{"id":"https://openalex.org/W1972957003","doi":"https://doi.org/10.2307/2273144","title":"Double jumps of minimal degrees over cardinals","display_name":"Double jumps of minimal degrees over cardinals","publication_year":1982,"publication_date":"1982-06-01","ids":{"openalex":"https://openalex.org/W1972957003","doi":"https://doi.org/10.2307/2273144","mag":"1972957003"},"language":"en","primary_location":{"id":"doi:10.2307/2273144","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273144","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/A5109116147","display_name":"C. T. Chong","orcid":null},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":true,"raw_author_name":"C. T. Chong","raw_affiliation_strings":["National University of Singapore, Singapore0511, Republic Of Singapore","National University of Singapore"],"affiliations":[{"raw_affiliation_string":"National University of Singapore, Singapore0511, Republic Of Singapore","institution_ids":["https://openalex.org/I165932596"]},{"raw_affiliation_string":"National University of Singapore","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5109116147"],"corresponding_institution_ids":["https://openalex.org/I165932596"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0840451,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"47","issue":"2","first_page":"329","last_page":"334"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9997000098228455,"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":0.9997000098228455,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9957000017166138,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9592999815940857,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/successor-cardinal","display_name":"Successor cardinal","score":0.9118189811706543},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7648562788963318},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7336297035217285},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7092431783676147},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.6366090774536133},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.5423600673675537},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.4696005582809448},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.422149121761322},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10244050621986389},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07654419541358948}],"concepts":[{"id":"https://openalex.org/C75306776","wikidata":"https://www.wikidata.org/wiki/Q7632662","display_name":"Successor cardinal","level":2,"score":0.9118189811706543},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7648562788963318},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7336297035217285},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7092431783676147},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.6366090774536133},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.5423600673675537},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.4696005582809448},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.422149121761322},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10244050621986389},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07654419541358948},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","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/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273144","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273144","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/1183741000","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741000","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":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W2001136729","https://openalex.org/W2068756392","https://openalex.org/W2144999713","https://openalex.org/W2167930025","https://openalex.org/W2747509665","https://openalex.org/W6811663084"],"related_works":["https://openalex.org/W2473542051","https://openalex.org/W2376029318","https://openalex.org/W3123320268","https://openalex.org/W2994519609","https://openalex.org/W2388301251","https://openalex.org/W2726593096","https://openalex.org/W2054269721","https://openalex.org/W4235459050","https://openalex.org/W4214635639","https://openalex.org/W2390822611"],"abstract_inverted_index":{"Jockusch":[0],"and":[1,121,189,212,274],"Posner":[2],"[4]":[3],"showed":[4],"that":[5,39,75,205,236,283],"every":[6,27,41,146,159,296],"minimal":[7,30,149,160,177,223,299],"\u03c9-degree":[8],"is":[9,14,48,141,158,208,219,290],"GL":[10,163],"2":[11,164,273],".":[12,200],"This":[13,234],"achieved":[15],"by":[16,230,281],"exhibiting":[17],"a":[18,51,76,80,92,143,191,209,220,267,291],"function":[19,28,33,77,144,221,241,292,297],"f":[20,34,44,105,242],"recursive":[21],"in":[22,79,186,203],"0":[23,54,172],"\u2032":[24],"which":[25],"dominates":[26],"of":[29,60,116,148,176,222,239,262,298],"\u03c9-degree.":[31],"The":[32,129],"has":[35],"the":[36,57,67,114,187,240,251,255,260],"peculiar":[37],"property":[38],"for":[40,96,174,284],"s":[42,46,62,69],",":[43,103,140,157,288],"(":[45,169],")":[47],"defined":[49,78,110],"after":[50,112],"search":[52],"(using":[53],"\u2032)":[55,173],"over":[56,83],"power":[58],"set":[59,147],"L":[61,199],"(G\u00f6del's":[63],"constructible":[64],"hierarchy":[65],"at":[66,125],"level":[68],").":[70],"It":[71],"can":[72],"be":[73,91,109,247],"seen":[74],"similar":[81],"manner":[82],"an":[84],"infinite":[85],"successor":[86,137,210],"cardinal":[87,211],"k":[88,99,127,139,150,156,161,178,207,215,224,287,300],"will":[89,107,123,245],"not":[90,108,228,246],"total":[93],"function,":[94],"since":[95],"example":[97],"if":[98,206],"=":[100,168,198],"\u03c1":[101,117],"+":[102],"then":[104,132,217],"(\u03c1)":[106],"until":[111],"all":[113],"subsets":[115],"have":[118],"been":[119],"examined,":[120],"this":[122,181,279],"take":[124],"least":[126],"steps.":[128],"following":[130],"questions":[131],"naturally":[133],"arise:":[134],"(i)":[135,185],"For":[136,153],"cardinals":[138,155,286],"there":[142,218,289],"dominating":[145,295],"-degree?":[151],"(ii)":[152,271],"arbitrary":[154],"-degree":[162,225],"(i.e.":[165],"b":[166,170,175],"\u2033":[167],"\u2228":[171],"-degree)?":[179],"In":[180],"paper":[182,280],"we":[183,258],"answer":[184,193,269],"negative":[188],"provide":[190,266],"positive":[192,268],"to":[194,249,265,270],"(ii),":[195],"assuming":[196],"V":[197],"We":[201,277],"show":[202],"fact":[204],"h":[213,231],"\u2264":[214],"0\u2032,":[216],"below":[226,293],"0\u2032":[227,294],"dominated":[229],"(Theorem":[232,272],"1).":[233,276],"implies":[235],"any":[237],"refinement":[238],"described":[243],"above":[244],"able":[248],"remove":[250],"difficulties":[252],"encountered.":[253],"On":[254],"other":[256],"hand,":[257],"introduce":[259],"notion":[261],"\u2018strong":[263],"domination\u2019":[264],"Corollary":[275],"end":[278],"indicating":[282],"limit":[285],"-degree.":[301]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}