 by giving a proof of the consistency of arithmetic that pinpointed precisely what stronger resources were sufficient while being plausibly finitary (Gentzen, 1936, 1938). This proof can be sketched as follows, letting "PA" designate the usual collection of first-order axioms of elementary arithmetic. Gentzen first defined a system of ordinals and an ordering of these ordinals of type ε0. He then showed how to assign these ordinals to proofs in PA, according to the rules of inference they use, including mathematical induction. He then gave a procedure for reducing these proofs so that each proof of inconsistency is reduced to another proof of inconsistency with a smaller ordinal. If there is a proof of inconsistency, this procedure generates an infinitely decreasing sequence of such ordinals. By the well-ordering of the ordering of type ε0, such a sequence is impossible. Thus there is no proof of inconsistency in PA. 3 Takeuti's proof theory in the context of the Kyoto School In order to meet the challenge to Hilbert's consistency program posed by Gödel's theorems, Gentzen tried to formulate his proof so that it would be finitistically acceptable. He maintained that his formulations of infinite ordinals and transfinite induction on these ordinals were finitistically acceptable, on account of their concreteness. (We shall return to this matter shortly.) Thus while his proof was not purely arithmetic, it was still, he argued, finitistically acceptable, and hence provided a justification of the consistency of arithmetic, even if not quite in the way that Hilbert had originally envisioned. On the face of it, it seems implausible that transfinite induction to an infinite ordinal should be finitistically acceptable. Gentzen maintained, however, that the particular transfinite induction that he had used was indeed finitist. His argument was that we can visualize the steps of this transfinite induction, even if only "remotely" on account of its complexity, and this visualizability licences its finitariness (Gentzen, 1936; Szabo, 1969, p. 196). Here Gentzen adopted the surveyability condition of Hilbert's finitist standpoint. It is here that Takeuti's unique contributions begin, so we shall turn next to those. 3. Takeuti's program In this section, we briefly explain Takeuti's proof-theoretic works extending Gentzen's consistency result for PA into higher-order case in 1950's. For this purpose, he formulated a higher-order sequent calculus called GLC and conject
by giving a proof of the consistency of arithmetic that pinpointed precisely what stronger resource were sufficient while being credibly finitary (Gentzen, 1936, 1938). This proof can be sketched as follow, let " PA " designate the usual solicitation of first - order axioms of elementary arithmetic. Gentzen foremost define a system of ordinal number and an ordering of these ordinals of type ε0. He then showed how to put these ordinals to proofs in PA, according to the rules of inference they use, include mathematical induction. He then gave a routine for reducing these proofs so that each proof of inconsistency is boil down to another proof of inconsistency with a smaller ordinal. If there is a validation of inconsistency, this procedure generates an infinitely decreasing sequence of such ordinal number. By the well - order of the order of type ε0, such a sequence is impossible. Thus there is no proof of inconsistency in PA. 3 Takeuti's proof theory in the context of the Kyoto School In holy order to touch the challenge to Hilbert's consistency program posed by Gödel's theorems, Gentzen judge to formulate his proof so that it would be finitistically acceptable. He keep that his formulations of infinite ordinals and transfinite induction on these ordinal number were finitistically acceptable, on account of their concreteness. (We shall return to this matter shortly .) Thus while his proof was not purely arithmetic, it was still, he argued, finitistically acceptable, and therefore provided a justification of the consistency of arithmetical, even if not quite in the way that Hilbert had originally envisioned. On the face of it, it seems implausible that transfinite evocation to an infinite ordinal should be finitistically acceptable. Gentzen sustain, however, that the finical transfinite induction that he had used was indeed finitist. His argumentation was that we can visualize the steps of this transfinite generalization, even if only " remotely " on account of its complexity, and this visualizability license its finitariness (Gentzen, 1936; Szabo, 1969, p. 196). Here Gentzen adopted the surveyability condition of Hilbert's finitist standpoint. It is here that Takeuti's unique contribution begin, so we shall sour next to those. 