{"id":"https://openalex.org/W2112190300","doi":"https://doi.org/10.2307/421130","title":"Singular Cardinals and the PCF Theory","display_name":"Singular Cardinals and the PCF Theory","publication_year":1995,"publication_date":"1995-12-01","ids":{"openalex":"https://openalex.org/W2112190300","doi":"https://doi.org/10.2307/421130","mag":"2112190300"},"language":"en","primary_location":{"id":"doi:10.2307/421130","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421130","pdf_url":null,"source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"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":"Bulletin 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/A5079801056","display_name":"Thomas Jech","orcid":null},"institutions":[{"id":"https://openalex.org/I130769515","display_name":"Pennsylvania State University","ror":"https://ror.org/04p491231","country_code":"US","type":"education","lineage":["https://openalex.org/I130769515"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Thomas Jech","raw_affiliation_strings":["Department of Mathematics, The Pennsylvania State University University Park, PA 16802.E-mail:","The Pennsylvania State University,#R##N#University Park"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The Pennsylvania State University University Park, PA 16802.E-mail:","institution_ids":["https://openalex.org/I130769515"]},{"raw_affiliation_string":"The Pennsylvania State University,#R##N#University Park","institution_ids":["https://openalex.org/I130769515"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5079801056"],"corresponding_institution_ids":["https://openalex.org/I130769515"],"apc_list":null,"apc_paid":null,"fwci":0.505,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.64093686,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"1","issue":"4","first_page":"408","last_page":"424"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9997000098228455,"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":0.9997000098228455,"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":0.9976000189781189,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9739999771118164,"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":0.745972216129303},{"id":"https://openalex.org/keywords/exponentiation","display_name":"Exponentiation","score":0.5110567808151245},{"id":"https://openalex.org/keywords/regular-cardinal","display_name":"Regular cardinal","score":0.4902549088001251},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.4620513319969177},{"id":"https://openalex.org/keywords/forcing","display_name":"Forcing (mathematics)","score":0.43146371841430664},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.41046571731567383},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4090960621833801},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.3836836814880371},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.34345871210098267},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3341969847679138},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15301606059074402},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08823490142822266}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.745972216129303},{"id":"https://openalex.org/C81539297","wikidata":"https://www.wikidata.org/wiki/Q33456","display_name":"Exponentiation","level":2,"score":0.5110567808151245},{"id":"https://openalex.org/C89507440","wikidata":"https://www.wikidata.org/wiki/Q1193137","display_name":"Regular cardinal","level":2,"score":0.4902549088001251},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.4620513319969177},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.43146371841430664},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.41046571731567383},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4090960621833801},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.3836836814880371},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.34345871210098267},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3341969847679138},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15301606059074402},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08823490142822266},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","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":5,"locations":[{"id":"doi:10.2307/421130","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421130","pdf_url":null,"source":{"id":"https://openalex.org/S130801046","display_name":"Bulletin of Symbolic Logic","issn_l":"1079-8986","issn":["1079-8986","1943-5894"],"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":"Bulletin of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.bsl/1181154879","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.bsl/1181154879","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:10.1.1.28.30","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.28.30","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.ucla.edu/~asl/bsl/0104/0104-002.ps","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.86.8103","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.86.8103","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.psu.edu/jech/preprints/singular-publ.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.92.9389","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.92.9389","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.psu.edu/jech/preprints/singular.