{"id":"https://openalex.org/W2015103007","doi":"https://doi.org/10.1017/s0022481200051550","title":"Adding a closed unbounded set","display_name":"Adding a closed unbounded set","publication_year":1976,"publication_date":"1976-06-01","ids":{"openalex":"https://openalex.org/W2015103007","doi":"https://doi.org/10.1017/s0022481200051550","mag":"2015103007"},"language":"en","primary_location":{"id":"doi:10.1017/s0022481200051550","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051550","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/A5001636447","display_name":"James E. Baumgartner","orcid":"https://orcid.org/0000-0002-9589-3504"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"J. E. Baumgartner","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102449669","display_name":"L. A. Harrington","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"L. A. Harrington","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034562840","display_name":"E. M. Kleinberg","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"E. M. Kleinberg","raw_affiliation_strings":["Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","Massachusetts Institute of Technology"],"affiliations":[{"raw_affiliation_string":"Massachusetts Institute of Technology, Cambridge, Massachusetts 02139","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute of Technology","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5001636447"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":3.3072,"has_fulltext":false,"cited_by_count":56,"citation_normalized_percentile":{"value":0.92658875,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"41","issue":"2","first_page":"481","last_page":"482"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9968000054359436,"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":0.9968000054359436,"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/T11727","display_name":"Advanced Algebra and Logic","score":0.9950000047683716,"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":0.972100019454956,"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/ultrafilter","display_name":"Ultrafilter","score":0.9467068910598755},{"id":"https://openalex.org/keywords/uncountable-set","display_name":"Uncountable set","score":0.8912631869316101},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7999343872070312},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.726207971572876},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.6245220899581909},{"id":"https://openalex.org/keywords/regular-cardinal","display_name":"Regular cardinal","score":0.5840684771537781},{"id":"https://openalex.org/keywords/closed-set","display_name":"Closed set","score":0.5709313750267029},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5345709919929504},{"id":"https://openalex.org/keywords/nowhere-dense-set","display_name":"Nowhere dense set","score":0.5003306865692139},{"id":"https://openalex.org/keywords/fubinis-theorem","display_name":"Fubini's theorem","score":0.455495685338974},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.44763901829719543},{"id":"https://openalex.org/keywords/infinite-set","display_name":"Infinite set","score":0.4298231601715088},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.39227521419525146},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.367410808801651},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.3431740403175354},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.278980016708374},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.24406325817108154},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11145225167274475}],"concepts":[{"id":"https://openalex.org/C87598191","wikidata":"https://www.wikidata.org/wiki/Q1571831","display_name":"Ultrafilter","level":2,"score":0.9467068910598755},{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.8912631869316101},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7999343872070312},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.726207971572876},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.6245220899581909},{"id":"https://openalex.org/C89507440","wikidata":"https://www.wikidata.org/wiki/Q1193137","display_name":"Regular cardinal","level":2,"score":0.5840684771537781},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.5709313750267029},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5345709919929504},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.5003306865692139},{"id":"https://openalex.org/C55526981","wikidata":"https://www.wikidata.org/wiki/Q1149022","display_name":"Fubini's theorem","level":2,"score":0.455495685338974},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.44763901829719543},{"id":"https://openalex.org/C199879729","wikidata":"https://www.wikidata.org/wiki/Q205140","display_name":"Infinite set","level":3,"score":0.4298231601715088},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.39227521419525146},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.367410808801651},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.3431740403175354},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.278980016708374},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.24406325817108154},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11145225167274475},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"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":2,"locations":[{"id":"doi:10.1017/s0022481200051550","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051550","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/1183739790","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183739790","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":4,"referenced_works":["https://openalex.org/W1967242789","https://openalex.org/W1987623954","https://openalex.org/W2024862302","https://openalex.org/W2155993241"],"related_works":["https://openalex.org/W2167408243","https://openalex.org/W3048792601","https://openalex.org/W3041130254","https://openalex.org/W4253831173","https://openalex.org/W2117235496","https://openalex.org/W3203074460","https://openalex.org/W2015159104","https://openalex.org/W2015103007","https://openalex.org/W4229892815","https://openalex.org/W1560971418"],"abstract_inverted_index":{"The":[0,44],"extreme":[1],"interest":[2],"of":[3,9,31,37,46,68,119,170,199,208],"set":[4,201],"theorists":[5],"in":[6,15,52],"the":[7,16,29,66,117,144,168,197,206],"notion":[8],"\u201cclosed":[10],"unbounded":[11,35,42,74,100],"set\u201d":[12],"is":[13,39,49,76,107,124,130,146,160,163,181,185,190],"epitomized":[14],"following":[17],"well-known":[18,161],"theorem:":[19],"Theorem":[20],"A.":[21],"For":[22],"any":[23,32,61,104],"regular":[24,63,85],"cardinal":[25,64],"\u03ba":[26,38,65,71,83,94,133,140],">":[27,134],"\u03c9,":[28],"intersection":[30,67],"two":[33],"closed":[34,40,73,77,99],"subsets":[36],"and":[41,51,78,225],".":[43,137],"proof":[45],"this":[47,129,162],"theorem":[48],"easy":[50],"fact":[53],"yields":[54],"a":[55,84,98,108,156,178,191],"stronger":[56],"result,":[57],"namely":[58],"that":[59,128],"for":[60,82,103,132,139],"uncountable":[62,86],"fewer":[69],"than":[70],"many":[72],"sets":[75],"unbounded.":[79],"Thus,":[80],"if,":[81],"cardinal,":[87],"we":[88,166],"let":[89],"denote":[90],"{":[91],"A":[92,96],"\u2286":[93],"\u2223":[95],"contains":[97],"subset},":[101],"then,":[102],"such":[105],"\u03ba,":[106],"\u03ba-additive":[109],"nonprincipal":[110],"filter":[111],"on":[112],"\u03ba.":[113],"Now":[114],"what":[115],"about":[116],"possibility":[118],"being":[120],"an":[121,150,221],"ultrafilter\u03ba":[122],"It":[123],"routine":[125],"to":[126,219],"see":[127],"impossible":[131,164],"\u2135":[135,142,152],"1":[136,143,153,189],"However,":[138],"=":[141],"situation":[145],"different.":[147],"If":[148],"were":[149],"ultrafilter,":[151],"would":[154],"be":[155,217,220],"measurable":[157,179,192],"cardinal.":[158],"As":[159],"if":[165,173],"assume":[167],"axiom":[169,207],"choice;":[171],"however":[172],"ZF":[174,186],"+":[175,187],"\u201cthere":[176],"exists":[177],"cardinal\u201d":[180,193],"consistent,":[182],"then":[183],"so":[184],"\u201c\u2135":[188],"[2].":[194],"Furthermore,":[195],"under":[196],"assumption":[198],"certain":[200],"theoretic":[202],"axioms":[203],"(such":[204],"as":[205],"determinateness":[209],"or":[210],"various":[211],"infinite":[212],"exponent":[213],"partition":[214],"relations)":[215],"can":[216],"proven":[218],"ultrafilter.":[222],"(See":[223],"[3]":[224],"[5].)":[226]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}