{"id":"https://openalex.org/W1551473042","doi":"https://doi.org/10.2307/2272350","title":"\u03a31 compactness for next admissible sets","display_name":"\u03a31 compactness for next admissible sets","publication_year":1974,"publication_date":"1974-03-01","ids":{"openalex":"https://openalex.org/W1551473042","doi":"https://doi.org/10.2307/2272350","mag":"1551473042"},"language":"en","primary_location":{"id":"doi:10.2307/2272350","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272350","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/A5004721342","display_name":"Judy Green","orcid":null},"institutions":[{"id":"https://openalex.org/I102322142","display_name":"Rutgers, The State University of New Jersey","ror":"https://ror.org/05vt9qd57","country_code":"US","type":"education","lineage":["https://openalex.org/I102322142"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Judy Green","raw_affiliation_strings":["Rutgers, The State University of New Jersey, Camden, New Jersey 08102"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Rutgers, The State University of New Jersey, Camden, New Jersey 08102","institution_ids":["https://openalex.org/I102322142"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5004721342"],"corresponding_institution_ids":["https://openalex.org/I102322142"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":1.0595,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.75039002,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"39","issue":"1","first_page":"105","last_page":"116"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9904000163078308,"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":0.9904000163078308,"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":0.9501000046730042,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9193000197410583,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/uncountable-set","display_name":"Uncountable set","score":0.9148067831993103},{"id":"https://openalex.org/keywords/compact-space","display_name":"Compact space","score":0.8910099267959595},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7525995969772339},{"id":"https://openalex.org/keywords/cofinality","display_name":"Cofinality","score":0.6681705713272095},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.5638088583946228},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5512519478797913},{"id":"https://openalex.org/keywords/relatively-compact-subspace","display_name":"Relatively compact subspace","score":0.5027244091033936},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.49711325764656067},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.48392611742019653},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.46142059564590454},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.44542351365089417},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.4355486333370209},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.4119272828102112},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22851309180259705},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16025254130363464}],"concepts":[{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.9148067831993103},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.8910099267959595},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7525995969772339},{"id":"https://openalex.org/C148070024","wikidata":"https://www.wikidata.org/wiki/Q1283623","display_name":"Cofinality","level":4,"score":0.6681705713272095},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.5638088583946228},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5512519478797913},{"id":"https://openalex.org/C33231828","wikidata":"https://www.wikidata.org/wiki/Q610232","display_name":"Relatively compact subspace","level":3,"score":0.5027244091033936},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.49711325764656067},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.48392611742019653},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.46142059564590454},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.44542351365089417},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4355486333370209},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.4119272828102112},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22851309180259705},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16025254130363464},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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/C171250308","wikidata":"https://www.wikidata.org/wiki/Q11468","display_name":"Nanotechnology","level":1,"score":0.0},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2272350","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2272350","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":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":8,"referenced_works":["https://openalex.org/W65053296","https://openalex.org/W1526907582","https://openalex.org/W1576560275","https://openalex.org/W2046306553","https://openalex.org/W2131495252","https://openalex.org/W2173438417","https://openalex.org/W3159824982","https://openalex.org/W4291439952"],"related_works":["https://openalex.org/W2103474290","https://openalex.org/W2973517705","https://openalex.org/W4288278828","https://openalex.org/W2965212342","https://openalex.org/W4302553137","https://openalex.org/W2994874415","https://openalex.org/W2108403101","https://openalex.org/W2017393767","https://openalex.org/W2128485936","https://openalex.org/W1971926932"],"abstract_inverted_index":{"Let":[0],"\u03c3":[1,32],"be":[2],"any":[3],"sequence":[4],"B0,":[5],"B1":[6],"\u2026,":[7],"Bn,":[8],"\u2026":[9],"of":[10,45,89,96,105,119,123,186,195,206,226,234,253,260],"transitive":[11],"sets":[12,76,93],"closed":[13],"under":[14],"pairs":[15],"with":[16,31],"for":[17,54],"each":[18],"n.":[19],"In":[20,128,166,213],"this":[21],"paper":[22],"we":[23,39,171,220],"show":[24],"that":[25,72,137],"the":[26,90,109,117,134,146,176,199,207,223,231],"smallest":[27],"admissible":[28,75,143],"set":[29,142,252,259],"A\u03c3":[30,34],"\u2208":[33],"is":[35,140,152,241,250],"\u03a31":[36,49,56,78,91,110,125,153],"compact.":[37,154],"Thus":[38],"have":[40],"an":[41],"entirely":[42],"new":[43,218],"class":[44,122],"explicitly":[46],"describable":[47],"uncountable":[48,55,124],"compact":[50,57,66,79,92,126],"sets.":[51,127],"The":[52],"search":[53],"languages":[58,233],"goes":[59],"back":[60],"to":[61,85,116,175,198,215,243],"Hanf's":[62],"negative":[63],"results":[64],"on":[65,84,245,257],"cardinals":[67],"[7].":[68],"Barwise":[69,130,177],"first":[70],"showed":[71,131],"all":[73],"countable":[74],"were":[77],"[1]":[80],"and":[81,150,181,228],"then":[82],"went":[83],"give":[86],"a":[87,183,204,239,246,251,258],"characterization":[88,102],"in":[94,107,237],"terms":[95],"strict":[97,147],"reflection":[98,148],"[2].":[99],"While":[100],"his":[101],"has":[103,114],"been":[104],"interest":[106],"understanding":[108],"compactness":[111,169],"phenomenon":[112],"it":[113,144],"led":[115],"identification":[118],"only":[120],"one":[121],"particular,":[129],"[2],":[132],"using":[133,162],"above":[135,211],"notation,":[136],"if":[138],"\u22c3nBn":[139],"power":[141],"satisfies":[145],"principle":[149],"hence":[151],"(This":[155],"result":[156,209,219],"was":[157],"obtained":[158],"independently":[159],"by":[160],"Karp":[161,235],"algebraic":[163],"methods":[164],"[9].)":[165],"proving":[167],"our":[168,217],"theorem":[170],"follow":[172],"Makkai's":[173,196],"approach":[174],"Compactness":[178],"Theorem":[179],"[12]":[180],"use":[182,229],"modified":[184],"version":[185],"Smullyan's":[187],"abstract":[188],"consistency":[189],"property":[190],"[14].":[191],"A":[192],"direct":[193],"generalization":[194],"method":[197],"cofinality":[200],"\u03c9":[201],"case":[202],"yields":[203],"proof":[205],"Barwise-Karp":[208],"mentioned":[210],"[6].":[212],"order":[214],"obtain":[216],"depart":[221],"from":[222],"usual":[224],"definition":[225],"language":[227],"instead":[230],"indexed":[232],"[9]":[236],"which":[238],"conjunction":[240],"considered":[242],"operate":[244],"function":[247],"whose":[248],"range":[249],"formulas":[254,261],"rather":[255],"than":[256],"itself.":[262]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}