{"id":"https://openalex.org/W2058873424","doi":"https://doi.org/10.2307/2273105","title":"Analytic sets having incomparable kleene degrees","display_name":"Analytic sets having incomparable kleene degrees","publication_year":1982,"publication_date":"1982-12-01","ids":{"openalex":"https://openalex.org/W2058873424","doi":"https://doi.org/10.2307/2273105","mag":"2058873424"},"language":"en","primary_location":{"id":"doi:10.2307/2273105","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273105","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/A5041872415","display_name":"Galen Weitkamp","orcid":null},"institutions":[{"id":"https://openalex.org/I117942362","display_name":"Western Illinois University","ror":"https://ror.org/02sshz518","country_code":"US","type":"education","lineage":["https://openalex.org/I117942362"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Galen Weitkamp","raw_affiliation_strings":["Western Illinois University, Macomb, Illinois 61455"],"affiliations":[{"raw_affiliation_string":"Western Illinois University, Macomb, Illinois 61455","institution_ids":["https://openalex.org/I117942362"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5041872415"],"corresponding_institution_ids":["https://openalex.org/I117942362"],"apc_list":null,"apc_paid":null,"fwci":2.1988,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.88315502,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"47","issue":"4","first_page":"860","last_page":"868"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9990000128746033,"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":0.9990000128746033,"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":0.9801999926567078,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9750000238418579,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/mathematics","display_name":"Mathematics","score":0.7178816795349121},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5812619924545288},{"id":"https://openalex.org/keywords/computability-theory","display_name":"Computability theory","score":0.5582413673400879},{"id":"https://openalex.org/keywords/semilattice","display_name":"Semilattice","score":0.5259026885032654},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.47166430950164795},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4580932557582855},{"id":"https://openalex.org/keywords/kleenes-recursion-theorem","display_name":"Kleene's recursion theorem","score":0.44874781370162964},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3700130581855774},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.16082483530044556},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1460249423980713}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7178816795349121},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5812619924545288},{"id":"https://openalex.org/C111142201","wikidata":"https://www.wikidata.org/wiki/Q818930","display_name":"Computability theory","level":2,"score":0.5582413673400879},{"id":"https://openalex.org/C132170107","wikidata":"https://www.wikidata.org/wiki/Q834585","display_name":"Semilattice","level":3,"score":0.5259026885032654},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.47166430950164795},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4580932557582855},{"id":"https://openalex.org/C123260079","wikidata":"https://www.wikidata.org/wiki/Q1933521","display_name":"Kleene's recursion theorem","level":2,"score":0.44874781370162964},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3700130581855774},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.16082483530044556},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1460249423980713},{"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/C207405024","wikidata":"https://www.wikidata.org/wiki/Q207348","display_name":"Semigroup","level":2,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273105","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273105","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/1183741145","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741145","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W138734646","https://openalex.org/W1221529125","https://openalex.org/W1565029977","https://openalex.org/W1978259458","https://openalex.org/W1986191839","https://openalex.org/W2004845705","https://openalex.org/W2033831013","https://openalex.org/W2085361497","https://openalex.org/W2090546112","https://openalex.org/W2905135108","https://openalex.org/W6659089980"],"related_works":["https://openalex.org/W4388310202","https://openalex.org/W2310420552","https://openalex.org/W2133902039","https://openalex.org/W1589207578","https://openalex.org/W2936847277","https://openalex.org/W167110062","https://openalex.org/W4247035339","https://openalex.org/W1161578261","https://openalex.org/W4597164","https://openalex.org/W90338445"],"abstract_inverted_index":{"One":[0],"concern":[1,139],"of":[2,9,12,28,37,46,49,62,90,98,114,121,129,132,135,140,149,154,174,206,213,252,305],"descriptive":[3,91,122],"set":[4,30,92,123,160,294],"theory":[5,31,93,124,131],"is":[6,83,143,182,195,208,229,295,302],"the":[7,26,35,59,66,87,95,112,119,130,146,201,211,222,303],"classification":[8],"definable":[10],"sets":[11,116,134,153,173],"reals.":[13,136],"Taken":[14],"loosely":[15],"\u2018definable\u2019":[16],"can":[17,40,261],"mean":[18],"anything":[19],"from":[20,264],"projective":[21],"to":[22,85,118,144,166,284],"formally":[23],"describable":[24],"in":[25,34,58,94,185,216,259,290,298,312],"language":[27],"Zermelo-Fraenkel":[29],"(ZF).":[32],"Recursiveness,":[33],"case":[36],"Kleene":[38,99,170,183,214,296],"recursion,":[39],"be":[41,74,262],"a":[42,109,127,196,253,273,277],"particularly":[43],"informative":[44],"notion":[45],"definability.":[47],"Sets":[48],"integers,":[50],"for":[51,68,169,267,319],"example,":[52],"are":[53],"categorized":[54],"by":[55],"their":[56,70,77],"positions":[57],"upper":[60,96],"semilattice":[61,97],"Turing":[63],"degrees,":[64],"and":[65,156,177,221,270,281,286,317],"algorithms":[67],"computing":[69,274],"characteristic":[71,202],"functions":[72],"may":[73],"taken":[75],"as":[76],"defining":[78],"presentations.":[79],"In":[80,101],"turn":[81],"it":[82,301],"interesting":[84],"position":[86],"common":[88,117],"fauna":[89],"degrees.":[100],"so":[102,199,233],"doing":[103],"not":[104],"only":[105],"do":[106],"we":[107],"gain":[108],"perspective":[110],"on":[111],"complexity":[113,148],"those":[115],"study":[120],"but":[125],"also":[126],"refinement":[128],"analytic":[133],"The":[137],"primary":[138],"this":[141],"paper":[142],"calculate":[145],"relative":[147],"several":[150,163],"notable":[151],"coanalytic":[152],"reals":[155,175],"display":[157],"(under":[158],"suitable":[159],"theoretic":[161],"hypothesis)":[162],"natural":[164],"solutions":[165],"Post's":[167],"problem":[168],"recursion.":[171],"For":[172],"A":[176,181,188,205,207,236,260,293],"B":[178,186,191,220,247,271,299,316],"one":[179],"says":[180],"recursive":[184,209,311],"(written":[187],"\u2264":[189],"K":[190],")":[192,239],"iff":[193],"there":[194,228],"real":[197,254,321],"y":[198,217,269,313,322],"that":[200,234,288],"function":[203,310],"X":[204,219,235,246,315],"(in":[210],"sense":[212],"[1959])":[215],",":[218,248,314],"existential":[223],"integer":[224,231,307],"quantifier":[225],"\u2203;":[226],"i.e.":[227],"an":[230,265,282,306],"e":[232,242],"(":[237],"x":[238,255],"\u2243":[240,318],"{":[241],"}(":[243],"x,":[244,268],"y,":[245],"\u2243).":[249],"Intuitively,":[250],"membership":[251],"\u03f5":[256],"\u03c9":[257,258],"decided":[263],"oracle":[266],"using":[272],"machine":[275],"with":[276],"countably":[278],"infinite":[279],"memory":[280,289],"ability":[283],"search":[285],"manipulate":[287],"finite":[291],"time.":[292],"semirecursive":[297],"if":[300],"domain":[304],"valued":[308],"partial":[309],"some":[320],".":[323]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
