{"id":"https://openalex.org/W2100629643","doi":"https://doi.org/10.2307/2274851","title":"A jump class of noncappable degrees","display_name":"A jump class of noncappable degrees","publication_year":1989,"publication_date":"1989-06-01","ids":{"openalex":"https://openalex.org/W2100629643","doi":"https://doi.org/10.2307/2274851","mag":"2100629643"},"language":"en","primary_location":{"id":"doi:10.2307/2274851","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274851","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/A5076862599","display_name":"S. Barry Cooper","orcid":"https://orcid.org/0000-0001-5103-0400"},"institutions":[{"id":"https://openalex.org/I130828816","display_name":"University of Leeds","ror":"https://ror.org/024mrxd33","country_code":"GB","type":"education","lineage":["https://openalex.org/I130828816"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"S. B. Cooper","raw_affiliation_strings":["Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England","(University of Leeds"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England","institution_ids":["https://openalex.org/I130828816"]},{"raw_affiliation_string":"(University of Leeds","institution_ids":["https://openalex.org/I130828816"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5076862599"],"corresponding_institution_ids":["https://openalex.org/I130828816"],"apc_list":null,"apc_paid":null,"fwci":0.9548,"has_fulltext":false,"cited_by_count":8,"citation_normalized_percentile":{"value":0.76972556,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"54","issue":"2","first_page":"324","last_page":"353"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9998999834060669,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9998999834060669,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.987500011920929,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9846000075340271,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.7452694773674011},{"id":"https://openalex.org/keywords/jump","display_name":"Jump","score":0.7141557931900024},{"id":"https://openalex.org/keywords/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.6452761888504028},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6008442640304565},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5977151989936829},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5557660460472107},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3320341110229492},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.26688504219055176},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.10982811450958252},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.07045552134513855}],"concepts":[{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.7452694773674011},{"id":"https://openalex.org/C2780695682","wikidata":"https://www.wikidata.org/wiki/Q4005959","display_name":"Jump","level":2,"score":0.7141557931900024},{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.6452761888504028},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6008442640304565},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5977151989936829},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5557660460472107},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3320341110229492},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.26688504219055176},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.10982811450958252},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.07045552134513855},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2274851","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274851","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/1183742908","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183742908","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":0.8799999952316284,"id":"https://metadata.un.org/sdg/14","display_name":"Life below water"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":13,"referenced_works":["https://openalex.org/W130517038","https://openalex.org/W1924816003","https://openalex.org/W1972685794","https://openalex.org/W1976647442","https://openalex.org/W2006726677","https://openalex.org/W2051601785","https://openalex.org/W2055051664","https://openalex.org/W2060075008","https://openalex.org/W2073407285","https://openalex.org/W2484395284","https://openalex.org/W2551720480","https://openalex.org/W2588945226","https://openalex.org/W4213156362"],"related_works":["https://openalex.org/W2072965782","https://openalex.org/W2132512348","https://openalex.org/W2140918173","https://openalex.org/W2132951762","https://openalex.org/W2000487993","https://openalex.org/W1972685794","https://openalex.org/W2156353169","https://openalex.org/W2078696977","https://openalex.org/W2142539551","https://openalex.org/W2007089758"],"abstract_inverted_index":{"Friedberg":[0],"[3]":[1],"showed":[2,20],"that":[3,21,72,161,190,238],"every":[4,193,239,292],"degree":[5,76],"of":[6,14,38,63,93,98,109,146,168,176,184,208,244,262],"unsolvability":[7],"above":[8,24],"0":[9,25,33,79,85,118,129,135,214,220,266,275,288],"\u2032":[10,26,34,80,86,119],"is":[11,74,82,89,123,234],"the":[12,22,36,39,47,51,59,91,107,143,165,173,205,227],"jump":[13,53,92,228,240,293],"some":[15],"degree,":[16],"and":[17,66,158,171,180,187,223],"Sacks":[18,52,251],"[9]":[19],"degrees":[23,111,115,149],"which":[27,50,81,88,121],"are":[28,35],"recursively":[29],"enumerable":[30],"(r.e.)":[31],"in":[32,84,204,219],"jumps":[37],"r.e.":[40,83,99,114,126,209,218],"degrees.":[41,100,210],"In":[42],"this":[43],"paper":[44],"we":[45],"examine":[46],"extent":[48],"to":[49,236],"theorem":[54,62],"can":[55],"be":[56],"combined":[57],"with":[58,130,164,172,216,291],"minimal":[60,96],"pair":[61,97],"Lachlan":[64],"[4]":[65],"Yates":[67],"[13].":[68],"We":[69],"prove":[70],"below":[71],"there":[73,122,253],"a":[75,95,116,131,255,257,271],"c":[77,212,217,225,231],"&gt;":[78,128,213],"but":[87],"not":[90],"half":[94],"This":[101],"extends":[102],"Yates'":[103],"result":[104],"[13]":[105],"proving":[106],"existence":[108],"noncappable":[110,169],"(that":[112],"is,":[113],"&lt;":[117],"for":[120,230],"no":[124],"corresponding":[125],"b":[127,133,273],"\u2229":[132],"=":[134,248,279],").":[136],"It":[137,151,233],"also":[138],"throws":[139],"more":[140],"light":[141],"on":[142],"class":[144,166,174,194,229,241],"PS":[145,162,191,245,280],"promptly":[147],"simple":[148],".":[150,232,281],"was":[152],"shown":[153],"by":[154,283],"Ambos-Spies,":[155],"Jockusch,":[156],"Shore":[157,186],"Soare":[159],"[1]":[160],"coincides":[163],"NC":[167,247],"degrees,":[170,179],"LC":[175,259,278],"all":[177],"low-cuppable":[178],"(using":[181],"earlier":[182],"work":[183],"Maass,":[185],"Stob":[188],"[5])":[189],"splits":[192],"H":[195],"n":[196,199,201],"or":[197],"L":[198],",":[200,260],"\u2265":[202],"0,":[203],"high-low":[206],"hierarchy":[207],"If":[211],"\u2032,":[215,221],"let":[222],"call":[224],"\u22121":[226],"easy":[235],"see":[237],"contains":[242],"members":[243],"(=":[246,268],"LC).":[249],"By":[250],"[8]":[252],"exists":[254],"low":[256],"\u2208":[258],"where":[261],"course":[263],"[":[264,286],"a,":[265,287],"\u2032]":[267,289],"{br.e.":[269],"\u2223":[270],"\u2264":[272,274],"\u2032})":[276],"\u2286":[277],"But":[282],"Robinson":[284],"[7]":[285],"intersects":[290],"class.":[294]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
