{"id":"https://openalex.org/W2148637706","doi":"https://doi.org/10.2307/421099","title":"An Elementary Approach to the Fine Structure of <i>L</i>","display_name":"An Elementary Approach to the Fine Structure of <i>L</i>","publication_year":1997,"publication_date":"1997-12-01","ids":{"openalex":"https://openalex.org/W2148637706","doi":"https://doi.org/10.2307/421099","mag":"2148637706"},"language":"en","primary_location":{"id":"doi:10.2307/421099","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421099","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/A5110295341","display_name":"Sy D. Friedman","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":true,"raw_author_name":"Sy D. Friedman","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA.E-mail:","Massachusetts Institute Of Technology Cambridge"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology Cambridge, MA 02139, USA.E-mail:","institution_ids":["https://openalex.org/I63966007"]},{"raw_affiliation_string":"Massachusetts Institute Of Technology Cambridge","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5078814566","display_name":"Peter Koepke","orcid":"https://orcid.org/0000-0002-2266-134X"},"institutions":[{"id":"https://openalex.org/I135140700","display_name":"University of Bonn","ror":"https://ror.org/041nas322","country_code":"DE","type":"education","lineage":["https://openalex.org/I135140700"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Peter Koepke","raw_affiliation_strings":["Mathematisches Institut, Universit\u00e4t Bonn, Beringstrasse 4, D-53115 Bonn, Germany.E-mail:","Universit\u00e4t Bonn"],"affiliations":[{"raw_affiliation_string":"Mathematisches Institut, Universit\u00e4t Bonn, Beringstrasse 4, D-53115 Bonn, Germany.E-mail:","institution_ids":["https://openalex.org/I135140700"]},{"raw_affiliation_string":"Universit\u00e4t Bonn","institution_ids":["https://openalex.org/I135140700"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5110295341"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":0.5397,"has_fulltext":false,"cited_by_count":14,"citation_normalized_percentile":{"value":0.65901751,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"3","issue":"4","first_page":"453","last_page":"468"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9973999857902527,"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":0.9973999857902527,"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":0.7701477408409119},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.5168648362159729},{"id":"https://openalex.org/keywords/transitive-relation","display_name":"Transitive relation","score":0.5045956373214722},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.49846911430358887},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.49353158473968506},{"id":"https://openalex.org/keywords/continuum-hypothesis","display_name":"Continuum hypothesis","score":0.4787244498729706},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4769255518913269},{"id":"https://openalex.org/keywords/elementary-proof","display_name":"Elementary proof","score":0.41951116919517517},{"id":"https://openalex.org/keywords/cantor-set","display_name":"Cantor set","score":0.41093266010284424},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1182565689086914}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7701477408409119},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5168648362159729},{"id":"https://openalex.org/C191399111","wikidata":"https://www.wikidata.org/wiki/Q64861","display_name":"Transitive relation","level":2,"score":0.5045956373214722},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.49846911430358887},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.49353158473968506},{"id":"https://openalex.org/C96488702","wikidata":"https://www.wikidata.org/wiki/Q208416","display_name":"Continuum hypothesis","level":2,"score":0.4787244498729706},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4769255518913269},{"id":"https://openalex.org/C2777478746","wikidata":"https://www.wikidata.org/wiki/Q5358910","display_name":"Elementary proof","level":2,"score":0.41951116919517517},{"id":"https://openalex.org/C36605576","wikidata":"https://www.wikidata.org/wiki/Q273188","display_name":"Cantor