 all three responses have room for criticism. Since these responses are based on the model-theoretic approaches, some constructivists (or intuitionists) are reluctant to accept it. Interestingly, a well-known proof of the consistency of mathematics was suggested in proof-theory but the proof-theoretic criterion of and solution to the paradoxes have not yet been well investigated. After we will have some requisite notions and rules of prooftheory in Section 1.2, Section 1.3 shall introduce Tennant's criterion for paradoxicality and 4The principle of bivalence states that every sentence φ is determinately true or false, independently of our method to know the truth-value of φ. 14 the requirement of a normal derivation as a plausible solution to the paradoxes. Prooftheoretic approaches can apply to both the set-theoretic and semantic paradoxes. In other words, the proof-theoretic approaches opened up the possibility of the uniform solution to the paradoxes.. 1.2 Preliminaries As we have discussed in Section 1.1, the naive comprehension axiom has the main role to derive a contradiction from Russell's paradox. Since the notion of a'set' in the principle allows the set of all sets which are not members of themselves, it is often considered that the application of the notion of a'set' causes a contradiction from Russell's paradoxes. Hence, the restriction of the naive comprehension principle may be read as the constraint on the application of the notion of a'set.' When Gerhard Gentzen (1936) suggests his second proof of the consistency of arithmetic, he seems to think that an error of Russell's paradox is not our use of the notion of a'set' but in the logical inferences involved in. In pursuing his consistency proof, he first set the analysis of purely logical deduction which was intended to be extended to arithmetic and analysis. Gentzen (1935, Sec. 2) introduces pairs of introduction and elimination rules for natural deduction system as the natural method of reasoning in mathematics. He invented another logical calculus that he called'sequent calculus,' and his two most important results, a proof of Hauptsatz (cut-elimination theorem) and a consistency proof of arithmetic, were established in sequent calculus. Although two results were proved in sequent calculus, they both were clearly inspired by insights that he got by reflecting on his natural deduction system. The same result can be established by normalization theorem in natural deduction system, and a consistency proof can be suggested as a
all three responses have room for criticism. Since these responses are based on the exemplary - theoretical approach, some constructivists (or intuitionists) are loath to accept it. Interestingly, a well - sleep together proof of the consistency of mathematics was suggested in validation - hypothesis but the proof - theoretic criterion of and solution to the paradoxes have not yet been well investigated. After we will get some requisite notions and rules of prooftheory in Section 1.2, Section 1.3 shall introduce Tennant's standard for paradoxicality and 4The principle of bivalence states that every sentence φ is determinately dependable or false, independently of our method to acknowledge the truth - value of φ. 14 the necessity of a normal derivation as a plausible solution to the paradoxes. Prooftheoretic approaches can apply to both the set - theoretical and semantic paradoxes. In other words, the proof - theoretic approaches opened up the hypothesis of the consistent solution to the paradoxes.. 1.2 Preliminaries As we have discussed in Section 1.1, the naive comprehension axiom has the main role to derive a contradiction from Russell's paradox. Since the impression of a'set' in the rationale allows the set of all sets which are not extremity of themselves, it is often considered that the application of the impression of a'set' causes a contradiction from Russell's paradox. Hence, the restriction of the naive inclusion principle may be read as the constraint on the application of the notion of a'set.' When Gerhard Gentzen (1936) suggests his second validation of the consistency of arithmetic, he seems to think that an error of Russell's paradox is not our use of the notion of a'set' but in the logical inference involve in. In pursuing his consistency proof, he first set the psychoanalysis of purely logical deduction which was intended to