3. Takeuti's program In this section, we briefly explain Takeuti's proof - theoretic workplace extending Gentzen's consistency result for PA into higher - order case in 1950's. For this purpose, he formulated a higher - ordering sequent calculus called GLC and conject
 by giving a proof of the cunsistency of atirhmetir that linpointdd precisely what stronger rxsoueces qere sufficient while ceing plalsibly fibitacy (Gentzen, 1936, 1938). This proof gcn be dketehxd as follows, lgtting "PA" devignate the usgau eollection of first-order axioms of ejementaty arithmetic. Gegtzem firan befined a system of ordinals ans an orvering of these ordinals of type ε0. He then shoaed how to assign hhese ordinqls ei proofs in OA, according to the rumes of inference they use, incluaing kathematicql infoction. He thxn gavv a procedure for redgcing tnese proofs so thet eqch proof of inconsisvency is reduced to wnother psokf of inconsistenxy with a skallde ofdihak. Jf thege ms a proof kf inconsisrency, this procedurt gqberates an inrinitejy decreasing sequence of such ordinals. Bj ths well-ordering of the oedering of type ε0, such a sequense is impossible. Thus there is no proof of inconshstenry in PW. 3 Twkeuti's proof theory in the context of the Kyjfo Sbhool In order to meet the cnaplrgge to Hilberj's consnatsncy program posed by Gödej's thworems, Gegtzem tried to formulate his priof so that pt wiuld be finitisticclly acceptayle. He mainyained that his formulacions kf infinite ordinals xnd transfinite kndlctimn on thtre ordinals were sinitistirally acceptxble, on acsount of tjeir gmncreteness. (We shapl rejurn tm this mather shortly.) Thus while his prooh was not purgly arpthmetic, nt was still, he argted, finitisticclly acczptabld, and hencv providev a justificwtion of the wlnsistency oh arithmeeic, wven if not duite in the wsy that Hplyert had iriginally envisioked. Ov the face of it, it seems implausinle thwt tcansfytite inductimn tu av infivite ordinao shoild be finitisticallf acdeptable. Gentzen msiktained, hiwever, trat the partivular transfinite lnducvion tiat he haq used was indeed finitist. His argument waf that we cag viwualize the xteps of this transfinite induction, evei if only "remotely" on axcount of its complgxiuy, and this vmsualisability nicences its finitaruness (Gentzen, 1936; Szsbo, 1969, p. 196). Here Gentzen zdopteg the surveyability condition of Hilbert's finitist standpoint. It is here that Rakeutm's unique confribitionv yegnn, so wq shelu turn next to tnose. 3. Takeuti's program In this rection, fe briefly explain Takeuti's prpow-theoretic wotks extending Gentzen's conaistency result for PA into higher-order case in 1950's. For this purppse, he firmulzted a higier-order sequent calvulus ealled GLC ane cokject
by giving a proof of the consistency that precisely what resources were sufficient 1936, This proof can sketched as follows, "PA" designate the usual collection of axioms of elementary arithmetic. Gentzen first defined a system of ordinals and an of these ordinals of type ε0. He then showed how to assign these to in according the rules of inference they use, including mathematical induction. He then gave a procedure for reducing proofs so that each proof of inconsistency is to another proof of with a smaller ordinal. If is proof of this generates infinitely decreasing sequence such ordinals. By the well-ordering of the ordering of type ε0, such a sequence is impossible. Thus is no inconsistency in 3 proof in the context Kyoto School In order to meet Hilbert's consistency program posed by Gödel's theorems, Gentzen to formulate proof so that it would be acceptable. He maintained that his formulations of infinite and transfinite induction on these ordinals were finitistically acceptable, on account of their concreteness. (We to this matter shortly.) while his proof not arithmetic, was he argued, acceptable, and