set","level":2,"score":0.41093266010284424},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1182565689086914},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","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},{"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:10.2307/421099","is_oa":false,"landing_page_url":"https://doi.org/10.2307/421099","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"}],"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/W2081899681","https://openalex.org/W2122092051","https://openalex.org/W2285371954","https://openalex.org/W4247372698"],"related_works":["https://openalex.org/W3037385063","https://openalex.org/W2122976707","https://openalex.org/W4311829300","https://openalex.org/W3014978195","https://openalex.org/W2779079174","https://openalex.org/W3126108959","https://openalex.org/W2391351106","https://openalex.org/W2963369738","https://openalex.org/W2025458971","https://openalex.org/W2148637706"],"abstract_inverted_index":{"We":[0,31],"present":[1],"here":[2],"an":[3,234],"approach":[4],"to":[5,109,179,291],"the":[6,34,40,52,56,81,103,111,151,181,191,195,204,212,218,242,293,304],"fine":[7],"structure":[8],"of":[9,25,37,42,55,71,74,106,114,124,153,158,166,185,199,203,217,257],"L":[10,29,83,105,118,129,133,138,146,159,176,208,219,231,263,296,308],"based":[11,210],"solely":[12],"on":[13,211,249],"elementary":[14,235],"model":[15,104,165,184],"theoretic":[16,197],"ideas,":[17],"and":[18,39,45,76,279,301],"illustrate":[19],"its":[20],"use":[21,70],"in":[22,28,51,194,207,295,297,307],"a":[23,72,122,252,255,267],"proof":[24,202],"Global":[26],"Square":[27],".":[30,97,161,287,309],"thereby":[32],"avoid":[33],"L\u00e9vy":[35],"hierarchy":[36,73],"formulas":[38],"subtleties":[41],"master":[43],"codes":[44],"projecta,":[46],"introduced":[47],"by":[48,247,266],"Jensen":[49],"[3]":[50],"original":[53],"form":[54],"theory.":[57,187],"Our":[58],"theory":[59,108,168],"could":[60],"appropriately":[61],"be":[62,170,180,245],"called":[63],"\u201dHyperfine":[64],"Structure":[65],"Theory\u201d,":[66],"as":[67,121,277],"we":[68],"make":[69],"structures":[75],"hull":[77],"operations":[78],"which":[79,127],"refines":[80],"traditional":[82],"\u03b1":[84,87,139,147,160,232,264],"-or":[85],"J":[86],"-sequences":[88],"with":[89],"their":[90],"\u03a3":[91],"n":[92],"-hull":[93],"operations.":[94],"\u00a71.":[95],"Introduction":[96],"In":[98],"1938,":[99],"K.":[100],"G\u00f6del":[101],"defined":[102,120],"set":[107,167,186,196],"show":[110],"relative":[112],"consistency":[113],"Cantor's":[115],"Continuum":[116,305],"Hypothesis.":[117],"is":[119,209,233,259,271],"union":[123],"initial":[125],"segments":[126],"satisfy:":[128],"0":[130],"=":[131,135,150],"\u2205,":[132],"\u03bb":[134],"\u222a":[136],"\u03b1&lt;\u03bb":[137],"for":[140],"limit":[141],"ordinals":[142],"\u03bb,":[143],"and,":[144],"crucially,":[145],"+":[148],"1":[149,286,300],"collection":[152],"1st":[154,173],"order":[155,174],"definable":[156,260,272],"subsets":[157],"Since":[162],"every":[163],"transitive":[164],"must":[169],"closed":[171],"under":[172],"definability,":[175],"turns":[177],"out":[178],"smallest":[182],"inner":[183],"Thus":[188],"it":[189,270],"occupies":[190],"central":[192],"place":[193],"spectrum":[198],"models.":[200],"The":[201,221],"continuum":[205],"hypothesis":[206],"very":[213],"uniform":[214],"hierarchical":[215],"definition":[216],"-hierarchy.":[220],"Condensation":[222],"Lemma":[223],"states":[224],"that":[225],"if":[226],"\u03c0":[227],":":[228],"M":[229,237,276],"\u2192":[230],"embedding,":[236],"transitive,":[238],"then":[239],"some":[240,262,274,282],";":[241],"lemma":[243],"can":[244],"proved":[246],"induction":[248],"\u03b1.":[250],"If":[251],"real,":[253],"i.e.,":[254],"subset":[256],"\u03c9,":[258],"over":[261,273,281],",then":[265],"L\u00f6wenheim-Skolem":[268],"argument":[269],"countable":[275],"above,":[278],"hence":[280],",":[283],"&lt;":[284],"\u03c9":[285,299],"This":[288],"allows":[289],"one":[290],"list":[292],"reals":[294],"length":[298],"therefore":[302],"proves":[303],"Hypothesis":[306]},"counts_by_year":[{"year":2019,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
