{"id":"https://openalex.org/W2014596272","doi":"https://doi.org/10.2307/2275705","title":"Grzegorcyk's hierarchy and <i>Iep</i>\u03a3<sub>1</sub>","display_name":"Grzegorcyk's hierarchy and <i>Iep</i>\u03a3<sub>1</sub>","publication_year":1994,"publication_date":"1994-12-01","ids":{"openalex":"https://openalex.org/W2014596272","doi":"https://doi.org/10.2307/2275705","mag":"2014596272"},"language":"en","primary_location":{"id":"doi:10.2307/2275705","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275705","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/A5081729792","display_name":"Gaisi Takeuti","orcid":null},"institutions":[{"id":"https://openalex.org/I2801919071","display_name":"University of Illinois System","ror":"https://ror.org/05e94g991","country_code":"US","type":"education","lineage":["https://openalex.org/I2801919071"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Gaisi Takeuti","raw_affiliation_strings":["Department of Mathematics, University of Illinois, Urbana, Illinois 61801, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Illinois, Urbana, Illinois 61801, E-mail:","institution_ids":["https://openalex.org/I2801919071"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5081729792"],"corresponding_institution_ids":["https://openalex.org/I2801919071"],"apc_list":null,"apc_paid":null,"fwci":0.4643,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.61530517,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"59","issue":"4","first_page":"1274","last_page":"1284"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9990000128746033,"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":0.9990000128746033,"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/T10126","display_name":"Logic, programming, and type systems","score":0.993399977684021,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9900000095367432,"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/mathematical-proof","display_name":"Mathematical proof","score":0.8366603255271912},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7171751260757446},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.6509772539138794},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.6169340014457703},{"id":"https://openalex.org/keywords/proof-theory","display_name":"Proof theory","score":0.6106491684913635},{"id":"https://openalex.org/keywords/characterization","display_name":"Characterization (materials science)","score":0.5702619552612305},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.5347805023193359},{"id":"https://openalex.org/keywords/direct-proof","display_name":"Direct proof","score":0.48461541533470154},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.47550448775291443},{"id":"https://openalex.org/keywords/isomorphism","display_name":"Isomorphism (crystallography)","score":0.4668017029762268},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4653988778591156},{"id":"https://openalex.org/keywords/structural-proof-theory","display_name":"Structural proof theory","score":0.44607239961624146},{"id":"https://openalex.org/keywords/primitive-recursive-function","display_name":"Primitive recursive function","score":0.4206888675689697},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.32906317710876465},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.32846149802207947},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.14728030562400818}],"concepts":[{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.8366603255271912},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7171751260757446},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.6509772539138794},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.6169340014457703},{"id":"https://openalex.org/C2318724","wikidata":"https://www.wikidata.org/wiki/Q852732","display_name":"Proof theory","level":3,"score":0.6106491684913635},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.5702619552612305},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.5347805023193359},{"id":"https://openalex.org/C161505775","wikidata":"https://www.wikidata.org/wiki/Q3240280","display_name":"Direct proof","level":2,"score":0.48461541533470154},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.47550448775291443},{"id":"https://openalex.org/C203436722","wikidata":"https://www.wikidata.org/wiki/Q902950","display_name":"Isomorphism (crystallography)","level":3,"score":0.4668017029762268},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4653988778591156},{"id":"https://openalex.org/C195653647","wikidata":"https://www.wikidata.org/wiki/Q7625047","display_name":"Structural proof theory","level":4,"score":0.44607239961624146},{"id":"https://openalex.org/C9986523","wikidata":"https://www.wikidata.org/wiki/Q1570472","display_name":"Primitive recursive function","level":2,"score":0.4206888675689697},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32906317710876465},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.32846149802207947},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.14728030562400818},{"id":"https://openalex.org/C171250308","wikidata":"https://www.wikidata.org/wiki/Q11468","display_name":"Nanotechnology","level":1,"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/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C115624301","wikidata":"https://www.wikidata.org/wiki/Q895901","display_name":"Crystal structure","level":2,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"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/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2275705","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275705","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":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W169256259","https://openalex.org/W1493890316","https://openalex.org/W1572451808","https://openalex.org/W1971856509","https://openalex.org/W2014082347","https://openalex.org/W2040404640"],"related_works":["https://openalex.org/W2922318846","https://openalex.org/W4234403895","https://openalex.org/W171706349","https://openalex.org/W2953934790","https://openalex.org/W2160215341","https://openalex.org/W153148314","https://openalex.org/W4389513820","https://openalex.org/W2626436722","https://openalex.org/W4387322292","https://openalex.org/W2020595717"],"abstract_inverted_index":{"A":[0],"proof-theoretic":[1,32],"characterization":[2,33],"of":[3,34,68,85,95,110,155,166,232],"the":[4,9,57,80,92,108,127,153,163,169,174,192,219,243,247],"primitive":[5,248],"recursive":[6,249],"functions":[7,13],"is":[8,19,30,91,104,143,200,214,228,239],"\u03a3":[10,16,140,147,160],"1":[11,17,141,148,161],"-definable":[12],"in":[14,21,77,79,89,98,158,168,216],"I":[15,146,204,233],"as":[18,162],"shown":[20],"Mints":[22],"[4],":[23],"Parsons":[24],"[5],":[25],"and":[26,46,171,246],"[8].":[27],"Then":[28,101,150],"what":[29],"a":[31,40,51,64,72,75,86,115,137,156,229],"Grzegorzyk's":[35],"hierarchy?":[36],"First":[37],"we":[38,49,135,151],"discuss":[39],"related":[41],"previous":[42],"work.":[43],"In":[44,132],"Clote":[45],"Takeuti":[47],"[2],":[48],"introduced":[50],"theory":[52,138,220],"TAC":[53,62,78,90,102,111,123,223],"that":[54,122,173,191,194,218],"corresponds":[55,125],"to":[56,74,106,126,145,181,203,222],"computational":[58],"complexity":[59],"class":[60,128,244],"AC.":[61],"has":[63,114,195],"very":[65],"weak":[66],"form":[67],"induction.":[69],"We":[70,120,188],"assign":[71],"rank":[73,84,116,154,177,198],"proof":[76,87,113,157,170],"following":[81],"way.":[82],"The":[83],"P":[88,99],"nesting":[93,164],"number":[94,165],"inductions":[96,167],"used":[97],".":[100,119,131,149,207,236],"i":[103,118,124,130,179,185],"defined":[105],"be":[107],"subtheory":[109],"whose":[112],"\u2264":[117,178],"proved":[121,215],"AC":[129,245],"this":[133],"paper":[134],"introduce":[136],"Iep":[139,159],"which":[142],"equivalent":[144,202],"define":[152],"prove":[172,190],"proofs":[175,196],"with":[176,197],"correspond":[180],"Grzegorcyk's":[182],"hierarchy":[183],"for":[184],"&gt;":[186],"0.":[187],"also":[189],"system":[193],"0":[199,206],"actually":[201],"\u0394":[205,234],"These":[208],"facts":[209],"are":[210],"interesting":[211],"since":[212],"it":[213],"[10]":[217],"isomorphic":[221],"\u2218":[224],"by":[225],"RSUV":[226],"isomorphism":[227],"conservative":[230],"extension":[231],"o":[235],"Therefore":[237],"there":[238],"some":[240],"analogy":[241],"between":[242],"functions.":[250]},"counts_by_year":[{"year":2014,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}