{"id":"https://openalex.org/W2152079305","doi":"https://doi.org/10.2307/2272750","title":"Jump equivalence of the \u0394<sub>2</sub><sup>0</sup> hyperimmune sets","display_name":"Jump equivalence of the \u0394<sub>2</sub><sup>0</sup> hyperimmune sets","publication_year":1972,"publication_date":"1972-09-01","ids":{"openalex":"https://openalex.org/W2152079305","doi":"https://doi.org/10.2307/2272750","mag":"2152079305"},"language":"en","primary_location":{"id":"doi:10.2307/2272750","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272750","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"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["GB","US"],"is_corresponding":true,"raw_author_name":"S. B. Cooper","raw_affiliation_strings":["University of California, Berkeley, California 94720","University of Leeds, Leeds, England"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of California, Berkeley, California 94720","institution_ids":["https://openalex.org/I95457486"]},{"raw_affiliation_string":"University of Leeds, Leeds, England","institution_ids":["https://openalex.org/I130828816"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5076862599"],"corresponding_institution_ids":["https://openalex.org/I130828816","https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":0.6563,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.70875684,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"37","issue":"3","first_page":"598","last_page":"600"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9955999851226807,"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":0.9955999851226807,"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":0.986299991607666,"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/T11567","display_name":"semigroups and automata theory","score":0.9660999774932861,"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/combinatorics","display_name":"Combinatorics","score":0.772344708442688},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7180168628692627},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.6936753988265991},{"id":"https://openalex.org/keywords/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.6487445831298828},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.5353192687034607},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.482086718082428},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.4735493063926697},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4667860269546509},{"id":"https://openalex.org/keywords/jump","display_name":"Jump","score":0.45767873525619507},{"id":"https://openalex.org/keywords/recursively-enumerable-set","display_name":"Recursively enumerable set","score":0.41246268153190613},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.123288094997406},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.07295092940330505}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.772344708442688},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7180168628692627},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.6936753988265991},{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.6487445831298828},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.5353192687034607},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.482086718082428},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.4735493063926697},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4667860269546509},{"id":"https://openalex.org/C2780695682","wikidata":"https://www.wikidata.org/wiki/Q4005959","display_name":"Jump","level":2,"score":0.45767873525619507},{"id":"https://openalex.org/C52703633","wikidata":"https://www.wikidata.org/wiki/Q676835","display_name":"Recursively enumerable set","level":3,"score":0.41246268153190613},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.123288094997406},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.07295092940330505},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2272750","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272750","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2093546192","https://openalex.org/W2101746971","https://openalex.org/W2106640617"],"related_works":["https://openalex.org/W2037833610","https://openalex.org/W1972481159","https://openalex.org/W1973052363","https://openalex.org/W1980641386","https://openalex.org/W2094468224","https://openalex.org/W2757989085","https://openalex.org/W4249600134","https://openalex.org/W2135152748","https://openalex.org/W1552949294","https://openalex.org/W2142539551"],"abstract_inverted_index":{"An":[0],"infinite":[1],"set":[2,38,92,226],"A":[3,22,176,190,212,220],"is":[4,127,243],"said":[5],"to":[6,25,78,85,98,128],"be":[7,79,86,99,196],"hyperhyperimmune":[8],"(h.h.i.)":[9],"if,":[10],"for":[11,83,205],"any":[12],"collection":[13],"of":[14,29,35,53,89,118,124,200,252],"disjoint":[15],"simultaneously":[16],"recursively":[17],"enumerable":[18],"(r.e.)":[19],"finite":[20,201],"sets,":[21],"must":[23],"fail":[24],"intersect":[26],"with":[27,56,64,156],"one":[28],"those":[30,152],"sets.":[31],"Thus":[32],"the":[33,51,61,87,108,119,136,250],"elements":[34],"an":[36,90,171],"h.h.i.":[37,54,91,120,137,148,172],"are,":[39],"in":[40,107],"a":[41,80,84,111,116,169,178,184,197,225],"sense,":[42],"very":[43],"elusive.":[44],"D.":[45],"A.":[46],"Martin":[47],"[3]":[48],"showed":[49],"that":[50,106,130,204],"degrees":[52,63,109,138,144,153],"sets":[55,149,202],"r.e.":[57,62],"complements":[58],"are":[59,150],"exactly":[60,151],"jump":[65,157],"0\u2033":[66,77,133,158,166,182],".":[67,141,159,187,254],"More":[68],"generally,":[69],"C.":[70],"G.":[71],"Jockusch":[72],"[2]":[73],"found":[74,94],"a\u2032":[75,95,131,164,180],"\u2265":[76,96,165,194],"sufficient":[81],"condition":[82,114],"degree":[88],"and":[93,183,218,240],"0\u2032":[97,140,146,155,186],"necessary.":[100],"However,":[101],"it":[102],"was":[103],"also":[104],"shown":[105],"as":[110],"whole":[112],"neither":[113],"gave":[115],"characterization":[117],"degrees.":[121],"The":[122,143],"purpose":[123],"this":[125],"note":[126],"prove":[129],"=":[132],"does":[134],"characterize":[135],"below":[139,145,154],"Theorem.":[142],"containing":[147],"Proof.":[160],"From":[161],"[2],":[162],"if":[163],",":[167,208,211,228],"then":[168],"contains":[170],"set.":[173],"Conversely,":[174],"let":[175,229,245],"\u2208":[177],"where":[179],"&lt;":[181,185],"Let":[188],"{":[189],"s":[191,193,210],"\u2223":[192,246,248],"0}":[195],"recursive":[198],"sequence":[199],"such":[203],"each":[206],"x":[207,213,215,222],"lim":[209],"(":[214,221],")":[216],"exists":[217],"equals":[219],").":[223],"For":[224],"B":[227,230,235,242,247,253],"[":[231],"m":[232,238],"]":[233],"denote":[234,249],"\u2229":[236],"[0,":[237],"],":[239],"(if":[241],"finite)":[244],"cardinality":[251]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
