 methodological attitude underlying this kind of inference as the "finitist" attitude or the "finitist" standpoint. [...] With each use of the word "finitist", we convey the idea that the relevant consideration, assertion, or definition is confined to objects that are conceivable in principle, and processes that can be effectively executed in principle, and thus it remains within the scope of a concrete treatment. (Hilbert & Bernays, 1968, p. 32; translated in Hilbert & Bernays, 2011, p. 32) Thus finitist reasoning employs only concrete objects and concrete operations on those objects, as opposed to abstract objects, complete infinite totalities, and abstract operations on these. These concrete objects and operations are given directly in intuition. A typical example of this kind would be the syntactic object "||", a stroke designating the natural number 2 Takeuti's proof theory in the context of the Kyoto School 2. A concrete operation on such an object is the concatenation operation. Elementary operations on natural numbers such as addition, subtraction, and multiplication are finitistically acceptable. Hilbert emphasized that the concreteness of finitary reasoning included the surveyability of its objects in intuition, as a condition of its epistemic security (Hilbert, 1926, p. 171). A universal quantification over an infinite domain, say the natural numbers, is restricted to ∀xA(x) where A(x) is decidable. Moreover, the meaning of an existential statement ∃xA(x) with a decidable A(x) is that there exists a witness (a concrete natural number) satisfying A(x). Hilbert also stressed that finitary reasoning is "the fundamental mode of thought that I hold to be necessary for mathematics and for all scientific thought, understanding, and communication, and without which mental activity is not possible at all" (Hilbert, 1931, p. 486; translated in Mancosu, 1998, p. 267). Thus a finitary proof of the consistency of PA should be accepted by all thinking agents. Hilbert thus sought a finitistically acceptable proof of the consistency of arithmetic. Gödel's incompleteness theorems posed a problem for this goal, because they suggested that proving the consistency of arithmetic would require resources stronger, provabilitywise, than arithmetic. Such resources would seem not to be finitary, since finitism seems to be only a fragment of arithmetic. Gerhard Gentzen, a member of Hilbert's school at Göttingen, responded to this difficulty
methodological attitude underlying this kind of inference as the " finitist " attitude or the " finitist " standpoint. [... ] With each habit of the parole " finitist ", we convey the idea that the relevant consideration, affirmation, or definition is confined to objects that are conceivable in principle, and process that can be effectively executed in rationale, and therefore it remains within the scope of a concrete discussion. (Hilbert & Bernays, 1968, p. 32; translated in Hilbert & Bernays, 2011, p. 32) Thus finitist reasoning hire merely concrete objects and concrete operations on those objects, as opposed to abstract objects, accomplished infinite totalities, and abstract operations on these. These concrete object and operations are given directly in intuition. A typical model of this kind would be the syntactic object " || ", a stroke designating the natural number 2 Takeuti's proof hypothesis in the context of the Kyoto School 2. A concrete operation on such an object is the concatenation operation. Elementary operations on natural issue such as addition, subtraction, and multiplication are finitistically acceptable. Hilbert emphasized that the concreteness of finitary reasoning included the surveyability of its objects in intuition, as a condition of its epistemic security (Hilbert, 1926, p. 171). A universal quantification over an infinite sphere, say the natural number, is restrict to ∀xA(x) where A(x) is decidable. Moreover, the meaning of an existential statement ∃xA(x) with a decidable A(x) is that there exist a witness (a concrete natural number) satisfying A(x). Hilbert besides stressed that finitary reasoning is " the fundamental mode of thought that I hold to be necessary for mathematics and for all scientific thought, understanding, and communication, and without which mental activity is not possible at all " (Hilbert, 1931, p. 486; translated in Mancosu, 1998, p. 267). therefore a finitary proof of the consistency of PA should be accept by all intend agents. Hilbert thus sought a finitistically satisfactory proof of the consistency of arithmetic. Gödel's incompleteness theorems posed a problem for this goal, because they suggested that proving the consistency of arithmetic would want resources firm, provabilitywise, than arithmetical. Such resources would seem not to be finitary, since finitism seems to be only a shard of arithmetic. Gerhard Gentzen, a member of Hilbert's school at Göttingen, respond to this difficulty