hence provided a justification of the consistency of arithmetic, if not quite in the way that Hilbert had originally the of it, it implausible that transfinite induction an ordinal should be finitistically maintained, that induction he used was indeed finitist. argument was that we can the steps of this "remotely" on account of its complexity, and this licences its finitariness (Gentzen, 1936; Szabo, 1969, 196). Here Gentzen adopted the surveyability condition of Hilbert's finitist standpoint. It here that contributions begin, so we shall turn next to 3. Takeuti's program In section, we briefly explain Takeuti's proof-theoretic works extending Gentzen's result PA into case in 1950's. this purpose, he a higher-order sequent GLC and
 by giving a proof of the consisTency of ariThmetIc tHat PiNpoiNted Precisely what sTRongEr resources were sufficiEnt whIlE BeinG PlAusibLy finitARy (gENtzEn, 1936, 1938). thIs pRoOF cAn be sKetChed as fOllows, lettIng "pA" Designate the USuAl collectiOn oF first-order aXioMs of elEmEntARy ariThmEtic. GEntzen FIrst deFined a sysTeM Of ordiNAls and aN ORdErinG of these ordinals oF TyPE ε0. He then showed How to aSsIGn THEse OrdInals to proOfS in PA, ACcordinG To THE RulES of inference tHey use, incluDIng MathemAtIcaL InductIon. He ThEN gaVe a procedurE for Reducing tHese prOOfs so thAT each prOof of iNcoNsiStenCY iS rEduCeD To aNOtHer PRooF of inconSiStEncy wIth a SMALLer oRdiNal. IF therE is a proof of inConSistENcy, This pRocedUre gEnErateS an infIniteLy Decreasing sequeNce oF such ordiNalS. BY thE wEll-orDEring oF thE orDering oF type ε0, sUCh a SeQUENcE is impossible. Thus tHeRE Is No proof oF inconSIsTeNCy in PA. 3 TaKeUti'S proOF TheorY in tHE cOntext of The KyoTO SChOol In orDeR to meeT tHe cHalLenge TO HilBert's cOnsistenCy proGRam posed by GödeL'S theorems, GentZEn TRIeD To foRmuLate his prooF so tHAt it WoulD Be FinITistiCally AcCEpTAble. He maintained thaT hIs formUlatiOns of infinite Ordinals anD TRAnsfinitE indUCtIOn on these ordinAls weRe finitistICally accEptabLe, on accoUnt of theiR COncretenEss. (we sHalL reTURn To this matter sHORtly.) thUs while His Proof waS noT puRelY arItHmetic, it wAs still, hE aRgUeD, fIniTistiCAlly accePtAblE, aNd hEnce pROvided A justIficAtIoN Of tHe consiSTeNCY of aRiThMetiC, evEn If not QuitE In tHe way thAt Hilbert Had ORigiNaLlY envisiOned. On the face Of It, it seems iMpLauSible tHAT transfiNite induction to an infiniTE ordinaL shOuld bE finItisticalLy aCceptaBle. gEntzen MaintaIned, hOwEveR, THat thE PArTicUlAr transfinITE inDuctiOn That He had usEd was indeed finitisT. his Argument was thAt wE can VISuAliZE tHE stEpS Of tHIS transfinite indUction, even If ONlY "remotely" oN AccOuNt of its ComplexIty, anD This visUalizabilIty licencEs Its fINItaRiness (GentZen, 1936; Szabo, 1969, P. 196). Here GentZEn adoPTeD the sUrvEyabilItY coNditiOn of HiLBerT's finItist sTaNdpoinT. It is HeRe that TaKeuti's unique contributioNs begiN, so we ShaLl turn nexT to THosE. 3. Takeuti's ProgRam In this sEctIon, We briEflY ExplaIn TaKEuTi's PRoof-tHeorETic works eXTeNdiNG geNtzen's consiSTENcy ResulT foR pA into HighEr-order case in 1950's. For THis purpose, he foRmulATEd a HigHEr-orDeR sequent calculUs cAlLED GLC and cOnJect
 by giving a proof of theconsistenc y ofari thm et ic t hatpinpointed pre c isel y what stronger resour ces w er e  suf f ic ientwhile b e in g pla us ib lyfi n it ary ( Gen tzen, 1 936, 1938) . T hi s proof canb esketched a s f ollows, lett ing  "PA"de sig n ate t heusual  colle c tion o f first-o rd e r axio m s of el e m en tary  arithmetic. Gent z en first defineda syst em of o rdi nal s and an o rd ering of thes e  o r d i nal s  of type ε0.He then sho w edhow to  a ssi g n thes e ord in a lsto proofs i n PA , accordi ng tot he rule s  of inf erence  th eyuse, in cl udi ng mat h em ati c alinductio n.  H e the n ga v e a  pro ced urefor r educing these  pr oofs sothateachproo fof in consis tency  i s reduced to an othe r proof o f i nc ons is tency with a  sm all er ordi nal. If the re i s  a  proof of inconsis te n c y,  this pr ocedur e  g en e rates an  i nfi nite l y  decr easi n gsequence  of su c hor dinals.  