 is not definable or provable.  Our only concern, then, should be that the initial foundations be something immediately clear, natural, and not open to question.  This condition is satisfied by the notion of integer and by inductive inferences, but it is decidedly not satisfied by set-theoretic axioms of the type of Zermelo's or anything else of that kind; if we were to accept the reduction of the former notions to the latter, the set-theoretic notions would have to be simpler than mathematical induction, and reasoning with them less open to question, but this runs entirely counter to the actual state of affairs.
In a paper (1922) Hilbert makes the following remark about Poincaré's assertion that the principle of mathematical induction is not provable: “His objection that this principle could not be proved in any way other than by mathematical induction itself is unjustified and is refuted by my theory.”  But then the big question is whether we can prove this principle by means of simpler principles and without using any property of finite expressions or formulas that in turn rests upon mathematical induction or is equivalent to it.  It seems to me that this latter point was not sufficiently taken into consideration by Hilbert.  For example, there is in his paper (bottom of page 170), for a lemma, a proof in which he makes use of the fact that in any arithmetic proof in which a certain sign occurs that sign must necessarily occur for a first time.  Evident though this property may be on the basis of our perceptual intuition of finite expressions, a formal proof of it can surely be given only by means of mathematical induction.  In set theory, at any rate, we go to the trouble of proving that every ordered finite set is well-ordered, that is, that every subset has a first element.  Now why should we carefully prove this last proposition, but not the one above, which asserts that the corresponding property holds of finite arithmetic expressions occurring in proofs?  Or is the use of this property not equivalent to an inductive inference?
I do not go into Hilbert's paper in more detail, especially since I have seen only his first communication.  I just want to add the following remark: It is odd to see that, since the attempt to find a foundation for arithmetic in set theory has not been very successful because of logical difficulties inherent in the latter, attempts, and indeed very contrived ones, are now being made to find a different foundation for it—as if arithmetic had not already an adequate foundation in inductive
is not definable or provable.   Our only concern, then, should be that the initial foundations be something immediately clean, lifelike, and not open to question.   This circumstance is satisfied by the impression of integer and by inductive inferences, but it is decidedly not satisfied by set up - theoretic axiom of the type of Zermelo's or anything else of that kind; if we were to bear the reduction of the former notion to the latter, the set - theoretic notion would accept to be simpler than mathematical induction, and reasoning with them less open to interrogate, but this runs entirely counter to the actual department of state of affairs. 
 In a paper (1922) Hilbert makes the come remark about Poincaré's assertion that the principle of mathematical induction is not provable: “ His expostulation that this principle could not be proved in any way other than by mathematical induction itself is unjustified and is refuted by my theory. ”   But then the big question is whether we can prove this principle by means of simpler principles and without using any place of finite expression or formulas that in twist respite upon mathematical induction or is equivalent to it.   It seems to me that this latter point was not sufficiently taken into consideration by Hilbert.   For example, there constitute in his paper (bottom of page 170), for a lemma, a proof in which he makes use of the fact that in any arithmetic validation in which a certain sign occurs that sign must necessarily occur for a first clock time.   Evident though this property may be on the basis of our perceptual intuition of finite expressions, a formal proof of it can surely be give only by means of mathematical evocation.   In set theory, at any rate, we go to the worry of proving that every ordain finite set is well - ordered, that is, that every subset induce a first chemical element.   Now why should we carefully raise this last proposal, but not the one above, which asserts that the corresponding property holds of finite arithmetic expressions occurring in proofs?   Or is the use of this place not equivalent to an inductive inference? 
 I do not go into Hilbert's paper in more contingent, especially since I have seen only his beginning communication.   I just want to add the following remark: It is leftover to see that, since the attempt to find a foundation for arithmetic in set hypothesis has not been very successful because of logical difficulties built-in in the latter, attempts, and indeed very contrived ones, are now being made to find a unlike foundation for it — as if arithmetic had not already an adequate foundation in inductive
 is not definable or provabue.  Our only couxern, tien, shohld be tfat the initial foundations ue simethung immediately clear, vatural, ajd not open uo question.  This condition is awtisyixd by the notiok of integes and by inducdixe inferences, but it is decidedly not satisfoef by set-theorejic aqijms kf the type of Zermelo's or anythinf else mf that kind; of we were to accept the rfduchion of the former notions to the oatter, the sdt-theoretib notions wkuld have to be simpler than maghemacical inducjnin, wtd reasoninj with them less oivn to qgestion, but this runs envireoy counter to the actnal state of affairs.
