212],"which":[22,62,102,193,222],"are":[23,32,63,72,103,180,194],"not":[24,64,104,195],"finite,":[25,65,105,196],"if":[26,66,106,125,139,197],"tf":[27,50,67,87,90],"(m)":[28,48,68,88,108,219],"N":[30,47,70,110,200],"(n)":[31,201],"least":[34,74,204],"Hartogs'":[35,75,205],"alephs":[36,76,206],"with":[37,77,207],"respect":[38,78,208],"to":[39,79,161,182,209,276],"respectively,":[43,83],"such":[45,85],"=":[49,54,147],"(n),":[51,91],"then":[52,92,217],"n.33.For":[55],"(rt)":[71,111],"rt":[82,192],">":[89,94,115],"n.E.":[95],"For":[96],"\u03b1rzy":[97],"#72*2?Tt":[101],"H":[107],"\u03b2^\u03afi":[109],"n,":[112,122,216],"\u03af^ew":[113],"R(m)":[114],"t^":[116],"(n).2).For":[117],"m,":[121],"tfrcd":[123],")o,":[124],"<":[127,130,220],"!jo,":[128],"\u00a3>,":[131],"\u03b2?z^":[132],"/or":[133],"\u03b2/2y":[134],"xand":[137],"],":[138],"+":[141],"ta":[142],"^":[143],"\u03b2wJ":[144],"n+":[145],"f":[146],")\u03b8,":[148],"\u03af/?ew":[149],"E=":[150],"f,":[151],"\u03af^en":[152],"either":[153],"m^":[154],"nor":[155],"n>":[156],"m.":[157],"is":[158,223,239,256,274],"inferentially":[159],"equivalent":[160],"axiom":[163,230,266,284],"choice.Concerning":[165],"these":[166,279],"formulas":[167,175],"should":[169],"noted:":[171],"1)":[172],"21,":[176],"33":[177],"(S":[179],"related":[181],"theorem:":[185],"Tl.For":[186],"N(m)":[198],"\u03b2re":[202],"respectively":[213],"m<":[215],"K":[218],"N(n).,":[221],"provable":[224],"without":[225],"aid":[227],"choice":[232,268],"and:":[233],"2)":[234,238],"formula":[237],"a":[240],"weaker":[241],"formulation":[242],"law":[245],"trichotomy":[247,255],"for":[248],"numbers,":[250],"since":[251],"in":[252,287],"preceded":[257],"by":[258],"an":[259],"antecedent":[260],"consisting":[261],"three":[263],"conditions.^Proof:(i)":[264],"The":[265],"implies":[269],"21,.33,":[270],"E,":[271],"\u00ae.It":[273],"easy":[275],"determine":[277],"theorems":[280],"follow":[281],"from":[282],"choice.Viz.,":[286],"order":[288]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}