{"id":"https://openalex.org/W2162272192","doi":"https://doi.org/10.2307/2275670","title":"The Sacks density theorem and \u03a3<sub>2</sub>-bounding","display_name":"The Sacks density theorem and \u03a3<sub>2</sub>-bounding","publication_year":1996,"publication_date":"1996-06-01","ids":{"openalex":"https://openalex.org/W2162272192","doi":"https://doi.org/10.2307/2275670","mag":"2162272192"},"language":"en","primary_location":{"id":"doi:10.2307/2275670","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275670","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/A5014153184","display_name":"Marcia J. Groszek","orcid":null},"institutions":[{"id":"https://openalex.org/I107672454","display_name":"Dartmouth College","ror":"https://ror.org/049s0rh22","country_code":"US","type":"education","lineage":["https://openalex.org/I107672454"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Marcia J. Groszek","raw_affiliation_strings":["Department of Mathematics, Dartmouth College, Hanover New Hampshire 03755, USA, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Dartmouth College, Hanover New Hampshire 03755, USA, E-mail:","institution_ids":["https://openalex.org/I107672454"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011939365","display_name":"Michael E. Mytilinaios","orcid":null},"institutions":[{"id":"https://openalex.org/I73142707","display_name":"Athens University of Economics and Business","ror":"https://ror.org/03s262162","country_code":"GR","type":"education","lineage":["https://openalex.org/I73142707"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Michael E. Mytilinaios","raw_affiliation_strings":["Department of Informatics, Athens University of Economics, Patission 76, Athens 104 34, Greece, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Informatics, Athens University of Economics, Patission 76, Athens 104 34, Greece, E-mail:","institution_ids":["https://openalex.org/I73142707"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056375116","display_name":"Theodore A. Slaman","orcid":"https://orcid.org/0000-0002-1719-2630"},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Theodore A. Slaman","raw_affiliation_strings":["Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, USA, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, USA, E-mail:","institution_ids":["https://openalex.org/I40347166"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.0672,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.87934137,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":95},"biblio":{"volume":"61","issue":"2","first_page":"450","last_page":"467"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":1.0,"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":1.0,"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":0.9944000244140625,"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/T11567","display_name":"semigroups and automata theory","score":0.9886000156402588,"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/recursively-enumerable-language","display_name":"Recursively enumerable language","score":0.705642580986023},{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.6840818524360657},{"id":"https://openalex.org/keywords/computability-theory","display_name":"Computability theory","score":0.653636634349823},{"id":"https://openalex.org/keywords/recursively-enumerable-set","display_name":"Recursively enumerable set","score":0.6389386653900146},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.629909336566925},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.6216681003570557},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.619438648223877},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5549530982971191},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4558393657207489},{"id":"https://openalex.org/keywords/turing-machine","display_name":"Turing machine","score":0.4165050685405731},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.21238228678703308},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.11952286958694458}],"concepts":[{"id":"https://openalex.org/C203702658","wikidata":"https://www.wikidata.org/wiki/Q1073063","display_name":"Recursively enumerable language","level":2,"score":0.705642580986023},{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.6840818524360657},{"id":"https://openalex.org/C111142201","wikidata":"https://www.wikidata.org/wiki/Q818930","display_name":"Computability theory","level":2,"score":0.653636634349823},{"id":"https://openalex.org/C52703633","wikidata":"https://www.wikidata.org/wiki/Q676835","display_name":"Recursively enumerable set","level":3,"score":0.6389386653900146},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.629909336566925},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.6216681003570557},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.619438648223877},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5549530982971191},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4558393657207489},{"id":"https://openalex.org/C29248071","wikidata":"https://www.wikidata.org/wiki/Q163310","display_name":"Turing machine","level":3,"score":0.4165050685405731},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.21238228678703308},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.11952286958694458},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2275670","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275670","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":[{"score":0.6899999976158142,"id":"https://metadata.un.org/sdg/14","display_name":"Life below water"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":12,"referenced_works":["https://openalex.org/W159724961","https://openalex.org/W1628042786","https://openalex.org/W1987623954","https://openalex.org/W2015329474","https://openalex.org/W2017254147","https://openalex.org/W2037416374","https://openalex.org/W2056414301","https://openalex.org/W2070390550","https://openalex.org/W2227416782","https://openalex.org/W2233217370","https://openalex.org/W2486165895","https://openalex.org/W4232277851"],"related_works":["https://openalex.org/W1408600649","https://openalex.org/W2001962035","https://openalex.org/W2467957161","https://openalex.org/W1924424002","https://openalex.org/W95138543","https://openalex.org/W2070295070","https://openalex.org/W2185077833","https://openalex.org/W2186864846","https://openalex.org/W102922957","https://openalex.org/W2359098868"],"abstract_inverted_index":{"Abstract":[0],"The":[1,36],"Sacks":[2],"Density":[3,22],"Theorem":[4,23],"[7]":[5],"states":[6],"that":[7,20,43],"the":[8,12,21],"Turing":[9],"degrees":[10],"of":[11,28,47,73,84],"recursively":[13,58],"enumerable":[14,59],"sets":[15],"are":[16],"dense.":[17],"We":[18],"show":[19],"holds":[24,66],"in":[25,44,78],"every":[26],"model":[27,46],"P":[29,48],"\u2212":[30,49],"+":[31,50],"B":[32,51,56,69],"\u03a3":[33,52,64],"2":[34,53],".":[35],"proof":[37],"has":[38],"two":[39],"components:":[40],"a":[41],"lemma":[42],"any":[45],",":[54],"if":[55],"is":[57],"and":[60,70],"incomplete":[61],"then":[62],"I":[63],"1":[65],"relative":[67],"to":[68,82],"an":[71],"adaptation":[72],"Shore's":[74],"[9]":[75],"blocking":[76],"technique":[77],"\u03b1":[79],"-recursion":[80],"theory":[81],"models":[83],"arithmetic.":[85]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
