{"id":"https://openalex.org/W1977614750","doi":"https://doi.org/10.2178/bsl/1102022662","title":"Transfinite Progressions: A Second Look at Completeness","display_name":"Transfinite Progressions: A Second Look at Completeness","publication_year":2004,"publication_date":"2004-09-01","ids":{"openalex":"https://openalex.org/W1977614750","doi":"https://doi.org/10.2178/bsl/1102022662","mag":"1977614750"},"language":"en","primary_location":{"id":"doi:10.2178/bsl/1102022662","is_oa":false,"landing_page_url":"https://doi.org/10.2178/bsl/1102022662","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":true,"oa_status":"green","oa_url":"http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-10793","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5036562220","display_name":"Torkel Franz\u00e9n","orcid":null},"institutions":[{"id":"https://openalex.org/I190632392","display_name":"Lule\u00e5 University of Technology","ror":"https://ror.org/016st3p78","country_code":"SE","type":"education","lineage":["https://openalex.org/I190632392"]}],"countries":["SE"],"is_corresponding":true,"raw_author_name":"Torkel Franz\u00e9n","raw_affiliation_strings":["Department of Computer Science and Electrical Engineering, Lule\u00e5 University of Technology, SE-97187, Lule\u00e5, SwedenE-mail:","Department of Computer Science and Electrical Engineering, Lule\u00e5 University of Technology, SE-97187, Lule\u00e5, Sweden E-mail/i: torkel@sm.ltu.se"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science and Electrical Engineering, Lule\u00e5 University of Technology, SE-97187, Lule\u00e5, SwedenE-mail:","institution_ids":["https://openalex.org/I190632392"]},{"raw_affiliation_string":"Department of Computer Science and Electrical Engineering, Lule\u00e5 University of Technology, SE-97187, Lule\u00e5, Sweden E-mail/i: torkel@sm.ltu.se","institution_ids":["https://openalex.org/I190632392"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5036562220"],"corresponding_institution_ids":["https://openalex.org/I190632392"],"apc_list":null,"apc_paid":null,"fwci":1.36625562,"has_fulltext":false,"cited_by_count":29,"citation_normalized_percentile":{"value":0.80395494,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"10","issue":"3","first_page":"367","last_page":"389"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9944999814033508,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9930999875068665,"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"}},{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9919999837875366,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/transfinite-number","display_name":"Transfinite number","score":0.8102123141288757},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.6416478157043457},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.637164831161499},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5849605798721313},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.5608601570129395},{"id":"https://openalex.org/keywords/g\u00f6dels-incompleteness-theorems","display_name":"G\u00f6del's incompleteness theorems","score":0.5052364468574524},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.4735240936279297},{"id":"https://openalex.org/keywords/turing","display_name":"Turing","score":0.4730890393257141},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.4614464342594147},{"id":"https://openalex.org/keywords/completeness","display_name":"Completeness (order theory)","score":0.42721235752105713},{"id":"https://openalex.org/keywords/consistency","display_name":"Consistency (knowledge bases)","score":0.4162468910217285},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3870048522949219},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.38012799620628357},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.36889004707336426},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.36380890011787415},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.31983238458633423},{"id":"https://openalex.org/keywords/g\u00f6del","display_name":"G\u00f6del","score":0.26423245668411255},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2569565176963806},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.09149309992790222}],"concepts":[{"id":"https://openalex.org/C147358099","wikidata":"https://www.wikidata.org/wiki/Q1069891","display_name":"Transfinite number","level":2,"score":0.8102123141288757},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.6416478157043457},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.637164831161499},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5849605798721313},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.5608601570129395},{"id":"https://openalex.org/C180937797","wikidata":"https://www.wikidata.org/wiki/Q200787","display_name":"G\u00f6del's incompleteness theorems","level":3,"score":0.5052364468574524},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.4735240936279297},{"id":"https://openalex.org/C9870796","wikidata":"https://www.wikidata.org/wiki/Q490481","display_name":"Turing","level":2,"score":0.4730890393257141},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.4614464342594147},{"id":"https://openalex.org/C17231256","wikidata":"https://www.wikidata.org/wiki/Q5156540","display_name":"Completeness (order theory)","level":2,"score":0.42721235752105713},{"id":"https://openalex.org/C2776436953","wikidata":"https://www.wikidata.org/wiki/Q5163215","display_name":"Consistency (knowledge