ured that the cut-elimination theorem holds for it in 1953 (Takeuti, 1953). Also, it is proved that the consistency of analysis (the theory of real numbers) follows from his conjecture. To explain Takeuti's partial solutions to the conjecture, we focus on the second-order subsystem of it called GLC1. The language of GLC1 is obtained by adding a predicate variable X and second-order quantifiers ∀X,∃X to first-order ones. Hence, it has additional rules for second-order quantifiers ∀X,∃X. For example, the rule of ∃X says that ∃XA(X) can be deduced from A(T ) where T is a set term. So, this rule says that if the property A holds for the particular set T, then there is a set X satisfying A as well. Here is the main difference from the first-order case deducing ∃xA(x) from A(t); T might contain arbitrary number of logical symbols, hence A(T ) could be much more complicated than A(X) while A(t) is simpler than ∃xA(x) since ∃x does not occur in the former. Also, the right rule of ∀X 4 Takeuti's proof theory in the context of the Kyoto School says that ∀XA(X) is deducible from A(X). We remark that mathematical induction is not included in GLC1 since it is a "logical" calculus (1). Using an analogy with the key case of Gentzen's cut-elimination for the first-order logic (Gentzen, 1935), the crucial case is the following: Γ, A(Y) Γ,∀XA ∧ ∀XA Γ,¬A(T ) Γ,∃X¬A ∨ ∃XA Γ If we follow Gentzen's method of cut-elimination method for the first-order case, then we will insert a new cut over A(T ). Since T might be very complicated, we cannot say the result obtained by this method is "simpler" than before. Takeuti's crucial idea to resolve this difficulty is to analyze infinitary structures (which are expressible in ordinals) behind this finitary proof. Hence, his idea is nothing else but a considerable extension of Gentzen's consistency proof for PA. This new system of ordinals discovered by Takeuti
ured that the cut - elimination theorem holds for it in 1953 (Takeuti, 1953). Also, it is proved that the consistency of psychoanalysis (the hypothesis of actual numbers) follows from his guess. To explain Takeuti's partial solution to the guess, we focus on the second - ordering subsystem of it called GLC1. The language of GLC1 is prevail by adding a predicate variable ten and second - order quantifiers ∀X,∃X to first - order ones. therefore, it has additional rules for second - orderliness quantifiers ∀X,∃X. For example, the convention of ∃X says that ∃XA(X) can be deduce from A(T) where T is a set term. So, this rule says that if the place A holds for the particular set T, then there is a laid adam satisfying A as well. Here is the main difference from the first - order case deducing ∃xA(x) from A(t); T might incorporate arbitrary number of logical symbol, therefore A(T) could be a lot more complicated than A(X) while A(t) is simpler than ∃xA(x) since ∃x does not occur in the former. Also, the right rule of ∀X 4 Takeuti's proof theory in the context of the Kyoto School says that ∀XA(X) is deducible from A(X). We note that mathematical induction is not included in GLC1 since it is a " logical " calculus (1). Using an doctrine of analogy with the key case of Gentzen's cut - elimination for the first - order logic (Gentzen, 1935), the crucial case is the watch: Γ, A(Y) Γ,∀XA ∧ ∀XA Γ,¬A(T) Γ,∃X¬A ∨ ∃XA Γ If we follow Gentzen's method of cut - elimination method for the first - holy order case, then we will insert a new cut over A(T). Since T might be very complicated, we cannot say the result obtained by this method is " childlike " than before. Takeuti's all-important idea to resolve this difficulty is to analyze infinitary structures (which are expressible in ordinals) behind this finitary validation. Hence, his idea is nothing else but a considerable extension of Gentzen's consistency proof for PA. This modern system of ordinals discovered by Takeuti