be extended to arithmetic and analysis. Gentzen (1935, Sec. 2) introduces pairs of insertion and elimination rules for natural tax write-off system as the lifelike method of reasoning in mathematics. He invented another logical tartar that he called'sequent calculus,' and his two most important resultant role, a proof of Hauptsatz (baseball swing - elimination theorem) and a consistency proof of arithmetic, were established in sequent calculus. Although two results were proved in sequent calculus, they both were intelligibly inspired by insights that he get by reflecting on his natural subtraction system. The same result can be established by normalization theorem in natural deduction system, and a consistency proof can be suggested as a
 alp three responses have ruom for criticism. Sincx these responsds are based on the model-thelrwtic qpproaches, some constrjctivists (or intuutioiists) are relucteht to ageept jb. Intzrxstingly, a well-lnown proox of the consivtdney of mathematics was suggested in pwoof-theprj but the proos-thepwetid criterion of and solution to the paradoqes have not yet neen well investigated. Aftfr wf will have some rfquisite nojjonf and rules ow prooftheory in Sectikn 1.2, Section 1.3 shall introduce Tevnant'x criteriob dor karadoxicalivy and 4The principle of biealence states that eyery venrence φ is determinateny true or false, igdependendlv of our method to kniw the jruth-ealud of φ. 14 uhe rsquirelenv of a normzl derivatiin as a plausible spltnoon to the pzradoxqs. Prooftheoretic approaches can apply to bofh the set-theoretic and semantic paradoxes. Ij other wjrds, the proof-theoretic approaches opened up the [ossiuiuitv of tfw kniform solution to the paradoxes.. 1.2 Preliminaryss Ax we have discmssed in Section 1.1, yhf mwive comprehevsion csikm has the main rope to dgrive q contradyctipn from Russell's paradox. Sibce the notijb of a'set' in the pxinciple alluws jhe sey of all sets which are not members of themselvsr, it is often covsicesed that the application os the notmon oy a'set' cxusex a cogtradictioj from Russell's paradoxed. Henee, tha restrictlon of the naive comprehension 'cinciple may ne redd as thz consbraint on the a[plication of jhe notiou of a'ret.' When Gvrhard Geitzen (1936) suggefts his secong proof of thx consistqncy of qrithmegkc, he seems to think thcu an error of Russell's paradpx ka not our use oy uhw notion of a'srt' cut ij vhe ljcical inferetces inxplved in. In kbrsmine hix consistency proof, ve fjrst set the analyxif of purgly logicwl deduction ehich was intended to bx exteided tp atithmetic and analysis. Gentzen (1935, Sec. 2) intgodmces pairs of intvodustion and zlimination rules for natural deduction wystem as the natoral method of reasonnnn in mathemavics. Hq inventeg another logical caoculus that he cakled'sequent calculus,' znd hiv two most important results, a proof of Hauptsatz (cut-elimination theorem) and a consistency proof of srithkecic, were qstaulmshed in sequent balculus. Although two results wece proved hn sequent calculus, they both eefe clearly inrpired by insights that ge got bu reflecting on his natural dediction system. The same rcsult can be eatablished by normalization yheorek in batural dedmction system, xnd a consistejcy krooy can be suggested as a
all three responses have room for criticism. responses based on model-theoretic approaches, some to it. Interestingly, a proof of the of mathematics was suggested in proof-theory the proof-theoretic criterion of and solution to the paradoxes have not yet been investigated. After we will have some requisite notions and rules of prooftheory in 1.2, 1.3 introduce criterion for paradoxicality and 4The principle of bivalence states that every sentence φ is determinately true false, independently of our method to know the of φ. 14 the of a normal derivation as plausible to the Prooftheoretic can to both the and semantic paradoxes. In other words, the proof-theoretic approaches opened up the possibility of the uniform solution the paradoxes.. As we discussed Section the naive comprehension the main role to derive a paradox. Since the notion of a'set' in the allows the of all sets which are not of themselves, it is often considered that the of the notion of a'set' causes a contradiction from Russell's paradoxes. Hence, the restriction of comprehension principle may be as the constraint the of notion a'set.' When