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":84,"referenced_works":["https://openalex.org/W40169474","https://openalex.org/W53237084","https://openalex.org/W157554669","https://openalex.org/W630314124","https://openalex.org/W656854703","https://openalex.org/W779054633","https://openalex.org/W978294532","https://openalex.org/W1487303443","https://openalex.org/W1536273738","https://openalex.org/W1562751575","https://openalex.org/W1579004575","https://openalex.org/W1582291305","https://openalex.org/W1784944400","https://openalex.org/W1963677918","https://openalex.org/W1964757436","https://openalex.org/W1967270974","https://openalex.org/W1983968672","https://openalex.org/W1996153963","https://openalex.org/W1998825076","https://openalex.org/W2016122582","https://openalex.org/W2022175888","https://openalex.org/W2024098812","https://openalex.org/W2025518919","https://openalex.org/W2025890236","https://openalex.org/W2026216307","https://openalex.org/W2028556155","https://openalex.org/W2029243463","https://openalex.org/W2035518801","https://openalex.org/W2037540270","https://openalex.org/W2038389031","https://openalex.org/W2041659058","https://openalex.org/W2046051057","https://openalex.org/W2051529371","https://openalex.org/W2051800246","https://openalex.org/W2054174194","https://openalex.org/W2056715322","https://openalex.org/W2056955426","https://openalex.org/W2061709388","https://openalex.org/W2075810781","https://openalex.org/W2081896365","https://openalex.org/W2083782312","https://openalex.org/W2086387381","https://openalex.org/W2088037688","https://openalex.org/W2088381392","https://openalex.org/W2089758466","https://openalex.org/W2090075222","https://openalex.org/W2092726141","https://openalex.org/W2096709794","https://openalex.org/W2101564260","https://openalex.org/W2115456387","https://openalex.org/W2122092051","https://openalex.org/W2125461239","https://openalex.org/W2130752785","https://openalex.org/W2136006142","https://openalex.org/W2136273669","https://openalex.org/W2275810460","https://openalex.org/W2314333333","https://openalex.org/W2314715369","https://openalex.org/W2321076316","https://openalex.org/W2325492036","https://openalex.org/W2475920342","https://openalex.org/W2477201441","https://openalex.org/W2487893106","https://openalex.org/W2489785608","https://openalex.org/W2496901757","https://openalex.org/W2498473121","https://openalex.org/W2500769916","https://openalex.org/W2572141284","https://openalex.org/W2588753921","https://openalex.org/W2963681040","https://openalex.org/W4205815734","https://openalex.org/W4206380161","https://openalex.org/W4213174560","https://openalex.org/W4213337468","https://openalex.org/W4243050438","https://openalex.org/W4248490242","https://openalex.org/W4251901424","https://openalex.org/W4256195314","https://openalex.org/W4256426709","https://openalex.org/W4388284830","https://openalex.org/W6621949684","https://openalex.org/W6675232589","https://openalex.org/W6680234567","https://openalex.org/W6722993700"],"related_works":["https://openalex.org/W2008443281","https://openalex.org/W416434181","https://openalex.org/W3003252680","https://openalex.org/W1534931658","https://openalex.org/W4302366117","https://openalex.org/W4246276930","https://openalex.org/W2963791786","https://openalex.org/W2330980320","https://openalex.org/W2037477540","https://openalex.org/W2319292934"],"abstract_inverted_index":{"\u00a71.":[0],"Introduction":[1],".":[2,142,231,359,392],"Among":[3],"the":[4,12,17,21,43,46,59,84,104,152,156,174,181,192,198,204,210,222,239,257,298,334,341,345,352,363,370,383],"most":[5,113],"remarkable":[6],"discoveries":[7],"in":[8,11,96,209,238],"set":[9,39,260,347,353,367,377,384],"theory":[10,99,106,368],"last":[13,199],"quarter":[14],"century":[15,242],"is":[16,303,373],"rich":[18],"structure":[19],"of":[20,23,35,40,45,100,151,197,206,221,228,248,250,252,259,264,269,277,285,297,344,354,357,366,378,385],"arithmetic":[22,178,227,251],"singular":[24,53,85,153,182,211],"cardinals,":[25,74],"and":[26,77,159,170,179,189,195,202,243,267,283,310,327],"its":[27],"deep":[28],"relationship":[29],"to":[30,90,116,176,180,256,290,316],"large":[31,73,207],"cardinals.":[32],"The":[33,81,112,226,262],"problem":[34,87],"finding":[36],"a":[37,97,148,274,376],"complete":[38],"rules":[41,249],"describing":[42],"behavior":[44],"continuum":[47],"function":[48],"2":[49,121,136,338],"\u2135":[50,54,122,125,137,140],"\u03b1":[51,55],"for":[52,109,127],"'s,":[56],"known":[57],"as":[58],"Singular":[60],"Cardinals":[61],"Problem":[62],",":[63,301,320,381],"has":[64,88],"been":[65],"attacked":[66],"by":[67,236],"many":[68,91],"different":[69],"techniques,":[70],"involving":[71],"forcing,":[72],"inner":[75],"models,":[76],"various":[78],"combinatorial":[79],"methods.":[80],"work":[82],"on":[83,280],"cardinals":[86,154,183,208,212,287],"led":[89,255],"often":[92],"surprising":[93],"results,":[94],"culminating":[95],"beautiful":[98],"Saharon":[101],"Shelah":[102],"called":[103],"pcf":[105,164,223],"(\u201cpcf\u201d":[107],"stands":[108],"\u201cpossible":[110],"cofinalities\u201d).":[111],"striking":[114],"result":[115],"date":[117],"states":[118],"that":[119,369],"if":[120],"n":[123,129],"<":[124,139],"\u03c9":[126,138],"every":[128],"=":[130],"0,":[131],"1,":[132],"2,":[133,168],"\u2026,":[134],"then":[135,305],"\u03c94":[141],"In":[143,166,214,323,333],"this":[144],"paper":[145],"we":[146,172,217,361],"present":[147,157,218],"brief":[149],"history":[150],"problem,":[155],"knowledge,":[158],"an":[160,219],"introduction":[161],"into":[162],"Shelah's":[163],"theory.":[165,224,261],"Sections":[167,185],"3":[169],"4":[171],"introduce":[173],"reader":[175],"cardinal":[177,229,253,271,342],"problems.":[184,332],"5,":[186],"6,":[187],"7":[188],"8":[190],"describe":[191],"main":[193],"results":[194],"methods":[196],"25":[200],"years":[201],"explain":[203],"role":[205],"problem.":[213],"Section":[215],"9":[216],"outline":[220],"\u00a72.":[225],"numbers":[230,233,254,272,299,388],"Cardinal":[232],"were":[234],"introduced":[235],"Cantor":[237],"late":[240],"19th":[241],"problems":[244],"arising":[245],"from":[246],"investigations":[247],"birth":[258],"operations":[263,279],"addition,":[265],"multiplication":[266,284],"exponentiation":[268,329],"infinite":[270,286,304],"are":[273,314],"natural":[275],"generalization":[276],"such":[278],"integers.":[281],"Addition":[282],"turns":[288],"out":[289],"be":[291],"simple:":[292],"when":[293],"at":[294],"least":[295],"one":[296],"\u03ba":[300,307,311,319,339,350,358,372,380,391],"\u03bb":[302,309,313,321],"both":[306],"+":[308,326],"\u00b7":[312],"equal":[315],"max":[317],"{":[318],"}.":[322],"contrast":[324],"with":[325,375],"\u00b7,":[328],"presents":[330],"fundamental":[331],"simplest":[335],"nontrivial":[336],"case,":[337],"represents":[340],"number":[343,371],"power":[346],"P":[348],"(":[349],"),":[351],"all":[355,386],"subsets":[356],"(Here":[360],"adopt":[362],"usual":[364],"convention":[365],"identified":[374],"cardinality":[379],"namely":[382],"ordinal":[387],"smaller":[389],"than":[390]},"counts_by_year":[{"year":2022,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-07-15T18:14:33.161393","created_date":"2025-10-10T00:00:00"}