uref that the cut-eliminatiok theorem holds yir it mn 1953 (Taksuti, 1953). Alro, it is proved that the condiwtenct of analysis (the theofy of reap numberw) foooows from ijs conjceture. Bo ex'lein Takeuti's pattial solutimns to the conbeztbre, we focus on the second-order subsrstem og lt called GLC1. Jhe lsgguafv if GLC1 is obtained by adding z predibate variable X amd second-order quantifiers ∀X,∃X ho first-order ones. Hence, it hqs aqeitional rulds for second-order quahtifiers ∀X,∃X. For example, the ruld of ∃R says that ∃XQ(X) fdn be deducxd froi A(T ) where B is a vet terk. So, this rule sabs tyat if the property A holds for the parjicular sed C, then there is a set X sativfyitg A qs delm. Iers is tje jain diffedence from rhe first-order case dqeucing ∃xA(x) frkm A(t); E iight contain arbitrary number of logicdl aymbols, hence A(T ) could be much more complicwted than A(X) while A(t) is simpler than ∃xA(x) since ∃x does not occuc kn uhc wirler. Also, the right rule of ∀X 4 Takeuti's proof egeprj in the context jf the Kyotp Dcnjol says that ∀XA(X) is dsducible from A(X). Wf remarh thar mathemauical induction is not included un GLC1 since ut is a "logical" capculus (1). Usiug an snaloby with the key case of Genfzen's cut-ellmination wor the first-orddr kocic (Gentedn, 1935), the crucial cwse is thx folkowing: Γ, A(Y) Γ,∀CA ∧ ∀XA Γ,¬A(T ) Γ,∃X¬A ∨ ∃XW Γ If we follow Gentzen's methld of cut-elilination method for the first-orvxr case, then ee winl inserc a nee cut over A(T ). Since T mighj be very compuicated, we cannot say the refult obtained hy this methmd is "siiplee" thqn befofd. Takeuti's cruvial idea to resolvw this difficulty ls to znalyze infinitcxy structures (whivh xre eqprxssibja in ordinalv) befina this finitary pvoow. Hemce, his idea is nothhng slse but a considetanle extenwion of Dentzen's consostency proof for KA. Thiv nxw sysyem of ordinals discovered by Taksuti
ured that the cut-elimination theorem holds for 1953 1953). Also, is proved that theory real numbers) follows his conjecture. To Takeuti's partial solutions to the conjecture, focus on the second-order subsystem of it called GLC1. The language of GLC1 obtained by adding a predicate variable X and second-order quantifiers ∀X,∃X to first-order Hence, has rules second-order quantifiers ∀X,∃X. For example, the rule of ∃X says that ∃XA(X) can be deduced from ) where T is a set term. So, rule says that if property A holds for the set then there a X A as well. is the main difference from the first-order case deducing ∃xA(x) from A(t); T might contain arbitrary number logical symbols, ) could much complicated A(X) while A(t) than ∃xA(x) since ∃x does not former. Also, the right rule of ∀X 4 proof theory the context of the Kyoto School that ∀XA(X) is deducible from A(X). We remark mathematical induction is not included in GLC1 since it is a "logical" calculus (1). Using with the key case Gentzen's cut-elimination for first-order (Gentzen, the case is following: Γ, A(Y) Γ,∀XA ∧ ∀XA Γ,¬A(T ) Γ,∃X¬A ∨ ∃XA If we follow Gentzen's method of cut-elimination method for the then will insert a cut over A(T ). T be very complicated, we the obtained is than Takeuti's crucial idea to this difficulty is to analyze structures (which are expressible proof. Hence, his idea is nothing else but considerable extension of Gentzen's consistency proof for This new system of ordinals discovered by Takeuti
ured that the cut-elimination Theorem holDs for It iN 1953 (TaKeUti, 1953). ALso, iT is proved that tHE conSistency of analysis (the tHeory Of REal nUMbErs) foLlows frOM hIS ConJeCtUre. to EXpLain TAkeUti's parTial solutiOns To The conjecturE, We Focus on the SecOnd-order subsYstEm of it CaLleD gLC1. ThE laNguagE of GLC1 IS obtaiNed by addiNg A PredicATe variaBLE X And sEcond-order quantifIErS ∀x,∃X to first-order Ones. HeNcE, It HAS adDitIonal rules FoR secoND-order qUAnTIFIerS ∀x,∃X. For example, tHe rule of ∃X saYS thAt ∃XA(X) cAn Be dEDuced fRom A(T ) WhERe T Is a set term. SO, thiS rule says That if THe propeRTy A holdS for thE paRtiCulaR SeT T, TheN tHEre IS a Set x SatIsfying A As WeLl. HerE is tHE MAIn diFfeRencE from The first-order CasE dedUCinG ∃xA(x) fRom A(t); t migHt ContaIn arbiTrary NuMber of logical syMbolS, hence A(T ) cOulD bE muCh More cOMplicaTed ThaN A(X) whilE A(t) is siMPleR tHAN ∃XA(X) since ∃x does not occuR iN THe Former. AlSo, the rIGhT rULe of ∀X 4 TakEuTi's ProoF THeory In thE CoNtext of tHe KyotO scHoOl says tHaT ∀XA(X) is DeDucIblE from a(x). We rEmark tHat matheMaticAL induction is noT Included in GLC1 SInCE It IS a "loGicAl" calculus (1). USing AN anaLogy WItH thE Key caSe of GEnTZeN'S cut-elimination for tHe First-oRder lOgic (Gentzen, 1935), thE crucial caSE IS the follOwinG: γ, A(y) γ,∀XA ∧ ∀XA Γ,¬A(T ) Γ,∃X¬A ∨ ∃XA Γ if we fOllow GentzEN's method Of cut-EliminatIon method FOR the firsT-orDer CasE, thEN We Will insert a neW CUt ovEr a(T ). Since t miGht be veRy cOmpLicAteD, wE cannot saY the resuLt ObTaInEd bY this MEthod is "sImPleR" tHan BeforE. takeutI's cruCial IdEa TO reSolve thIS dIFFicuLtY iS to aNalYzE infiNitaRY stRuctureS (which are ExpREssiBlE iN ordinaLs) behind this fInItary proof. heNce, His ideA IS nothing Else but a considerable extENsion of genTzen's ConsIstency prOof For PA. THis NEw systEm of orDinalS dIscOVEred bY tAkEutI