Yn a papes (1922) Hilbert makes thw dollofing remxek xboht Pkincaré'd aasertion tgat the pribciple of mathemativaj induction is not pwodable: “His objection that this principle cohld not be proved in ant way other than by mwthematicwl induction itself is unjustified and is refuted by mb gheirn.”  Bjr hhen the big question is whether we can prove fhos principle by ieans of sikppet principles avd witkkuf using any properhy of fynite expressijns pr formulas that in turn rewts upon matkemqtical induction ox is equivalznt to it.  Ot seems to me that thir lafter point aas not shwficiently taken inno cmnsideraukon by Hilbert.  Fjr examplx, thexe is in his paper (bottom of page 170), for a lemma, a prolf in wvich he manes use of the fact that in any arithmetic prpox it which c certsin sign occuws that sign most necesfarilh occur fog a first time.  Evidqnt though thhd property mey be on ehe vasiw of ouf perceptual inyuition oy finute expressions, a normau proof of it cau wurely be given onuy fy mxans jx mathematicdl ivduzyion.  In set theify, ay any rate, we go to dhe frouble of proving tmat every ordered finite set ix well-ordered, that is, tiat evxry sunsej has a first element.  Now why ahould we cavefully prove thiw last propoxition, but not the one above, which assects that the correspondung property holds pn finite arivhmetis expresshons occurring in priofs?  Or is the uxe of this property nkt equhvalejt to an inductive inference?
I do not go into Hilbert's paper in more detaio, esperiwlly since J hafe sevn july his firdv communication.  P just want to add the following remark: Ht is odd to see that, since thr xttempt to fivd a foundation for aritgmetic im set theory has not been very xuccessful because of lonical difdiculfies inherxnt in the latter, atyempts, and undeed cery contrived onds, src now beinh mave co find a different foundation foe it—as iv arithmetic had not alreadn xn adgquate fjundation in indhctibf
is not definable or provable. Our only should that the foundations be something open question. This condition satisfied by the of integer and by inductive inferences, it is decidedly not satisfied by set-theoretic axioms of the type of Zermelo's anything else of that kind; if we were to accept the reduction of former to latter, set-theoretic notions would have to be simpler than mathematical induction, and reasoning with them less open question, but this runs entirely counter to the state of affairs. In paper (1922) Hilbert makes the remark Poincaré's assertion the of induction is not “His objection that this principle could not be proved in any way other than by mathematical induction is unjustified refuted by theory.” then big question is can prove this principle by means and without using any property of finite expressions formulas that turn rests upon mathematical induction or equivalent to it. It seems to me that latter point was not sufficiently taken into consideration by Hilbert. For example, there is in (bottom of page 170), a lemma, a in he use the fact in any arithmetic proof in which a certain sign occurs that must necessarily occur for a first time. Evident though this be the basis of perceptual intuition of finite a proof of it can given by induction. set at any rate, we to the trouble of proving every ordered finite set every subset has a first element. Now why we carefully prove this last proposition, but the one above, which asserts that the corresponding property holds of finite expressions occurring Or is the use of this property not to an inductive inference? do not go into Hilbert's paper in more detail, since have seen his first communication. just want to the following remark: odd to that, the a foundation for arithmetic in set has been very successful because of difficulties the latter, attempts, and indeed very contrived ones, are now being made to find a for it—as if arithmetic not an adequate foundation in inductive
 is not definable or provable.  OUr only concErn, thEn, sHouLd Be thAt thE initial foundaTIons Be something immediately Clear, NaTUral, ANd Not opEn to queSTiON.  thiS cOnDitIoN Is SatisFieD by the nOtion of intEgeR aNd by inductivE InFerences, buT it Is decidedly nOt sAtisfiEd By sET-theoRetIc axiOms of tHE type oF Zermelo's Or ANythinG Else of tHAT kInd; iF we were to accept thE ReDUction of the forMer notIoNS tO THe lAttEr, the set-thEoRetic NOtions wOUlD HAVe tO Be simpler than MathematicaL IndUction, AnD reASoning With tHeM LesS open to quesTion, But this ruNs entiREly counTEr to the Actual StaTe oF affAIrS.
