{"id":"https://openalex.org/W2159847132","doi":"https://doi.org/10.2178/jsl/1278682215","title":"The consistency strength of choiceless failures of SCH","display_name":"The consistency strength of choiceless failures of SCH","publication_year":2010,"publication_date":"2010-07-09","ids":{"openalex":"https://openalex.org/W2159847132","doi":"https://doi.org/10.2178/jsl/1278682215","mag":"2159847132"},"language":"en","primary_location":{"id":"doi:10.2178/jsl/1278682215","is_oa":false,"landing_page_url":"https://doi.org/10.2178/jsl/1278682215","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":"The 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/A5066099291","display_name":"Arthur W. Apter","orcid":"https://orcid.org/0000-0002-7091-3628"},"institutions":[{"id":"https://openalex.org/I121847817","display_name":"The Graduate Center, CUNY","ror":"https://ror.org/00awd9g61","country_code":"US","type":"education","lineage":["https://openalex.org/I121847817"]},{"id":"https://openalex.org/I141810926","display_name":"Baruch College","ror":"https://ror.org/023qavy03","country_code":"US","type":"education","lineage":["https://openalex.org/I141810926"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Arthur W. Apter","raw_affiliation_strings":["Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA","The Cuny Graduate Center, Mathematics 365 Fifth Avenue, New York, New York 10016, USA. E-mail:","Department of Mathematics, Baruch College of CUNY, New York, New York 10010 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Baruch College of Cuny, New York, New York 10010, USA","institution_ids":["https://openalex.org/I141810926"]},{"raw_affiliation_string":"The Cuny Graduate Center, Mathematics 365 Fifth Avenue, New York, New York 10016, USA. E-mail:","institution_ids":["https://openalex.org/I121847817"]},{"raw_affiliation_string":"Department of Mathematics, Baruch College of CUNY, New York, New York 10010 USA","institution_ids":["https://openalex.org/I141810926"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078814566","display_name":"Peter Koepke","orcid":"https://orcid.org/0000-0002-2266-134X"},"institutions":[{"id":"https://openalex.org/I135140700","display_name":"University of Bonn","ror":"https://ror.org/041nas322","country_code":"DE","type":"education","lineage":["https://openalex.org/I135140700"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Peter Koepke","raw_affiliation_strings":["Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit\u00e4t, Endenicher Allee 60, D-53115 Bonn, Germany. E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universit\u00e4t, Endenicher Allee 60, D-53115 Bonn, Germany. E-mail:","institution_ids":["https://openalex.org/I135140700"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3678,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.61828712,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"75","issue":"3","first_page":"1066","last_page":"1080"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9998000264167786,"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":0.9998000264167786,"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":0.9961000084877014,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.989799976348877,"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/cofinality","display_name":"Cofinality","score":0.8987419605255127},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7940763831138611},{"id":"https://openalex.org/keywords/surjective-function","display_name":"Surjective function","score":0.7417706251144409},{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.7236971259117126},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.6786535978317261},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5812870860099792},{"id":"https://openalex.org/keywords/injective-function","display_name":"Injective function","score":0.5782148838043213},{"id":"https://openalex.org/keywords/regular-cardinal","display_name":"Regular cardinal","score":0.5539506673812866},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.5019896030426025},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44649800658226013},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4317353367805481},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.3772103786468506},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.3312312960624695},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.27101200819015503},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.09986144304275513}],"concepts":[{"id":"https://openalex.org/C148070024","wikidata":"https://www.wikidata.org/wiki/Q1283623","display_name":"Cofinality","level":4,"score":0.8987419605255127},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7940763831138611},{"id":"https://openalex.org/C119238805","wikidata":"https://www.wikidata.org/wiki/Q229102","display_name":"Surjective function","level":2,"score":0.7417706251144409},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.7236971259117126},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.6786535978317261},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5812870860099792},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.5782148838043213},{"id":"https://openalex.org/C89507440","wikidata":"https://www.wikidata.org/wiki/Q1193137","display_name":"Regular cardinal","level":2,"score":0.5539506673812866},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.5019896030426025},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44649800658226013},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4317353367805481},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3772103786468506},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.3312312960624695},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.27101200819015503},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.09986144304275513},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2178/jsl/1278682215","is_oa":false,"landing_page_url":"https://doi.org/10.2178/jsl/1278682215","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":"The Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1278682215","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1278682215","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.533.1479","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.533.1479","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://faculty.baruch.cuny.edu/aapter/papers/ak3b.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth","score":0.5699999928474426}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W157554669","https://openalex.org/W1139854568","https://openalex.org/W1536273738","https://openalex.org/W1580079495","https://openalex.org/W1582291305","https://openalex.org/W1594179826","https://openalex.org/W1963677918","https://openalex.org/W1972177384","https://openalex.org/W1983968672","https://openalex.org/W1984145698","https://openalex.org/W2049932494","https://openalex.org/W2051800246","https://openalex.org/W2056715322","https://openalex.org/W2088381392","https://openalex.org/W2579613946","https://openalex.org/W2763704976","https://openalex.org/W4213337468","https://openalex.org/W4388284830"],"related_works":["https://openalex.org/W4289676933","https://openalex.org/W2885719937","https://openalex.org/W4298219619","https://openalex.org/W2567038806","https://openalex.org/W2032612126","https://openalex.org/W4288278828","https://openalex.org/W2079040423","https://openalex.org/W3166038717","https://openalex.org/W2963932623","https://openalex.org/W4296957153"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"determine":[2],"exact":[3],"consistency":[4,96],"strengths":[5,97],"for":[6],"various":[7],"failures":[8,44,85,107],"of":[9,18,27,34,45,56,63,67,100,103,108,113,129],"the":[10,16,19,25,31,61,64,110],"Singular":[11],"Cardinals":[12],"Hypothesis":[13],"(SCH)":[14],"in":[15,98],"setting":[17],"Zermelo-Fraenkel":[20],"axiom":[21],"system":[22],"ZF":[23],"without":[24],"Axiom":[26],"Choice":[28],"(AC).":[29],"By":[30],"new":[32],"notion":[33],"parallel":[35],"Prikry":[36],"forcing":[37],"that":[38,83,118],"we":[39,41,81],"introduce,":[40],"obtain":[42],"surjective":[43,54],"SCH":[46],"using":[47],"only":[48],"one":[49],"measurable":[50,104],"cardinal,":[51],"including":[52],"a":[53,124],"failure":[55],"Shelah's":[57],"pcf":[58],"theorem":[59,112,117],"about":[60],"size":[62],"power":[65],"set":[66],"\u2135":[68,75,87],"\u03c9":[69,76,88],".":[70],"Using":[71],"symmetric":[72],"collapses":[73],"to":[74,146],",":[77,78,80,89,90],"or":[79,91],"show":[82],"injective":[84],"at":[86,123],"can":[92],"have":[93],"relatively":[94],"mild":[95],"terms":[99],"Mitchell":[101],"orders":[102],"cardinals.":[105],"Injective":[106],"both":[109],"aforementioned":[111],"Shelah":[114],"and":[115,143,148],"Silver's":[116],"GCH":[119],"cannot":[120],"first":[121],"fail":[122],"singular":[125],"strong":[126],"limit":[127],"cardinal":[128],"uncountable":[130],"cofinality":[131],"are":[132,137],"also":[133],"obtained.":[134],"Lower":[135],"bounds":[136],"shown":[138],"by":[139],"core":[140],"model":[141],"techniques":[142],"methods":[144],"due":[145],"Gitik":[147],"Mitchell.":[149]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}