bases)","level":2,"score":0.4162468910217285},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3870048522949219},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38012799620628357},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.36889004707336426},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.36380890011787415},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.31983238458633423},{"id":"https://openalex.org/C2781302003","wikidata":"https://www.wikidata.org/wiki/Q3294091","display_name":"G\u00f6del","level":2,"score":0.26423245668411255},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2569565176963806},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.09149309992790222},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","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},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","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}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.2178/bsl/1102022662","is_oa":false,"landing_page_url":"https://doi.org/10.2178/bsl/1102022662","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"},{"id":"pmh:oai:CULeuclid:euclid.bsl/1102022662","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.bsl/1102022662","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.686.8236","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.686.8236","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://lukemuehlhauser.com/wp-content/uploads/Franzen-Transfinite-progressions-a-second-look-at-completeness.pdf","raw_type":"text"},{"id":"pmh:oai:DiVA.org:ltu-10793","is_oa":true,"landing_page_url":"http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-10793","pdf_url":null,"source":{"id":"https://openalex.org/S4306401559","display_name":"KTH Publication Database DiVA (KTH Royal Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:DiVA.org:ltu-10793","is_oa":true,"landing_page_url":"http://urn.kb.se/resolve?urn=urn:nbn:se:ltu:diva-10793","pdf_url":null,"source":{"id":"https://openalex.org/S4306401559","display_name":"KTH Publication Database DiVA (KTH Royal Institute of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":11,"referenced_works":["https://openalex.org/W1571003765","https://openalex.org/W1572451808","https://openalex.org/W1974965205","https://openalex.org/W2000071514","https://openalex.org/W2005337919","https://openalex.org/W2008131652","https://openalex.org/W2028227692","https://openalex.org/W2075471377","https://openalex.org/W2116361905","https://openalex.org/W2977904728","https://openalex.org/W4250417020"],"related_works":["https://openalex.org/W126781605","https://openalex.org/W1574825745","https://openalex.org/W1572152362","https://openalex.org/W2151251236","https://openalex.org/W2765595756","https://openalex.org/W4249126973","https://openalex.org/W2016661762","https://openalex.org/W645268937","https://openalex.org/W3047205667","https://openalex.org/W2111771095"],"abstract_inverted_index":{"\u00a71.":[0],"Iterated":[1],"G\u00f6delian":[2],"extensions":[3,115,235],"of":[4,9,15,36,46,114,119,133,141,147,153,165,185,200,202,228,231,242],"theories":[5,134,142,203],".":[6],"The":[7,156,210],"idea":[8,72],"iterating":[10,206],"ad":[11],"infinitum":[12],"the":[13,65,68,110,151,163,196,214,229],"operation":[14],"extending":[16],"a":[17,23,26,34,124,144,148,172,183],"theory":[18],"T":[19,30,38],"by":[20,85,116,162,205],"adding":[21],"as":[22],"new":[24],"axiom":[25],"G\u00f6del":[27],"sentence":[28],"for":[29],",":[31],"or":[32],"equivalently":[33],"formalization":[35],"\u201c":[37],"is":[39,82,179],"consistent\u201d,":[40],"thus":[41],"obtaining":[42],"an":[43,131,221],"infinite":[44],"sequence":[45,184],"theories,":[47],"arose":[48],"naturally":[49],"when":[50,62],"Godel's":[51],"incompleteness":[52],"theorem":[53],"first":[54],"appeared,":[55],"and":[56,93,96,138,167,225],"occurs":[57],"today":[58],"to":[59,135,181,192,220,236],"many":[60],"non-specialists":[61],"they":[63],"ponder":[64],"theorem.":[66],"In":[67],"logical":[69],"literature":[70],"this":[71,247],"has":[73,190],"been":[74],"thoroughly":[75],"explored":[76],"through":[77,143],"two":[78,243],"main":[79],"approaches.":[80],"One":[81],"that":[83],"initiated":[84],"Turing":[86],"in":[87,101,150,160,223,239],"his":[88],"\u201cordinal":[89],"logics\u201d":[90],"(see":[91],"Gandy":[92],"Yates":[94],"[2001])":[95],"taken":[97],"very":[98],"much":[99],"further":[100],"Feferman's":[102],"work":[103,164],"on":[104,213],"transfinite":[105],"progressions,":[106],"which":[107,120,178],"also":[108],"introduced":[109],"more":[111],"general":[112],"study":[113],"reflection":[117,208,234],"principles,":[118],"consistency":[121],"statements":[122],"are":[123],"special":[125],"case.":[126],"This":[127,187],"approach":[128,189],"starts":[129,169],"from":[130,171],"assignment":[132],"ordinal":[136,154],"notations,":[137],"extracts":[139],"sequences":[140,201],"suitable":[145],"choice":[146],"path":[149],"set":[152],"notations.":[155],"second":[157,188],"approach,":[158,212],"illustrated":[159],"particular":[161,240],"Schmerl":[166],"Beklemishev,":[168],"instead":[170],"suitably":[173],"well-behaved":[174],"primitive":[175],"recursive":[176],"well-ordering,":[177],"used":[180],"define":[182],"theories.":[186],"led":[191],"precise":[193],"results":[194],"about":[195],"relative":[197],"proof-theoretical":[198],"strength":[199],"obtained":[204],"different":[207],"principles.":[209],"Turing-Feferman":[211],"other":[215],"hand,":[216],"lends":[217],"itself":[218],"well":[219],"investigation":[222],"qualitative":[224],"philosophical":[226],"terms":[227],"relevance":[230],"such":[232],"iterated":[233],"mathematical":[237],"knowledge,":[238],"because":[241],"developments":[244],"associated":[245],"with":[246],"approach.":[248]},"counts_by_year":[{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2}],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}
