{"id":"https://openalex.org/W1963589974","doi":"https://doi.org/10.2307/2270133","title":"The substitution method","display_name":"The substitution method","publication_year":1965,"publication_date":"1965-06-01","ids":{"openalex":"https://openalex.org/W1963589974","doi":"https://doi.org/10.2307/2270133","mag":"1963589974"},"language":"en","primary_location":{"id":"doi:10.2307/2270133","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2270133","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/A5074571537","display_name":"W. W. Tait","orcid":null},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"W. W. Tait","raw_affiliation_strings":["Stanford University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Stanford University","institution_ids":["https://openalex.org/I97018004"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5074571537"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":null,"apc_paid":null,"fwci":5.7746,"has_fulltext":false,"cited_by_count":30,"citation_normalized_percentile":{"value":0.9436865,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"30","issue":"2","first_page":"175","last_page":"192"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9959999918937683,"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"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9959999918937683,"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/T12170","display_name":"History and Theory of Mathematics","score":0.9853000044822693,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"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":0.9750999808311462,"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/substitution","display_name":"Substitution (logic)","score":0.7404539585113525},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.6049902439117432},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6023297905921936},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.5085350275039673},{"id":"https://openalex.org/keywords/predicate","display_name":"Predicate (mathematical logic)","score":0.4836522340774536},{"id":"https://openalex.org/keywords/statement","display_name":"Statement (logic)","score":0.4621013402938843},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.41057395935058594},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.36814212799072266},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3544354736804962},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.34040939807891846},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22331157326698303},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.15210700035095215}],"concepts":[{"id":"https://openalex.org/C2778220771","wikidata":"https://www.wikidata.org/wiki/Q1522579","display_name":"Substitution (logic)","level":2,"score":0.7404539585113525},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.6049902439117432},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6023297905921936},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.5085350275039673},{"id":"https://openalex.org/C140146324","wikidata":"https://www.wikidata.org/wiki/Q1144319","display_name":"Predicate (mathematical logic)","level":2,"score":0.4836522340774536},{"id":"https://openalex.org/C2777026412","wikidata":"https://www.wikidata.org/wiki/Q2684591","display_name":"Statement (logic)","level":2,"score":0.4621013402938843},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.41057395935058594},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.36814212799072266},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3544354736804962},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.34040939807891846},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22331157326698303},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.15210700035095215},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","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/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"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/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2270133","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2270133","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"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183735110","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183735110","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"}],"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/W1975810242","https://openalex.org/W1987888400","https://openalex.org/W2035851396","https://openalex.org/W2072163715","https://openalex.org/W4233221743","https://openalex.org/W4255132490"],"related_works":["https://openalex.org/W2379444625","https://openalex.org/W2808944898","https://openalex.org/W2393147081","https://openalex.org/W2575869988","https://openalex.org/W3196207352","https://openalex.org/W2951724202","https://openalex.org/W3084261076","https://openalex.org/W1996614773","https://openalex.org/W4387820098","https://openalex.org/W2125274893"],"abstract_inverted_index":{"This":[0],"paper":[1,198],"deals":[2],"with":[3,213,239,254,277],"Hilbert's":[4,113,149],"substitution":[5],"method":[6],"for":[7,42,215,266],"eliminating":[8],"bound":[9,36,162],"variables":[10,37],"from":[11],"first":[12,17],"order":[13,18,211],"proofs.":[14],"With":[15],"a":[16,31,55,209,294],"system":[19,32,212],"S":[20,33,46,50,60,84,141,152,194,203,235,244,267,273,276,298],"framed":[21],"in":[22,45,49,59,79,140,174,234,262],"the":[23,26,88,96,117,121,132,144,179,218,228,260,269],"\u03b5-calculus":[24],"[2]":[25],"problem":[27],"is":[28,61,72,205,245,293],"to":[29,62,158,170,207],"associate":[30],"'":[34],"without":[35,248],"and":[38,102],"an":[39,64,278],"effective":[40],"procedure":[41,165],"transforming":[43],"derivations":[44,48],"into":[47,193],"\u2032.":[51,85,195,236],"The":[52,109,161],"transform":[53,122],"of":[54,67,83,90,99,116,123,134,181,220,230,275,281,283,297],"formula":[56],"A":[57,68,80,100,105,124,182],"derived":[58],"be":[63,126,156,168,171,191,208,225],"\u201c\u03b5-substitution":[65],"instance\u201d":[66],",":[69,101],"i.e.":[70,119,250],"it":[71,147],"obtained":[73],"by":[74,81,129,227,264,285],"replacing":[75],"terms":[76,82,92],"\u03b5x":[77],"B(x)":[78],"In":[86,196],"general":[87],"choice":[89],"these":[91],"will":[93],"depend":[94],"on":[95,104,143,289],"particular":[97],"derivation":[98,180],"not":[103,155],"alone.":[106],"Cf.":[107],"[4].":[108],"present":[110],"formulation":[111,150],"sharpens":[112],"original":[114],"statement":[115],"problem,":[118],"that":[120,201],"should":[125],"finitistically":[127],"verifiable,":[128],"making":[130],"explicit":[131],"methods":[133],"verification":[135],"used,":[136],"namely":[137],"those":[138],"formalized":[139],"\u2032;":[142],"other":[145],"hand,":[146],"generalizes":[148],"since":[151],"\u2032":[153,204,268],"need":[154],"restricted":[157],"finitist":[159],"systems.":[160],"variable":[163,271],"elimination":[164],"can":[166,189,224,258],"always":[167],"taken":[169,206],"primitive":[172,187],"recursive":[173],"(the":[175],"G\u00f6del":[176],"number":[177,246],"of)":[178],".":[183],"Constructions":[184],"which":[185,292],"transcend":[186],"recursion":[188],"simply":[190],"built":[192],"this":[197],"we":[199,257],"show":[200],"if":[202],"second":[210],"constants":[214],"functionals,":[216],"then":[217,256],"existence":[219],"suitable":[221],"\u03b5-substitution":[222],"instances":[223],"expressed":[226],"solvability":[229],"certain":[231],"functional":[232],"equations":[233,261],"We":[237],"deal":[238],"two":[240],"cases":[241,286],"here.":[242],"If":[243],"theory":[247],"induction,":[249],"essentially":[251],"predicate":[252],"calculus":[253],"identity,":[255],"solve":[259],"question":[263],"taking":[265],"free":[270],"part":[272],"*":[274],"added":[279],"rule":[280],"definition":[282,288],"functionals":[284],"(recursive":[287],"finite":[290],"ordinals),":[291],"conservative":[295],"extension":[296],"*.":[299]},"counts_by_year":[{"year":2023,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2012,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
