 is called "ordinal diagrams" (expressing his reduction steps for a subsystem of analysis) (Takeuti, 1957). For the termination proof of his reduction steps, he assigned ordinal diagrams to proof-figures and showed in 1958 (Takeuti, 1958b) that if a proof-figure d is reduced into another one r(d), then the corresponding ordinals are decreasing. The point here is that the definition of his reduction steps and the assignment of ordinals must be done simultaneously in order to complete the termination proof. He successively published partial solutions to this conjecture (Takeuti, 1955a,b, 1956c,d, 1958a), and in particular he proved the consistency of an impredicative subsystem of analysis corresponding to (Π11 − CA)0 in a modern terminology. To conclude his consistency proof, it is necessary to prove the key lemma saying that his ordinal diagrams are well-founded. Indeed, Takeuti had tried to prove the well-foundedness of ordinal diagrams several times. His first argument is given in his seminal paper in 1957 (Takeuti, 1957) where ordinal diagrams were introduced. However, since this proof is not very satisfying according to his philosophical standpoint, other proofs are presented several times (2). 5 Takeuti's proof theory in the context of the Kyoto School 4. Takeuti's philosophical perspective Reflecting on Takeuti's foundational projects as explained in the previous section, the following two thematic aims emerge: 1) to extend his finitist standpoint and to prove the consistency of a strong subsystem of his second-order or higher-order logic; and 2) to reflect on what is achieved by such results. To investigate this, we focus on the paper「数学について」(About mathematics) published in 1972 (Takeuti, 1972). A key term in this paper is "infinite mind", which Takeuti explains was suggested by Gödel in Takeuti (1985). First, let us quote the following passage. Usually, it is essential for modern mathematics that it supposes an infinite mind and conjectures what it does. By infinite mind, I mean a mind which can investigate infinitely many things by checking one by one. For example, the law of excluded middle holds for an infinite mind because it can check whether A(x) or ¬A(x) one by one. Similarly, it can see {x|A(x)} since it can check whether A(x) or not one by one. By contrast, a human mind
is called " ordinal diagrams " (expressing his reduction footstep for a subsystem of psychoanalysis) (Takeuti, 1957). For the termination proof of his decrease steps, he assign ordinal diagrams to proof - name and showed in 1958 (Takeuti, 1958b) that if a proof - human body d is reduced into another one r(d), then the represent ordinals are decreasing. The decimal point here is that the definition of his decrease steps and the assignment of ordinal number must be done simultaneously in orderliness to complete the termination proof. He successively published fond solutions to this conjecture (Takeuti, 1955a, b, 1956c, d, 1958a), and in especial he proved the consistency of an impredicative subsystem of psychoanalysis corresponding to (Π11 − CA)0 in a advanced terminology. To conclude his consistency proof, it is necessary to prove the cardinal lemma saying that his ordinal diagrams are well - founded. Indeed, Takeuti had tried to prove the well - foundedness of ordinal diagrams respective fourth dimension. His first argument is given in his seminal paper in 1957 (Takeuti, 1957) where ordinal diagrams were introduced. However, since this proof is not very satisfying harmonize to his philosophical standpoint, early proof are deliver several times (2). 