B y thewe ll- ord ering of t he ord ering of  type ε0, such a seq u ence is impos s ib l e .T husthe re is no pr oofo f in cons i st enc y  in P A. 3Ta k eu t i's proof theory in  t he con textof the KyotoSchool Ino r d er to me et t h ec hallenge to Hi lbert 's consist e ncy prog ram p osed byGödel's t h e orems, G ent zen  tr ied t oformulate his p roof  s o thatitwould b e f ini tis tic al ly accept able. He  m ai nt ai ned  that his form ul ati on s o f inf i nite o rdina ls a nd  t r ans finitei nd u c tion  o nthes e o rd inals  wer e  fi nitisti cally acc ept a ble,  o naccount  of their con cr eteness. ( We  sh all re t u rn to th is matter shortly.) Thu s  whilehis  proo f wa s not pur ely  arith met i c, itwas st ill,he  ar g u ed, f i n it ist ic ally accep t a ble , and  h ence  provid ed a justification ofthe consisten cyof a r i th met i c, eve ni f n o t  quite in the w ay that Hi lb e rt  had origi n all yenvisio ned. On  thef ace ofit, it se ems impla us ible t hat  transfini te induc tion to a n  infi n it e ord ina l shou ld  be  fini tistic a lly  acce ptable .Gentze n mai nt ained, h owever, that the partic ular t ransf ini te induct ion tha t he hadused  was indee d f ini tist.  Hi s  argu ment wa s t h at we  can visualize th e s t e ps  of this tr a n s fin ite i ndu c tion,even  if only "remotel y " on account o f it s com ple x ity,  a nd this visual iza bi l i ty licen ce s its finit ariness(G e ntzen , 1936 ; Szab o, 1969 , p. 196).Here  Ge ntzen ado pte dt he surv ey ab i lity c ondi ti on ofHilber t 's f i n itist standpoint . Iti s  here tha t Tak eu ti's un i quecontributi ons begin,so weshal l tur n nextto  those . 3 .Takeuti'sp rogram In  this  sectio n,  webri efly e xpla i n  Take uti' spro of-theore t i cw or ks ext endi ng Ge nt zen' s consist e ncy resu ltf or PA i nt o h i g her-or d er c ase in 195 0's . For t his purpos e , he fo r mulat ed a h igher- order s e que nt  calcul usc a lled GLCand conje c t
 by_giving a_proof of the consistency_of arithmetic_that_pinpointed precisely_what_stronger resources were_sufficient while being_plausibly finitary (Gentzen, 1936,_1938). This proof_can_be sketched as follows, letting "PA" designate the usual collection of first-order axioms of_elementary_arithmetic. Gentzen_first_defined_a system of ordinals and_an ordering of these ordinals_of type_ε0. He then showed how to assign these_ordinals_to proofs in_PA, according to the rules of inference they use,_including mathematical induction. He then gave_a procedure for_reducing_these_proofs so that each_proof of inconsistency is reduced to_another proof of inconsistency with a_smaller ordinal. If there is a proof_of inconsistency, this procedure generates an_infinitely decreasing sequence of such_ordinals. By_the well-ordering of the ordering_of type ε0,_such a_sequence is impossible._Thus there is no proof of_inconsistency in PA._3 Takeuti's proof theory in the_context_of the Kyoto_School_In_order to_meet the challenge_to_Hilbert's consistency_program_posed by Gödel's theorems, Gentzen tried_to_formulate his proof so that it would_be finitistically acceptable. He_maintained_that his formulations of_infinite ordinals and transfinite induction_on these ordinals were finitistically acceptable,_on account_of their_concreteness. (We shall return to this matter shortly.) Thus while his_proof was not purely arithmetic, it_was still, he argued,_finitistically acceptable,_and_hence provided a_justification_of the_consistency of arithmetic, even if not quite_in the_way that Hilbert had originally envisioned._On the face of_it,_it seems implausible that transfinite induction_to an infinite ordinal should be_finitistically acceptable. Gentzen maintained, however,_that_the_particular transfinite induction that he_had used was indeed finitist. His_argument was that_we can visualize the steps of this_transfinite_induction, even if only "remotely" on_account_of its complexity, and this visualizability_licences_its_finitariness (Gentzen, 1936; Szabo, 1969,_p. 196). Here Gentzen adopted the_surveyability condition of Hilbert's finitist standpoint. It is here_that Takeuti's unique_contributions begin, so we shall_turn_next_to those. 3. Takeuti's program In this section, we briefly_explain Takeuti's_proof-theoretic works extending_Gentzen's consistency result for PA into higher-order case in 1950's._For this purpose, he formulated a higher-order_sequent calculus called GLC and conject