{"id":"https://openalex.org/W1558043875","doi":"https://doi.org/10.1017/s0022481200051860","title":"On \u03a3<sub>1</sub> well-orderings of the universe","display_name":"On \u03a3<sub>1</sub> well-orderings of the universe","publication_year":1976,"publication_date":"1976-03-01","ids":{"openalex":"https://openalex.org/W1558043875","doi":"https://doi.org/10.1017/s0022481200051860","mag":"1558043875"},"language":"en","primary_location":{"id":"doi:10.1017/s0022481200051860","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051860","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/A5103437344","display_name":"Leo Harrington","orcid":null},"institutions":[{"id":"https://openalex.org/I63190737","display_name":"University at Buffalo, State University of New York","ror":"https://ror.org/01y64my43","country_code":"US","type":"education","lineage":["https://openalex.org/I63190737"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Leo Harrington","raw_affiliation_strings":["State University of New York at Buffalo, Amherst, New York 14226","State Univ. of New York at Buffalo"],"affiliations":[{"raw_affiliation_string":"State University of New York at Buffalo, Amherst, New York 14226","institution_ids":["https://openalex.org/I63190737"]},{"raw_affiliation_string":"State Univ. of New York at Buffalo","institution_ids":["https://openalex.org/I63190737"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079801056","display_name":"Thomas Jech","orcid":null},"institutions":[{"id":"https://openalex.org/I63190737","display_name":"University at Buffalo, State University of New York","ror":"https://ror.org/01y64my43","country_code":"US","type":"education","lineage":["https://openalex.org/I63190737"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Thomas Jech","raw_affiliation_strings":["State University of New York at Buffalo, Amherst, New York 14226","State Univ. of New York at Buffalo"],"affiliations":[{"raw_affiliation_string":"State University of New York at Buffalo, Amherst, New York 14226","institution_ids":["https://openalex.org/I63190737"]},{"raw_affiliation_string":"State Univ. of New York at Buffalo","institution_ids":["https://openalex.org/I63190737"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5103437344"],"corresponding_institution_ids":["https://openalex.org/I63190737"],"apc_list":null,"apc_paid":null,"fwci":0.5691,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.57598221,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"1","first_page":"167","last_page":"170"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9972000122070312,"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"}},"topics":[{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9972000122070312,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9961000084877014,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9498999714851379,"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/countable-set","display_name":"Countable set","score":0.8565077185630798},{"id":"https://openalex.org/keywords/universe","display_name":"Universe","score":0.6833000183105469},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6815696954727173},{"id":"https://openalex.org/keywords/forcing","display_name":"Forcing (mathematics)","score":0.6246893405914307},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.5822477340698242},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.5459555387496948},{"id":"https://openalex.org/keywords/partially-ordered-set","display_name":"Partially ordered set","score":0.5119737386703491},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.4986279010772705},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.49706795811653137},{"id":"https://openalex.org/keywords/order-type","display_name":"Order type","score":0.4905499815940857},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49000921845436096},{"id":"https://openalex.org/keywords/tree","display_name":"Tree (set theory)","score":0.48724880814552307},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.4826298654079437},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46522021293640137},{"id":"https://openalex.org/keywords/ordered-set","display_name":"Ordered set","score":0.4494372308254242},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17415472865104675},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.10323882102966309},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08625629544258118}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.8565077185630798},{"id":"https://openalex.org/C84999194","wikidata":"https://www.wikidata.org/wiki/Q1","display_name":"Universe","level":2,"score":0.6833000183105469},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6815696954727173},{"id":"https://openalex.org/C197115733","wikidata":"https://www.wikidata.org/wiki/Q1003136","display_name":"Forcing (mathematics)","level":2,"score":0.6246893405914307},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.5822477340698242},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.5459555387496948},{"id":"https://openalex.org/C180645754","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Partially ordered set","level":2,"score":0.5119737386703491},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.4986279010772705},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.49706795811653137},{"id":"https://openalex.org/C138890893","wikidata":"https://www.wikidata.org/wiki/Q620006","display_name":"Order type","level":2,"score":0.4905499815940857},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49000921845436096},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.48724880814552307},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.4826298654079437},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46522021293640137},{"id":"https://openalex.org/C2985459651","wikidata":"https://www.wikidata.org/wiki/Q474715","display_name":"Ordered set","level":2,"score":0.4494372308254242},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17415472865104675},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.10323882102966309},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08625629544258118},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","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},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C44870925","wikidata":"https://www.wikidata.org/wiki/Q37547","display_name":"Astrophysics","level":1,"score":0.0},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","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},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1017/s0022481200051860","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0022481200051860","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":6,"referenced_works":["https://openalex.org/W1487175521","https://openalex.org/W1522350611","https://openalex.org/W2022040382","https://openalex.org/W2122092051","https://openalex.org/W3211875283","https://openalex.org/W6673102121"],"related_works":["https://openalex.org/W2093159991","https://openalex.org/W2947054308","https://openalex.org/W4288346128","https://openalex.org/W3161339150","https://openalex.org/W4226492519","https://openalex.org/W2949843706","https://openalex.org/W2000176012","https://openalex.org/W1599440049","https://openalex.org/W2907457351","https://openalex.org/W2072370430"],"abstract_inverted_index":{"The":[0,123,148,167],"constructible":[1],"universe":[2],"L":[3,12,47,106,140,155],"of":[4,17,35,40,42,71,85,101,113,126,135,154,173,189,242],"G\u00f6del":[5],"[2]":[6],"has":[7,65,77,92,108,142],"a":[8,19,29,50,58,69,78,99,109,133,143,151,158,161,170,207,210,228,232,236,243,246],"natural":[9],"well-ordering":[10,25,52,82,112,146,213],"&lt;":[11],"given":[13],"by":[14,28,57,156,179,231],"the":[15,114,121,182,239],"order":[16],"construction;":[18],"closer":[20],"look":[21],"reveals":[22],"that":[23,67,107,178,201,214],"this":[24,118,127],"is":[26,55,88,129,132,150,169,215],"definable":[27,51,56],"\u03a3":[30,79,110,144,211],"1":[31,80,111,145,175,212],"formula.":[32,61],"Cohen's":[33],"method":[34],"forcing":[36],"provides":[37],"several":[38,197],"examples":[39],"models":[41],"ZF":[43,102,136],"+":[44,103,137],"V":[45,104,138],"\u2260":[46,105,139],"which":[48,141],"have":[49,186],"but":[53],"none":[54],"relatively":[59],"simple":[60],"Recently,":[62],"Mansfield":[63],"[7]":[64],"shown":[66],"if":[68],"set":[70,220,237],"reals":[72],"(or":[73],"hereditarily":[74],"countable":[75],"sets)":[76],",":[81],"then":[83],"each":[84],"its":[86],"elements":[87],"constructible.":[89],"A":[90],"question":[91,119],"thus":[93],"arisen":[94],"whether":[95],"one":[96,224],"can":[97,193,205],"find":[98],"model":[100,134,149,183,208],"universe.":[115],"We":[116,199],"answer":[117],"in":[120,196,202],"affirmative.":[122],"main":[124],"result":[125],"paper":[128,230],"Theorem.":[130],"There":[131],".":[147,176,222],"generic":[152],"extension":[153],"adjoining":[157],"branch":[159,168],"through":[160],"Suslin":[162],"tree":[163],"with":[164,209],"certain":[165],"properties.":[166],"nonconstructible":[171,187],"subset":[172],"\u2135":[174],"Note":[177],"Mansfield's":[180],"theorem,":[181],"must":[184],"not":[185,216],"subsets":[188],"\u03c9.":[190],"Our":[191],"results":[192],"be":[194],"generalized":[195],"directions.":[198],"note":[200],"particular,":[203],"we":[204],"get":[206],"L[X]":[217],"for":[218],"any":[219],"X":[221],"As":[223],"might":[225],"expect":[226],"from":[227],"joint":[229],"recursion":[233],"theorist":[234],"and":[235,245],"theorist,":[238],"proof":[240],"consists":[241],"construction":[244],"computation.":[247]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2016-06-24T00:00:00"}