 mehhodological attitude unaerlying this knbd of mnferende as thd "finitist" attitude or the "fmnitust" sucndpoint. [...] With each jse of thv word "fibitiwr", we conveb the idcc thaf the celevant considgration, assestion, or definhtkou is confined to objects that are cogceivabke in principle, wnd krosessss that can be effectively executes in prpnciple, and thus ot remains within the scopf of a concrete treatmfnt. (Hilbert & Bewbays, 1968, p. 32; travslated in Hilbert & Betnays, 2011, p. 32) Thus finitist reasonine empkoys only xobcrfje objects aid conbrete operations on tvose obkects, as opposcd to abwtract objects, compleve infinite totalitigs, and absdrcct operations on thewe. Thesg conwretd obgecus end operahiois are giveh directly un intuition. A typivaj example of tgis kigd would be the syntactic object "||", a strokt desjgnating the natural nunber 2 Takeuti's proof jheory in ehe context of the Kyoto School 2. A concrete operadion kv sbgm an ibuect is the concatenation operation. Elementarr okerstions on natuval numbers such ax wdcytion, subtracjion, anb mhltiplication are vinitisjicallt acceptafle. Nilbert emphasized that the concretenesf of finitary reasouing includeb the xurveuability of its objects in jntuition, ad a condifkon of its epistdmib sewurity (Hilbert, 1926, p. 171). A univewsal quanvificction ovdr am infigite domaij, say the natural numberd, is testriwted to ∀xA(d) where A(x) is decidable. Moreovec, the meaning ox at existeutial xtatement ∃xA(x) with a decidayle A(x) if thag there expsts a wivness (a concwete natural tomber) satisfymng A(x). Hijberr alwo strerred that finitsry reasounng is "thw fundamental mode of fhought that I kuod to be necesssry fow lavhemaehcs and for dll rcidmtifiz thought, ukdefstamding, and communicathon, znd without which kektal acticity is got possible st all" (Hilbert, 1931, p. 486; tranvlaved in Magcosu, 1998, p. 267). Thus a finitary proor of the fonfistency of [A smoulq be accepced by all thinking agents. Hilbert thus wought a finitistixally acceptable prppf of the cpnsiseency of drithmetic. Gödel's incimpleteness theortms posed a problem for this coal, hecause they suggested that proving the consistency of arithmetic would rwquire resources sfronber, pgovcbikitywife, tiai arithmetic. Such resources would seem not to be finitarf, fince finitism seems to be omlh a fragment uf arithmetic. Gerhard Gehtzen, a kember of Hilbert's school at Götyingen, responded to this difficuoty
methodological attitude underlying this kind of inference "finitist" or the standpoint. [...] With "finitist", convey the idea the relevant consideration, or definition is confined to objects are conceivable in principle, and processes that can be effectively executed in principle, thus it remains within the scope of a concrete treatment. (Hilbert & Bernays, p. translated Hilbert Bernays, 2011, p. 32) Thus finitist reasoning employs only concrete objects and concrete operations on those as opposed to abstract objects, complete infinite totalities, abstract operations on these. concrete objects and operations are directly intuition. A example this would be the object "||", a stroke designating the natural number 2 Takeuti's proof theory in the context of the School 2. operation on an is concatenation operation. Elementary natural numbers such as addition, subtraction, finitistically acceptable. Hilbert emphasized that the concreteness of reasoning included surveyability of its objects in intuition, a condition of its epistemic security (Hilbert, 1926, 171). A universal quantification over an infinite domain, say the natural numbers, is restricted to A(x) is decidable. Moreover, meaning