{"id":"https://openalex.org/W2112673210","doi":"https://doi.org/10.2307/2273040","title":"Encodability of Kleene's <i>O</i>","display_name":"Encodability of Kleene's <i>O</i>","publication_year":1973,"publication_date":"1973-09-01","ids":{"openalex":"https://openalex.org/W2112673210","doi":"https://doi.org/10.2307/2273040","mag":"2112673210"},"language":"en","primary_location":{"id":"doi:10.2307/2273040","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273040","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/A5008417331","display_name":"Carl G. Jockusch","orcid":null},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Carl G. Jockusch","raw_affiliation_strings":["University of Illinois at Urbana-Champaign, Urbana, Illinois 61801"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana-Champaign, Urbana, Illinois 61801","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023422267","display_name":"Robert I. Soare","orcid":null},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Robert I. Soare","raw_affiliation_strings":["University of Illinois at Chicago Circle, Chicago, Illinois 60680","University of Illinois at Chicago Circle Chicago Illinois 60680"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Chicago Circle, Chicago, Illinois 60680","institution_ids":["https://openalex.org/I39422238"]},{"raw_affiliation_string":"University of Illinois at Chicago Circle Chicago Illinois 60680","institution_ids":["https://openalex.org/I39422238"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.55,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.68467153,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"38","issue":"3","first_page":"437","last_page":"440"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9968000054359436,"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":0.9968000054359436,"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":0.9936000108718872,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9772999882698059,"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/mathematics","display_name":"Mathematics","score":0.7711074352264404},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6810093522071838},{"id":"https://openalex.org/keywords/order-type","display_name":"Order type","score":0.5772873163223267},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5750331878662109},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.5457940697669983},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5401121973991394},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5275682210922241},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5226606726646423},{"id":"https://openalex.org/keywords/existential-quantification","display_name":"Existential quantification","score":0.5191318392753601},{"id":"https://openalex.org/keywords/partial-function","display_name":"Partial function","score":0.5142332911491394},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08027973771095276}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7711074352264404},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6810093522071838},{"id":"https://openalex.org/C138890893","wikidata":"https://www.wikidata.org/wiki/Q620006","display_name":"Order type","level":2,"score":0.5772873163223267},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5750331878662109},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.5457940697669983},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5401121973991394},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5275682210922241},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5226606726646423},{"id":"https://openalex.org/C136979486","wikidata":"https://www.wikidata.org/wiki/Q773483","display_name":"Existential quantification","level":2,"score":0.5191318392753601},{"id":"https://openalex.org/C182359184","wikidata":"https://www.wikidata.org/wiki/Q1756942","display_name":"Partial function","level":2,"score":0.5142332911491394},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08027973771095276},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","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},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2273040","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273040","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":16,"referenced_works":["https://openalex.org/W596977029","https://openalex.org/W1554860413","https://openalex.org/W1968812492","https://openalex.org/W2007915800","https://openalex.org/W2033831013","https://openalex.org/W2046274763","https://openalex.org/W2058683290","https://openalex.org/W2060142264","https://openalex.org/W2100003518","https://openalex.org/W2114698937","https://openalex.org/W2530006810","https://openalex.org/W4210930452","https://openalex.org/W4231526177","https://openalex.org/W4291439952","https://openalex.org/W6633264645","https://openalex.org/W6807913826"],"related_works":["https://openalex.org/W3193217268","https://openalex.org/W2001063456","https://openalex.org/W4237089393","https://openalex.org/W2084893822","https://openalex.org/W2341789161","https://openalex.org/W2062000137","https://openalex.org/W2052758143","https://openalex.org/W1996227329","https://openalex.org/W2528006471","https://openalex.org/W2112673210"],"abstract_inverted_index":{"Let":[0,209,246],"\u03c9":[1,18,27,30,33,60,170],"be":[2,112,247,268],"the":[3,36,68,89,94,202,212,234,248,254],"nonnegative":[4],"integers.":[5],"G.":[6],"E.":[7],"Sacks":[8],"once":[9],"asked":[10],"whether":[11],"there":[12,61,172],"exists":[13,62,173],"an":[14],"infinite":[15,56,166],"X":[16,25,58,65,168,176],"\u2286":[17,24,59,64,169,175],"such":[19,66,177],"that,":[20],"for":[21,54,103,164,227],"all":[22,104],"Y":[23,29,63,84,174,182,186],",":[26,135,171],"1":[28,31,34,232,263],"where":[32,85],"is":[35,74,81,88,130,159,233,252],"first":[37,69],"nonrecursive":[38],"ordinal.":[39,72],"In":[40,117],"this":[41,46],"note":[42],"we":[43,119],"negatively":[44],"answer":[45],"question":[47],"by":[48,76,240],"giving":[49],"a":[50,100,156,193,199,207,220],"simple":[51],"proof":[52],"that":[53,67,78,155,178],"every":[55,165,228],"set":[57,96,157,167,190],"recursively":[70],"inaccessible":[71],"This":[73],"accomplished":[75],"proving":[77],"H":[79,86],"\u03b1":[80,87,90],"hyper-arithmetic":[82],"in":[83,99,114,133,261,270],"th":[91],"hyperjump":[92,203],"of":[93,204,214,238,257],"empty":[95],"\u2205,":[97],"defined":[98],"suitable":[101],"sense":[102],"ordinals":[105],"Background":[106],"information":[107],"and":[108,136,146,273],"undefined":[109],"notation":[110],"can":[111,267],"found":[113,269],"Rogers":[115],"[11].":[116],"particular,":[118],"write":[120],"A":[121,129,137,142,150,158,179,191,196,205,216],"\u2264":[122,125,143,148,180,184],"h":[123,139,144,149,181],"B(A":[124],"T":[126,185],"B)":[127],"if":[128,141,223],"hyperarithmetical":[131],"(recursive)":[132],"B":[134,140,145,147],"\u2261":[138],".":[151,230,264],"We":[152],"will":[153],"say":[154],"hyperarithmetically":[160],"(recursively)":[161],"encodable":[162],"if,":[163],"(A":[183],").":[187,208],"For":[188],"any":[189,258],"(hyperdegree":[192],")":[194],"let":[195],"\u2032":[197],"(":[198,206],"\u2032)":[200],"denote":[201,211],"0":[210],"hyperdegree":[213],"\u2205.":[215],"function":[217,221],"f":[218],"majorizes":[219],"g":[222],"f(n)":[224],"\u2265":[225],"g(n)":[226],"n":[229],"E":[231,262],"representing":[235],"(type-2)":[236],"functional":[237],"introduced":[239],"Tugu\u00e9":[241],"[13]":[242],"(also":[243],"Kleene":[244],"[6]).":[245],"smallest":[249],"ordinal":[250],"which":[251],"not":[253],"order":[255],"type":[256],"well-ordering":[259],"recursive":[260],"Information":[265],"on":[266],"Richter":[271],"[9]":[272],"[10].":[274]},"counts_by_year":[{"year":2019,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}