{"id":"https://openalex.org/W2130593600","doi":"https://doi.org/10.2307/2273294","title":"Weakly compact cardinals: A combinatorial proof","display_name":"Weakly compact cardinals: A combinatorial proof","publication_year":1979,"publication_date":"1979-12-01","ids":{"openalex":"https://openalex.org/W2130593600","doi":"https://doi.org/10.2307/2273294","mag":"2130593600"},"language":"en","primary_location":{"id":"doi:10.2307/2273294","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273294","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/A5024923217","display_name":"Saharon Shelah","orcid":"https://orcid.org/0000-0003-0462-3152"},"institutions":[{"id":"https://openalex.org/I197251160","display_name":"Hebrew University of Jerusalem","ror":"https://ror.org/03qxff017","country_code":"IL","type":"education","lineage":["https://openalex.org/I197251160"]},{"id":"https://openalex.org/I95457486","display_name":"University of California, Berkeley","ror":"https://ror.org/01an7q238","country_code":"US","type":"education","lineage":["https://openalex.org/I95457486"]}],"countries":["IL","US"],"is_corresponding":true,"raw_author_name":"S. Shelah","raw_affiliation_strings":["Hebrew University, Jerusalem, Israel","University of Californiaat Berkeley, Berkeley, California 94720"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Hebrew University, Jerusalem, Israel","institution_ids":["https://openalex.org/I197251160"]},{"raw_affiliation_string":"University of Californiaat Berkeley, Berkeley, California 94720","institution_ids":["https://openalex.org/I95457486"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5024923217"],"corresponding_institution_ids":["https://openalex.org/I197251160","https://openalex.org/I95457486"],"apc_list":null,"apc_paid":null,"fwci":0.637,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.71965748,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"44","issue":"4","first_page":"559","last_page":"562"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.998199999332428,"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":0.998199999332428,"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":0.9581999778747559,"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":0.9556000232696533,"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":0.740166425704956},{"id":"https://openalex.org/keywords/combinatorial-proof","display_name":"Combinatorial proof","score":0.6904922127723694},{"id":"https://openalex.org/keywords/order-type","display_name":"Order type","score":0.6732344627380371},{"id":"https://openalex.org/keywords/cofinality","display_name":"Cofinality","score":0.6467949151992798},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.6411005258560181},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6254061460494995},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.5328418016433716},{"id":"https://openalex.org/keywords/regular-cardinal","display_name":"Regular cardinal","score":0.5150139331817627},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.48264366388320923},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.4709514081478119},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.46163320541381836},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.26509737968444824},{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.18306848406791687},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.11913555860519409},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.10568675398826599}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.740166425704956},{"id":"https://openalex.org/C118539577","wikidata":"https://www.wikidata.org/wiki/Q3402784","display_name":"Combinatorial proof","level":2,"score":0.6904922127723694},{"id":"https://openalex.org/C138890893","wikidata":"https://www.wikidata.org/wiki/Q620006","display_name":"Order type","level":2,"score":0.6732344627380371},{"id":"https://openalex.org/C148070024","wikidata":"https://www.wikidata.org/wiki/Q1283623","display_name":"Cofinality","level":4,"score":0.6467949151992798},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.6411005258560181},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6254061460494995},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.5328418016433716},{"id":"https://openalex.org/C89507440","wikidata":"https://www.wikidata.org/wiki/Q1193137","display_name":"Regular cardinal","level":2,"score":0.5150139331817627},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48264366388320923},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.4709514081478119},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.46163320541381836},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26509737968444824},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.18306848406791687},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.11913555860519409},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.10568675398826599},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2273294","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2273294","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/1183740464","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183740464","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":2,"referenced_works":["https://openalex.org/W1602067282","https://openalex.org/W2027753458"],"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/W2963932623","https://openalex.org/W2024803908","https://openalex.org/W2257297416","https://openalex.org/W4296957153","https://openalex.org/W2130593600"],"abstract_inverted_index":{"We":[0,104],"give":[1],"here":[2],"a":[3,14,65,129,150,170,175,201,210,215],"direct":[4,61],"purely":[5],"combinatorial":[6,15],"proof":[7,44,63],"that":[8],"weak":[9],"compactness":[10],"is":[11,21,72,86,169,188,196,206],"equivalent":[12,30],"to":[13,52],"property":[16,19],"(2).":[17],"This":[18],"(2)":[20],"apparently":[22],"stronger,":[23],"and":[24,32,77,93,114,120,203],"from":[25],"it,":[26],"all":[27],"other":[28],"usual":[29,33],"definitions":[31],"properties":[34],"of":[35,64,110,162,200,209,213],"weakly":[36,54,69],"compact":[37,55,70],"cardinals":[38],"can":[39],"be":[40,46,146,149,154,159],"deduced.":[41],"So":[42],"this":[43],"may":[45],"useful":[47],"for":[48,173],"books":[49],"which":[50,85],"want":[51],"present":[53],"cardinals,":[56],"but":[57],"not":[58,73],"logic.":[59],"A":[60,164],"simple":[62],"weaker":[66],"implication":[67],"(e.g.,":[68],"\u03bc":[71,148],"the":[74,96,108,160,198,207],"first":[75],"inaccessible,":[76],"every":[78],"stationary":[79],"set":[80,133,167,208],"has":[81],"an":[82],"initial":[83],"segment":[84],"stationary)":[87],"was":[88,127],"given":[89],"by":[90,95,107,121],"Kunen":[91],"[K]":[92],"independently":[94],"author":[97],"[Sh].":[98],"Baumgartner":[99],"[B]":[100],"had":[101],"another":[102],"proof.":[103],"were":[105],"motivated":[106],"manuscript":[109],"Erd\u00f6s,":[111],"Hajnal,":[112],"Mate":[113],"Rado's":[115],"book":[116,130],"on":[117,131],"partition":[118],"calculus,":[119],"conversations":[122],"with":[123],"A.":[124],"Levi":[125],"who":[126],"writing":[128],"naive":[132],"theory.":[134],"Notation":[135],".":[136,216],"Let":[137,156],"i,":[138],"j":[139],",":[140,178,202],"\u03b1,":[141],"\u03b2,":[142],"\u03b3,":[143],"\u03b7,":[144],"\u03c3":[145],"ordinals,":[147],"cardinal,":[151],"f,":[152],"g":[153],"functions.":[155],"cf":[157],"\u03b1":[158,205],"cofinality":[161],"\u03b1.":[163],"partially":[165],"ordered":[166],"T":[168,177,186,204,212],"tree":[171],"if":[172],"any":[174],"\u2208":[176,185,211],"{":[179],"b:":[180],"b":[181,184],"<":[182],"a,":[183],"}":[187],"well":[189],"ordered;":[190],"its":[191],"order":[192],"type":[193],"(an":[194],"ordinal)":[195],"called":[197],"level":[199,214]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2015,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}