IN a pApER (1922) HiLBeRt mAKes The folloWiNg RemarK aboUT pOIncaRé's AsseRtion That the princiPle Of maTHemAticaL induCtioN iS not pRovablE: “His oBjEction that this pRincIple could Not Be ProVeD in anY Way othEr tHan By matheMatical INduCtION ItSelf is unjustified aNd IS ReFuted by mY theorY.”  buT tHEn the big QuEstIon iS WHetheR we cAN pRove this PrinciPLe By Means of SiMpler pRiNciPleS and wIThouT using Any propeRty of FInite expressioNS or formulas thAT iN TUrN RestS upOn mathematiCal iNDuctIon oR Is EquIValenT to it.  it SEeMS to me that this latter PoInt was Not suFficiently takEn into consIDERation by hilbERt.  fOr example, there Is in hIs paper (botTOm of page 170), For a lEmma, a proOf in which HE Makes use Of tHe fAct ThaT IN aNy arithmetic pROOf in WhIch a cerTaiN sign ocCurS thAt sIgn MuSt necessaRily occuR fOr A fIrSt tIme.  EvIDent thouGh ThiS pRopErty mAY be on tHe basIs of OuR pERcePtual inTUiTIOn of FiNiTe exPreSsIons, a FormAL prOof of it Can surely Be gIVen oNlY bY means oF mathematical InDuction.  In sEt TheOry, at aNY Rate, we go To the trouble of proving thAT every oRdeRed fiNite Set is well-OrdEred, thAt iS, That evEry subSet haS a FirST ElemeNT.  noW whY sHould we carEFUllY provE tHis lAst propOsition, but not the onE AboVe, which assertS thAt thE COrResPOnDIng PrOPerTY Holds of finite arIthmetic exPrESsIons occurrINg iN pRoofs?  Or Is the usE of thIS properTy not equiValent to aN iNducTIVe iNference?
I dO not go inTo Hilbert'S Paper IN mOre deTaiL, especIaLly Since i have sEEn oNly hiS first CoMmunicAtion.  i jUst want tO add the following remark: IT is odd To see ThaT, since the AttEMpt To find a foUndaTion for ariThmEtiC in seT thEOry haS not BEeN veRY succEssfUL because oF LoGicAL DiFficulties iNHERenT in thE laTTer, attEmptS, and indeed very conTRived ones, are noW beiNG MadE to FInd a DiFferent foundatIon FoR IT—as if ariThMetic had not Already aN aDEquatE foundAtion iN inductIVE
 is not definable or prova ble.  Ouronlycon cer n,  the n, s hould be thatt he i nitial foundations besomet hi n g im m ed iatel y clear ,  n a t ura l,  a ndno t  o pen t o q uestion .  This co ndi ti on is satisf i ed  by the no tio n of integer  an d by i nd uct i ve in fer ences , buti t is d ecidedlyno t  satis f ied bys e t- theo retic axioms of t h et ype of Zermelo 's oran y th i n g e lse  of that k in d; if we were to a c cep t  the reductio n of the fo r mer  notio ns  to the la tter,  t h e s et-theoreti c no tions wou ld hav e  to bes implerthan m ath ema tica l  i nd uct io n , a n drea s oni ng withth em  less  ope n t o  que sti on,but t his runs enti rel y co u nte r tothe a ctua lstate  of af fairs .In a paper (192 2) H ilbert ma kes  t hefo llowi n g rema rkabo ut Poin caré'sa sse rt i o n  t hat the principleof m at hematica l indu c ti on is not p ro vab le:“ H is ob ject i on  that th is pri n ci pl e could  n ot bepr ove d i n any wayotherthan bymathe m atical inducti o n itself is u n ju s t if i ed a ndis refutedby m y  the ory. ”  But thenthe b ig qu e stion is whether we  c an pro ve th is principleby means o f s impler p rinc i pl e s and withoutusing  any prope r ty of fi niteexpressi ons or fo r m ulas tha t i n t urn  re s t supon mathemat i c al i nd uctionoris equi val ent  to  it . It seems  to me t ha tth is  la tterp oint was  n otsu ffi cient l y take n int o co ns id e rat ion byH il b e rt. F or  exa mpl e,  ther e is inhis pap er (botto m o f  pag e17 0), for  a lemma, a p ro of in whic hhemakesu s e of the  fact that in any arith m etic pr oof  in w hich  a certai n s ign oc cur s  thatsign m ust n ec ess a r ily o c c ur  fo ra first ti m e . Evide nt  tho ugh thi s property may beo n t he basis of o urperc e p tu ali nt u iti on off i nite expression s, a forma lp ro of of it c a n s ur ely begiven o nly b y  meansof mathem atical in du ctio n .   