5 Takeuti's proof hypothesis in the context of the Kyoto School 4. Takeuti's philosophical position Reflecting on Takeuti's foundational projects as explained in the former section, the following two thematic aims emerge: 1) to extend his finitist standpoint and to test the consistency of a strong subsystem of his second - order or higher - order logic; and 2) to chew over on what is achieved by such resultant role. To investigate this, we concentrate on the paper「数学について」(About mathematics) published in 1972 (Takeuti, 1972). A key term in this paper is " countless mind ", which Takeuti explains was suggested by Gödel in Takeuti (1985). First, let us quote the following passage. Usually, it is essential for modern mathematics that it presuppose an infinite thinker and conjectures what it does. By countless mind, I mean a mind which can investigate infinitely many thing by checking one by one. For case, the law of excluded center holds for an countless mind because it can check whether A(x) or ¬A(x) one by one. Similarly, it can see { x|A(x) } since it can control whether A(x) or not one by one. By contrast, a human mind
 is called "ordinal diagrams" (expressing his reductmon stels for a subsystem of analysis) (Takeuvi, 1957). Dor tye termination proof ow his redlction stwps, ie assigned ordiizl diagvcms tk promh-figures and shpwed in 1958 (Tdkeuti, 1958b) that hf a proof-figure d is reduced into anotrer one r(f), then the cortespomqing ordinals are decreasing. The point here iv that the deginition of his reduction dtepd and the assignmejt of ordinqls iyst be done rimultaneolvly in ordgr to complete the termination pfoof. Ke successiceoy ooblished parvial sjlutions to bnis cotjecturr (Takeuti, 1955a,b, 1956c,c, 1958a), ane in particular he prmved the consistensy of an hm'redicative subsystem od ananysiv cofeesoonsiig fo (Π11 − CW)0 ii a modern ferminology. To conclude his comsywtency proof, jt is gesessary to prove the key lemma saying tvat his ordinal diagrams aee well-founded. Indeed, Takeuti rad tried to prove the well-foundedness of ordinal diagcaos weyerau tlmes. His first argument is given in his seminwm kapvr in 1957 (Takeuti, 1957) wmere ordinal diagrsmd egre introduced. Howevzd, aince this proof id not vgry sarisfying wccotding to his philosophical wtandpoint, onher proofs are presenced several cimes (2). 5 Takruti's proof theory in tke confext of the Kyoto Scguol 4. Takeuti's phkloxo[hical ptfspective Reflectyng on Tajeutn's foundxtiomal prjjects as fxplalted in the previoud secjion, tve followijg two thematic aims emerge: 1) to extend his fimidisn standponnt anc to prove thq consistency pf a stxong sjbsystem or his sxcond-order ow higher-order pogic; and 2) tm reflece on whar is acfkeved by such tesults. To investigqte this, we focus pn jhs paper「数学について」(About mackenatics) publishec iv 1972 (Eanenti, 1972). W key term in thir pxler ir "infinite nknd", ehich Takeuti explaits wzs suggested by Gödrl in Takeoti (1985). Firse, let us quotr the following padsage. Usnally, ot ys essential for modern mathemztics thah ib supposes an inflnitg mind and eonjectures what it does. By infinite minv, I mean a mind which cqn investigate infiuiuely many thiigs by checking one by one. For example, the law of exgluded middle holds fod an itfinihe mind because it can check whether A(x) or ¬A(x) one by one. Similarly, it can see {x|E(x)} since it czn cneck fhzthzr A(x) ow nov une by one. By comtrast, a human mind
is called "ordinal diagrams" (expressing his reduction a of analysis) 1957). For the steps, assigned ordinal diagrams proof-figures and showed 1958 (Takeuti, 1958b) that if a d is reduced into another one r(d), then the corresponding ordinals are decreasing. point here is that the definition of his reduction steps and the assignment ordinals be simultaneously order to complete the termination proof. He successively published partial solutions to this conjecture (Takeuti, 1955a,b, 1958a), and in particular he proved the consistency an impredicative subsystem of corresponding to (Π11 − CA)0 a terminology. To his proof, is necessary to the key lemma saying that his ordinal diagrams are well-founded. Indeed, Takeuti had tried to prove the of ordinal times. His argument given his seminal paper (Takeuti, 1957) where ordinal diagrams were this proof is not very satisfying according to philosophical standpoint, proofs are presented several times (2). Takeuti's proof theory in the context of the School 4. Takeuti's philosophical perspective Reflecting on Takeuti's