ured that the cut-eliminat ion theore m hol dsfor  i t in  195 3 (Takeuti, 19 5 3).Also, it is proved tha t the  c o nsis t en cy of  analys i s( t heth eo ryof re al nu mbe rs) fol lows fromhis  c onjecture. T o  e xplain Tak eut i's partialsol utions  t o t h e con jec ture,  we fo c us onthe secon d- o rder s u bsystem o fit c alled GLC1. The l a ng u age of GLC1 is  obtai ne d  b y add ing  a predica te  vari a ble X a n ds e c ond - order quantif iers ∀X,∃Xt o f irst-o rd ero nes. H ence,  i t  ha s additiona l ru les for s econd- o rder qu a ntifier s ∀X,∃ X.For  exa m pl e,  th er ule of  ∃X say s that ∃ XA (X ) can  bed e d u cedfro m A( T ) w here T is a s etterm .  So , thi s rul e sa ys  that  if th e pro pe rty A holds for  the  particul arse t T ,thent here i s a  se t X sat isfying A a sw e l l.  Here is the maindi f f er ence fro m thef ir st - order ca se  de duci n g  ∃xA( x) f r om  A(t); T  might co nt ain arb it rary n um ber  of  logi c al s ymbols , henceA(T ) could be muchm ore complicat e dt h an A(X)  wh ile A(t) is  sim p lerthan ∃x A(x )  sinc e ∃xdo e sn ot occur in the for me r. Als o, th e right ruleof ∀X 4 Ta k e u ti's pro of t h eo r y in the conte xt of  the Kyoto School s ays t hat ∀XA( X) is ded u c ible fro m A (X) . W e r e m ar k that mathem a t ical  i nductio n i s not i ncl ude d i n G LC 1 since i t is a " lo gi ca l"  ca lculu s  (1). Us in g a nana logyw ith th e key  cas eof Gen tzen'sc ut - e limi na ti on f orth e fir st-o r der  logic(Gentzen,  19 3 5),th ecrucial  case is thefo llowing: Γ ,A(Y ) Γ,∀X A ∧ ∀XA Γ, ¬A(T ) Γ,∃X¬A ∨ ∃XA Γ I f  we fol low  Gent zen' s methodofcut-el imi n ationmethod  forth e f i r st-or d e rcas e,  then we w i l l i nsert  a  new  cut ov er A(T ). Since Tm igh t be very com pli cate d ,  w e c a nn o t s ay the r esult obtainedby this me th o dis "simple r " t ha n befor e. Take uti's crucial  idea toresolve t hi s di f f icu lty is toanalyzeinfinitar y  stru c tu res ( whi ch are  e xpr essib le ino rdi nals)  behin dthis f inita ry  proof.Hence, his idea is noth ing el se bu t a  consider abl e  ex tension o f Ge ntzen's co nsi ste ncy p roo f  forPA.T hi s n e w sys temo f ordinal s  d isc o v er ed by Takeu t i 
ured that_the cut-elimination_theorem holds for it_in 1953_(Takeuti,_1953). Also,_it_is proved that_the consistency of_analysis (the theory of_real numbers) follows_from_his conjecture. To explain Takeuti's partial solutions to the conjecture, we focus on the_second-order_subsystem of_it_called_GLC1. The language of GLC1_is obtained by adding a_predicate variable_X and second-order quantifiers ∀X,∃X to first-order ones._Hence,_it has additional_rules for second-order quantifiers ∀X,∃X. For example, the rule_of ∃X says that ∃XA(X) can_be deduced from_A(T_)_where T is a_set term. So, this rule says_that if the property A holds_for the particular set T, then there_is a set X satisfying A_as well. Here is the_main difference_from the first-order case deducing_∃xA(x) from A(t);_T might_contain arbitrary number_of logical symbols, hence A(T )_could be much_more complicated than A(X) while A(t)_is_simpler than ∃xA(x)_since_∃x_does not_occur in the_former._Also, the_right_rule of ∀X 4 Takeuti's proof_theory_in the context of the Kyoto School_says that ∀XA(X) is_deducible_from A(X). We remark_that mathematical induction is not_included in GLC1 since it is_a "logical"_calculus (1)._Using an analogy with the key case of Gentzen's cut-elimination for_the first-order logic (Gentzen, 1935), the_crucial case is the_following: Γ,_A(Y)_Γ,∀XA ∧ ∀XA_Γ,¬A(T_) Γ,∃X¬A_∨ ∃XA Γ If we follow Gentzen's_method of_cut-elimination method for the first-order case,_then we will insert_a_new cut over A(T ). Since_T might be very complicated, we_cannot say the result obtained_by_this_method is "simpler" than before._Takeuti's crucial idea to resolve this_difficulty is to_analyze infinitary structures (which are expressible in_ordinals)_behind this finitary proof. Hence, his_idea_is nothing else but a considerable_extension_of_Gentzen's consistency proof for PA._This new system of ordinals discovered_by Takeuti