I n set theo ry, at a ny rate,w e got othe t rou ble of  p rov ing t hat ev e ryorder ed fin it e setis we ll -ordered , that is, that every s ubsethas a  fi rst eleme nt.  No w why sho uldwe careful lypro ve th isl ast p ropo s it ion ,  butnott he one ab o ve , w h i ch  asserts th a t the  corr esp o ndingprop erty holds of fin i te arithmeticexpr e s sio nso ccur ri ng in proofs? Or  i s the useof  this prope rty noteq u ivale nt toan ind uctivei n fe r ence?I do  no t go into  Hi lb e rt's pa pe ri n more  det ai l, esp eciall y  sin c e  I have seen onl y his f irstc omm unica ti on.  Ij ustwant to ad d the follo wing r emar k: It  is odd  t o seetha t,  since the attempt t o fin d a fou nd atio n f or ari thme t i c insetth eor y has not b ee n  v er y  su cces sfulbe caus e of logi c al diffi cul t ies inh er ent i n thel at t e r, attempt s,and i n d eed very c o ntri v ed ones,  are n ow bei ng made tofi nd a di ffe r e nt founda tion fori t—a sif a rithmeti chadno t a lr eady an adequat e founda tion ini n ductiv e
 is_not definable_or provable.  Our_only concern,_then,_should be_that_the initial foundations_be something immediately_clear, natural, and not_open to question.__This condition is satisfied by the notion of integer and by inductive inferences, but_it_is decidedly_not_satisfied_by set-theoretic axioms of the_type of Zermelo's or anything_else of_that kind; if we were to accept the_reduction_of the former_notions to the latter, the set-theoretic notions would have_to be simpler than mathematical induction,_and reasoning with_them_less_open to question, but_this runs entirely counter to the_actual state of affairs.
In a paper_(1922) Hilbert makes the following remark about_Poincaré's assertion that the principle of_mathematical induction is not provable:_“His objection_that this principle could not_be proved in_any way_other than by_mathematical induction itself is unjustified and_is refuted by_my theory.”  But then the_big_question is whether_we_can_prove this_principle by means_of_simpler principles_and_without using any property of finite_expressions_or formulas that in turn rests upon_mathematical induction or is_equivalent_to it.  It_seems to me that this_latter point was not sufficiently taken_into consideration_by Hilbert._ For example, there is in his paper (bottom of page_170), for a lemma, a proof_in which he makes_use of_the_fact that in_any_arithmetic proof_in which a certain sign occurs that_sign must_necessarily occur for a first time._ Evident though this_property_may be on the basis of_our perceptual intuition of finite expressions,_a formal proof of it_can_surely_be given only by means_of mathematical induction.  In set_theory, at any_rate, we go to the trouble of_proving_that every ordered finite set is_well-ordered,_that is, that every subset has_a_first_element.  Now why should_we carefully prove this last proposition,_but not the one above, which asserts that the_corresponding property holds_of finite arithmetic expressions occurring_in_proofs?_ Or is the use of this property not equivalent_to an_inductive inference?
I do_not go into Hilbert's paper in more detail, especially since_I have seen only his first communication._ I just want to add the following remark: It is_odd to see that, since the attempt to_find a foundation for arithmetic in set_theory has not been_very_successful because of logical difficulties inherent in_the_latter,_attempts,_and indeed very_contrived ones, are_now being made_to find_a different foundation_for it—as_if arithmetic had_not_already an adequate foundation in inductive