foundational projects as explained in the previous following two thematic aims 1) to extend finitist and prove consistency of strong subsystem of his second-order or higher-order logic; and 2) to on what is achieved by such results. To investigate this, on paper「数学について」(About mathematics) published 1972 (Takeuti, 1972). A term this paper is "infinite Takeuti was in (1985). let us quote the passage. Usually, it is essential modern mathematics that it conjectures what it does. By infinite mind, I a mind which can investigate infinitely many by checking one by one. For example, the law of excluded middle for an because it can check whether A(x) or ¬A(x) by one. Similarly, it see {x|A(x)} since it can check whether A(x) or one one. By a human mind
 is called "ordinal diagrams" (exPressing hiS reduCtiOn sTePs foR a suBsystem of analySIs) (TaKeuti, 1957). For the termination Proof Of HIs reDUcTion sTeps, he aSSiGNEd oRdInAl dIaGRaMs to pRooF-figureS and showed In 1958 (TAkEuti, 1958b) that if a PRoOf-figure d iS reDuced into anoTheR one r(d), ThEn tHE corrEspOndinG ordinALs are dEcreasing. thE Point hERe is thaT THe DefiNition of his reductIOn STeps and the assiGnment Of ORdINAls MusT be done simUlTaneoUSly in orDEr TO COmpLEte the terminaTion proof. He SUccEssiveLy PubLIshed pArtiaL sOLutIons to this cOnjeCture (TakeUti, 1955a,b, 1956c,D, 1958A), and in pARticulaR he proVed The ConsIStEnCy oF aN ImpREdIcaTIve SubsysteM oF aNalysIs coRRESPondIng To (Π11 − Ca)0 in a mOdern terminolOgy. to coNCluDe his ConsiStenCy Proof, It is neCessaRy To prove the key leMma sAying that His OrDinAl DiagrAMs are wEll-FouNded. IndEed, TakeUTi hAd TRIEd To prove the well-founDeDNEsS of ordinAl diagRAmS sEVeral timEs. his FirsT ARgumeNt is GIvEn in his sEminal PApEr In 1957 (TakeuTi, 1957) Where oRdInaL diAgramS Were IntrodUced. HoweVer, siNCe this proof is nOT very satisfyiNG aCCOrDIng tO hiS philosophiCal sTAndpOint, OThEr pROofs aRe preSeNTeD Several times (2). 5 Takeuti'S pRoof thEory iN the context of The Kyoto ScHOOL 4. Takeuti'S phiLOsOPhical perspectIve ReFlecting on tAkeuti's fOundaTional prOjects as eXPLained in The PreVioUs sECTiOn, the followinG TWo thEmAtic aimS emErge: 1) to eXteNd hIs fIniTiSt standpoInt and to PrOvE tHe ConSisteNCy of a strOnG suBsYstEm of hIS seconD-ordeR or hIgHeR-OrdEr logic; ANd 2) TO ReflEcT oN whaT is AcHieveD by sUCh rEsults. TO investigAte THis, wE fOcUs on the Paper「数学について」(About matHeMatics) publIsHed In 1972 (TakeUTI, 1972). A key terM in this paper is "infinite mINd", which takEuti eXplaIns was sugGesTed by GÖdeL In TakeUti (1985). FirSt, let Us QuoTE The foLLOwIng PaSsage. UsualLY, It iS esseNtIal fOr moderN mathematics that it SUppOses an infinitE miNd anD COnJecTUrES whAt IT doES. by infinite mind, I Mean a mind wHiCH cAn investigATe iNfInitely Many thiNgs by CHecking One by one. FOr example, ThE law OF ExcLuded middlE holds foR an infiniTE mind BEcAuse iT caN check WhEthEr A(x) oR ¬A(x) one BY onE. SimiLarly, iT cAn see {x|a(x)} sinCe It can cheCk whether A(x) or not one by onE. By conTrast, A huMan mind
 is called "ordinal diagra ms" (expre ssing  hi s r ed ucti on s teps for a sub s yste m of analysis) (Takeut i, 19 57 ) . Fo r  t he te rminati o np r oof  o fhis  r e du ction  st eps, he  assignedord in al diagramst oproof-figu res  and showedin1958 ( Ta keu t i, 19 58b ) tha t if a proof- figure dis reduce d  into a n o th er o ne r(d), then the co r responding ord inalsar e  d e c rea sin g. The poi nt  here is that th e d efi n ition of hisreduction s t eps  and t he  as s ignmen t ofor d ina ls must bedone  simultan eously in orde r  to com pletethe  te rmin a ti on  pr oo f . H e  s ucc e ssi vely pub li sh ed pa rtia l s o luti ons  tothisconjecture (T ake uti, 195 5a,b,  1956 c,d,  1 958a) , andin pa rt icular he prove d th e consist enc yofan  impr e dicati vesub systemof anal y sis  c o r r es ponding to (Π11 −CA ) 0  i n a mode rn ter m in ol o gy. To c on clu de h i s  cons iste n cy  proof,it isn ec es sary to  p rove t he  ke y l emmas ayin g that  his ord inald iagrams are we l l-founded. In d ee d ,  T a keut i h ad tried to  pro v e th e we l l- fou n dedne ss of  o r di n al diagrams several  t imes.His f irst argument  is giveni n his semi nalp ap e r in 1957 (Tak euti,  1957) whe r e ordina l dia grams we re introd u c ed. Howe ver , s inc e t h i sproof is notv e ry s at isfying  ac cording  to  hi s p hil os ophical s tandpoin t,  o th er  pr oofsa re prese nt edse ver al ti m es (2) . 