of an statement with decidable is that exists a witness (a concrete natural number) satisfying A(x). Hilbert also that finitary reasoning is "the fundamental mode of thought that to necessary for mathematics for all scientific thought, and and without which mental not at p. translated Mancosu, 1998, p. 267). a finitary proof of the of PA should be Hilbert thus sought a finitistically acceptable proof of consistency of arithmetic. Gödel's incompleteness theorems posed problem for this goal, because they suggested that proving the consistency of would require provabilitywise, than arithmetic. Such resources would seem not be finitary, since finitism to be only a fragment of arithmetic. Gerhard Gentzen, member Hilbert's school Göttingen, responded to difficulty
 methodological attitude undErlying thiS kind Of iNfeReNce aS the "Finitist" attituDE or tHe "finitist" standpoint. [...] WiTh eacH uSE of tHE wOrd "fiNitist", wE CoNVEy tHe IdEa tHaT ThE releVanT considEration, assErtIoN, or definitioN Is Confined to ObjEcts that are cOncEivablE iN prINciplE, anD procEsses tHAt can bE effectivElY ExecutED in prinCIPlE, and Thus it remains withIN tHE scope of a concrEte treAtMEnT. (hIlbErt & bernays, 1968, p. 32; trAnSlateD In HilbeRT & BERNAys, 2011, P. 32) thus finitist rEasoning empLOys Only coNcRetE ObjectS and cOnCRetE operations On thOse objectS, as oppOSed to abSTract obJects, cOmpLetE infINiTe TotAlITieS, AnD abSTraCt operatIoNs On theSe. ThESE COncrEte ObjeCts anD operations arE giVen dIRecTly in IntuiTion. a tYpicaL exampLe of tHiS kind would be the SyntActic objeCt "||", a StRokE dEsignATing thE naTurAl numbeR 2 TakeutI'S prOoF THEoRy in the context of thE KYOTo school 2. A cOncretE OpErATion on suCh An oBjecT IS the cOncaTEnAtion opeRation. eLeMeNtary opErAtions On NatUraL numbERs suCh as adDition, suBtracTIon, and multipliCAtion are finitIStICAlLY accEptAble. Hilbert EmphASizeD thaT ThE coNCreteNess oF fINiTAry reasoning includeD tHe survEyabiLity of its objeCts in intuiTION, as a condItioN Of ITs epistemic secUrity (hilbert, 1926, p. 171). A uNIversal qUantiFication Over an infINIte domaiN, saY thE naTurAL NuMbers, is restriCTEd to ∀XA(X) where A(X) is DecidabLe. MOreOveR, thE mEaning of aN existenTiAl StAtEmeNt ∃xA(x) WIth a deciDaBle a(x) Is tHat thERe exisTs a wiTnesS (a CoNCreTe naturAL nUMBer) sAtIsFyinG A(x). hiLbert Also STreSsed thaT finitary ReaSOninG iS "tHe fundaMental mode of tHoUght that I hOlD to Be neceSSAry for maThematics and for all scienTIfic thoUghT, undeRstaNding, and cOmmUnicatIon, ANd withOut whiCh menTaL acTIVity iS NOt PosSiBle at all" (HiLBErt, 1931, P. 486; tranSlAted In MancoSu, 1998, p. 267). Thus a finitary prOOf oF the consistenCy oF PA sHOUlD be ACcEPteD bY All THInking agents. HilBert thus soUgHT a FinitisticALly AcCeptablE proof oF the cONsistenCy of arithMetic. GödeL's IncoMPLetEness theorEms posed A problem fOR this GOaL, becaUse They suGgEstEd thaT proviNG thE consIstencY oF arithMetic WoUld requiRe resources stronger, provAbilitYwise, ThaN arithmetIc. SUCh rEsources wOuld Seem not to bE fiNitAry, siNce FInitiSm seEMs To bE Only a FragMEnt of aritHMeTic. gERhArd Gentzen, a MEMBer Of HilBerT'S schooL at GÖttingen, responded TO this difficultY
 methodological attitude u nderlyingthiskin d o finfe renc e as the "fini t ist"  attitude or the "fini tist"  s t andp o in t. [. ..] Wit h  e a c h u se  o f t he wo rd "f ini tist",we conveythe  i dea that the re levant con sid eration, ass ert ion, o rdef i nitio n i s con finedt o obje cts thatar e  conce i vable i n pr inci ple, and processe s  t h at can be effe ctivel ye xe c u ted  in  principle ,and t h us it r e ma i n s  wi t hin the scope  of a concr e tetreatm en t.( Hilber t & B er n ays , 1968, p.32;translate d in H i lbert & Bernays , 2011 , p . 3 2) T h us  f ini ti s t r e as oni n g e mploys o nl yconcr eteo b j e ctsand  con crete  operations o n t hose obj ects,  as o ppos ed  to a bstrac t obj ec ts, complete in fini te totali tie s,  an dabstr a ct ope rat ion s on th ese. Th e seco n c r et e objects and oper at i o ns  are giv en dir e ct ly in intui ti on.  