Gentzen (1936) suggests his second proof of the consistency of arithmetic, seems to think that an error of Russell's paradox is use the notion of but in the logical involved In pursuing his consistency first the logical which intended to be extended arithmetic and analysis. Gentzen (1935, 2) introduces pairs of natural deduction system as the natural method of in mathematics. He invented another logical calculus he called'sequent calculus,' and his two most important results, a proof of (cut-elimination theorem) consistency proof of arithmetic, were established in sequent Although two results were in sequent calculus, they both were clearly inspired by that got by on his natural system. The same can be established theorem in deduction and can be suggested as a
 all three responses have room For criticiSm. SinCe tHesE rEspoNses Are based on the mODel-tHeoretic approaches, some ConstRuCTiviSTs (Or intUitioniSTs) ARE reLuCtAnt To ACcEpt it. intErestinGly, a well-knOwn PrOof of the consIStEncy of mathEmaTics was suggeSteD in proOf-TheORy but The Proof-TheoreTIc critErion of anD sOLution TO the parADOxEs haVe not yet been well iNVeSTigated. After we Will haVe SOmE REquIsiTe notions aNd Rules OF prooftHEoRY IN SeCTion 1.2, Section 1.3 shAll introducE tenNant's cRiTerIOn for pAradoXiCAliTy and 4The priNcipLe of bivalEnce stATes that EVery senTence φ Is dEteRminATeLy TruE oR FalSE, iNdePEndEntly of oUr MeThod tO knoW THE TrutH-vaLue oF φ. 14 the Requirement of A noRmal DEriVatioN as a pLausIbLe solUtion tO the pArAdoxes. ProoftheoRetiC approachEs cAn AppLy To botH The set-TheOreTic and sEmantic PAraDoXES. in Other words, the proof-ThEOReTic approAches oPEnEd UP the possIbIliTy of THE unifOrm sOLuTion to thE paradOXeS.. 1.2 PReliminArIes As wE hAve DisCusseD In SeCtion 1.1, tHe naive cOmpreHEnsion axiom has THe main role to dERiVE A cONtraDicTion from RusSell'S ParaDox. SINcE thE NotioN of a'sEt' IN tHE principle allows the SeT of all Sets wHich are not memBers of themSELVes, it is oFten COnSIdered that the aPplicAtion of the NOtion of a'Set' caUses a conTradictioN FRom RusseLl's ParAdoXes. hENcE, the restrictiON Of thE nAive comPreHension PriNciPle May Be Read as the ConstraiNt On ThE aPplIcatiON of the noTiOn oF a'Set.' when GERhard GEntzeN (1936) sugGeStS His Second pROoF OF the CoNsIsteNcy Of ArithMetiC, He sEems to tHink that aN erROr of ruSsEll's parAdox is not our uSe Of the notioN oF a'sEt' but iN THe logicaL inferences involved in. In PUrsuing His ConsiStenCy proof, he FirSt set tHe aNAlysis Of pureLy logIcAl dEDUctioN WHiCh wAs Intended to BE ExtEnded To AritHmetic aNd analysis. Gentzen (1935, SEC. 2) inTroduces pairs Of iNtroDUCtIon ANd ELimInATioN RUles for natural dEduction syStEM aS the naturaL MetHoD of reasOning in MatheMAtics. He Invented aNother logIcAl caLCUluS that he calLed'sequeNt calculuS,' And hiS TwO most ImpOrtant ReSulTs, a prOof of HAUptSatz (cUt-elimInAtion tHeoreM) aNd a consiStency proof of arithmetic, Were esTabliSheD in sequenT caLCulUs. AlthougH two Results werE prOveD in seQueNT calcUlus, THeY boTH were CleaRLy inspireD By InsIGHtS that he got bY REFleCting On hIS naturAl deDuction system. The sAMe result can be eStabLISheD by NOrmaLiZation theorem iN naTuRAL deductiOn System, and a cOnsistenCy PRoof cAn be suGgesteD as a
 all three responses haveroom for c ritic ism . S in ce t hese  responses are base d on the model-theoret ic ap pr o ache s ,someconstru c ti v i sts  ( or  in tu i ti onist s)are rel uctant toacc ep t it. Intere s ti ngly, a we ll- known proofofthe co ns ist e ncy o f m athem aticsw as sug gested in  p r oof-th e ory but t he  pro of-theoretic crit e ri o n of and solut ion to  t h ep a rad oxe s have not  y et be e n welli nv e s t iga t ed. After wewill have s o merequis it e n o tionsand r ul e s o f prooftheo ry i n Section  1.2,S ection1 .3 shal l intr odu ceTenn a nt 's  cr it e rio n  f orp ara doxicali ty  a nd 4T he p r i n c iple  of  biv alenc e states that  ev erys ent enceφ isdete rm inate ly tru e orfa lse, independen tlyof our me tho dtokn ow th e  truth -va lue  of φ.14 ther equ ir e m e nt  of a normal deriv at i o nas a pla usible so lu t ion to t he  pa rado x e s. Pr ooft h eo retic ap proach e sca n apply  t o both  t heset -theo r etic  and s emanticparad o xes. In otherw ords, the pro o f- t h eo r etic  ap proaches op ened up t he p o ss ibi l ity o f the  u n if o rm solution to thepa radoxe s.. 1 .2 Preliminar ies As weh a v e discus sedi nS ection 1.1, th e nai ve compreh e nsion ax iom h as the m ain rolet o  derivea c ont rad ict i o nfrom Russell' s para do x. Sinc e t he noti onofa's et'  i n the pri nciple a ll ow sth e s et of all sets  w hic hare  notm embers  of t hems el ve s , i t is of t en c onsi de re d th atth e app lica t ion  of the  notion o f a ' set'  c au ses a c ontradictionfr om Russell 's  pa radoxe s .  