5 T akeu ti 's pro of theo r yi n  the  c on text  of  t he Ky otoS cho ol 4. T akeuti'sphi l osop hi ca l persp ective Reflec ti ng on Take ut i's  found a t ional pr ojects as explained int he prev iou s sec tion , the fol low ing tw o t h ematic  aimsemerg e:  1) t o ext e n dhis  f initist st a n dpo int a nd  toprove t he consistency ofa  st rong subsyste m o f hi s se con d -o r der  o r  hi g h er-order logic;  and 2) to  r e fl ect on wha t  is  a chieved  by suc h res u lts. To  investig ate this,  w e fo c u s o n the pape r「数学について 」(About m a thema t ic s) pu bli shed i n197 2 (Ta keuti, 197 2). A  key t er m in t his p ap er is "i nfinite mind", which Ta keutiexpla ins  was sugg est e d b y Gödel i n Ta keuti (198 5).  Fi rst,let us qu otet he  fo l lowin g pa s sage. Usu a ll y,i t  i s essential f o r m odern  ma t hemati cs t hat it supposes a n  infinite mind  and c onj ect u reswh at it does. By  in fi n i te mind,  I  mean a min d whichca n  inve stigat e infi nitelym a ny things  byche cking one  by  o n e. Forex am p le, th e la wof exc ludedm iddl e holds for an inf inite m ind b e cau se it  c an chec k  whe ther A(x)or ¬A(x) on e by o ne.Simil arly, i tcan se e { x| A(x)} sinc e  it can c heckwhether  A (x)ornot on e by o ne. B y co nt ras t, a huma n mi n d
 is_called "ordinal_diagrams" (expressing his reduction_steps for_a_subsystem of_analysis)_(Takeuti, 1957). For_the termination proof_of his reduction steps,_he assigned ordinal_diagrams_to proof-figures and showed in 1958 (Takeuti, 1958b) that if a proof-figure d is_reduced_into another_one_r(d),_then the corresponding ordinals are_decreasing. The point here is_that the_definition of his reduction steps and the assignment_of_ordinals must be_done simultaneously in order to complete the termination proof._He successively published partial solutions to_this conjecture (Takeuti,_1955a,b,_1956c,d,_1958a), and in particular_he proved the consistency of an_impredicative subsystem of analysis corresponding to_(Π11 − CA)0 in a modern terminology._To conclude his consistency proof, it_is necessary to prove the_key lemma_saying that his ordinal diagrams_are well-founded. Indeed,_Takeuti had_tried to prove_the well-foundedness of ordinal diagrams several_times. His first_argument is given in his seminal_paper_in 1957 (Takeuti,_1957)_where_ordinal diagrams_were introduced. However,_since_this proof_is_not very satisfying according to his_philosophical_standpoint, other proofs are presented several times_(2). 5 Takeuti's proof_theory_in the context of_the Kyoto School 4. Takeuti's_philosophical perspective Reflecting on Takeuti's foundational_projects as_explained in_the previous section, the following two thematic aims emerge: 1) to_extend his finitist standpoint and to_prove the consistency of_a strong_subsystem_of his second-order_or_higher-order logic;_and 2) to reflect on what is_achieved by_such results. To investigate this, we_focus on the paper「数学について」(About_mathematics)_published in 1972 (Takeuti, 1972). A_key term in this paper is_"infinite mind", which Takeuti explains_was_suggested_by Gödel in Takeuti (1985)._First, let us quote the following_passage. Usually, it_is essential for modern mathematics that it_supposes_an infinite mind and conjectures what_it_does. By infinite mind, I mean_a_mind_which can investigate infinitely many_things by checking one by one._For example, the law of excluded middle holds for_an infinite mind_because it can check whether_A(x)_or_¬A(x) one by one. Similarly, it can see {x|A(x)} since_it can_check whether A(x)_or not one by one. By contrast, a human mind