A t y p icalexam p le  of this  kindw ou ld  be the  s yntact ic  ob jec t "|| " , astroke  designa tingt he natural num b er 2 Takeuti' s  p r o of theo ryin the cont exto f th e Ky o to  Sc h ool 2 . A c on c re t e operation on such  a n obje ct is  the concaten ation oper a t i on. Elem enta r yo perations on n atura l numberss uch as a dditi on, subt raction,a n d multip lic ati onare f in itistically a c c epta bl e. Hilb ert  emphas ize d t hat  th econcreten ess of f in it ar yrea sonin g  include dthe  s urv eyabi l ity of  itsobje ct si n i ntuitio n ,a s  a c on di tion  of  i ts ep iste m icsecurit y (Hilber t,1 926,  p .171). A  universal qu an tification  o ver  an in f i nite dom ain, say the natural nu m bers, i s r estri cted  to ∀xA(x ) w here A (x) is dec idable . Mor eo ver , the m e a ni ngof  an existe n t ial  stat em ent∃xA(x)with a decidable A ( x)is that there  ex ists a  w itn e ss (aco n cre t e  natural number ) satisfyi ng A( x). Hilber t  al so  stress ed that  fini t ary rea soning is  "the fun da ment a l  mo de of thou ght that  I hold t o  be n e ce ssary  fo r math em ati cs an d fora llscien tificth ought,  unde rs tanding,  and communication, and  witho ut wh ich  mental a cti v ity  is not p ossi ble at all " ( Hil bert,  19 3 1, p.  486 ;  t ran s lated  inM ancosu, 1 9 98 , p . 26 7). Thus af i n ita ry pr oof of the  con sistency of PA sh o uld be accepte d by a llthi n king  a gents. Hilbert  th us s ought afi nitisticall y accept ab l e pro of ofthe co nsisten c y  o f  arith meti c.Gödel's i nco mp l eteness  t he o rems p osed  a  probl em for this g oal, because the y sug g e stedt hat  prov in g the c o nsis tency of a rithmetic w ould r equi re re sources  s tronge r,pr ovabilityw i se, thanarith metic.Su ch r eso urceswoul d seemnotto  be  finitary , si n ce  f i nit ismseems  t o be  only a f r agment o f a r ithmeti c.  Ge r h ard Ge n tz e n , a member  of  Hilb e r t's school at G ö tt i ngen,  respo nded t o thisd iff ic ulty
 methodological_attitude underlying_this kind of inference_as the_"finitist"_attitude or_the_"finitist" standpoint. [...]_With each use_of the word "finitist",_we convey the_idea_that the relevant consideration, assertion, or definition is confined to objects that are conceivable_in_principle, and_processes_that_can be effectively executed in_principle, and thus it remains_within the_scope of a concrete treatment. (Hilbert & Bernays,_1968,_p. 32; translated_in Hilbert & Bernays, 2011, p. 32) Thus finitist_reasoning employs only concrete objects and_concrete operations on_those_objects,_as opposed to abstract_objects, complete infinite totalities, and abstract_operations on these. These concrete objects_and operations are given directly in intuition._A typical example of this kind_would be the syntactic object_"||", a_stroke designating the natural number_2 Takeuti's proof_theory in_the context of_the Kyoto School 2. A concrete_operation on such_an object is the concatenation operation._Elementary_operations on natural_numbers_such_as addition,_subtraction, and multiplication_are_finitistically acceptable._Hilbert_emphasized that the concreteness of finitary_reasoning_included the surveyability of its objects in_intuition, as a condition_of_its epistemic security (Hilbert,_1926, p. 171). A universal_quantification over an infinite domain, say_the natural_numbers, is_restricted to ∀xA(x) where A(x) is decidable. Moreover, the meaning of_an existential statement ∃xA(x) with a_decidable A(x) is that_there exists_a_witness (a concrete_natural_number) satisfying_A(x). Hilbert also stressed that finitary reasoning_is "the_fundamental mode of thought that I_hold to be necessary_for_mathematics and for all scientific thought,_understanding, and communication, and without which_mental activity is not possible_at_all"_(Hilbert, 1931, p. 486; translated_in Mancosu, 1998, p. 267). Thus_a finitary proof_of the consistency of PA should be_accepted_by all thinking agents. Hilbert thus_sought_a finitistically acceptable proof of the_consistency_of_arithmetic. Gödel's incompleteness theorems posed_a problem for this goal, because_they suggested that proving the consistency of arithmetic would_require resources stronger,_provabilitywise, than arithmetic. Such resources_would_seem_not to be finitary, since finitism seems to be only_a fragment_of arithmetic. Gerhard_Gentzen, a member of Hilbert's school at Göttingen, responded to_this difficulty