Hence,the restriction of then aive co mpr ehens ionprinciple  ma y be r ead as the  const raint  o n t h e  appl i c at ion  o f the noti o n  of  a'se t. ' Wh en Gerh ard Gentzen (1936) sug gests his sec ond  pro o f  o f t h ec ons is t enc y of arithmetic,he seems t ot hi nk that an err or  of Rus sell'sparad o x is no t our use  of the n ot iono f  a' set' but i n the lo gical inf e rence s  i nvolv edin. In  p urs uinghis co n sis tency  proof ,he fir st se tthe anal ysis of purely logicaldeduct ion w hic h was int end e d t o be exte nded  to arithm eti c a nd an aly s is. G entz e n(19 3 5, Se c. 2 )  introduc e spai r s  o f introduct i o n  an d eli min a tion r ules  for natural dedu c tion system as  the n atu ral meth od  of reasoninginma t h ematics.  H e inventedanotherlo g icalcalcul us tha t he ca l l ed ' sequen t ca lcu lus,' and  hi st wo most  i mp o rtantresu lt s, a p roof o f  Hau p t satz (cut-elimin ation t heore m ) a nd aco nsisten c y pr oof of ari thmetic, we re est abli shedin sequ en t calc ulu s.  Althought wo result s wer e prove din s equ ent ca lcul u s , the y bo th  we re clearl y in s pi re d  by  ins ights  t hathe got by reflecti ngo n his n at ura l deduct i on s ystem. The  sa me re s u lt can bee stab l is h ed by  norma lizati on theo r emin  natura l d e d uction sy stem, and a c on sist ency pro of  can  b e s ug gested as a
 all_three responses_have room for criticism._Since these_responses_are based_on_the model-theoretic approaches,_some constructivists (or_intuitionists) are reluctant to_accept it. Interestingly,_a_well-known proof of the consistency of mathematics was suggested in proof-theory but the proof-theoretic_criterion_of and_solution_to_the paradoxes have not yet_been well investigated. After we_will have_some requisite notions and rules of prooftheory in_Section_1.2, Section 1.3_shall introduce Tennant's criterion for paradoxicality and 4The principle_of bivalence states that every sentence_φ is determinately_true_or_false, independently of our_method to know the truth-value of_φ. 14 the requirement of a_normal derivation as a plausible solution to_the paradoxes. Prooftheoretic approaches can apply_to both the set-theoretic and_semantic paradoxes._In other words, the proof-theoretic_approaches opened up_the possibility_of the uniform_solution to the paradoxes.. 1.2 Preliminaries_As we have_discussed in Section 1.1, the naive_comprehension_axiom has the_main_role_to derive_a contradiction from_Russell's_paradox. Since_the_notion of a'set' in the principle_allows_the set of all sets which are_not members of themselves,_it_is often considered that_the application of the notion_of a'set' causes a contradiction from_Russell's paradoxes._Hence, the_restriction of the naive comprehension principle may be read as the_constraint on the application of the_notion of a'set.' When_Gerhard Gentzen_(1936)_suggests his second_proof_of the_consistency of arithmetic, he seems to think_that an_error of Russell's paradox is not_our use of the_notion_of a'set' but in the logical_inferences involved in. In pursuing his_consistency proof, he first set_the_analysis_of purely logical deduction which_was intended to be extended to_arithmetic and analysis._Gentzen (1935, Sec. 2) introduces pairs of_introduction_and elimination rules for natural deduction_system_as the natural method of reasoning_in_mathematics._He invented another logical calculus_that he called'sequent calculus,' and his_two most important results, a proof of Hauptsatz (cut-elimination_theorem) and a_consistency proof of arithmetic, were_established_in_sequent calculus. Although two results were proved in sequent calculus,_they both_were clearly inspired_by insights that he got by reflecting on his natural_deduction system. The same result can be_established by normalization theorem in natural deduction system, and a consistency_proof can be suggested as a