Bibliography Index

7/28/2019 Bibliography Index

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Bibliogr:aphical emarksanclFurther lleacling

PreliminariesWe try Lo indicate for. each important resti l t or notion its author and the correspondingpublication, and possibly a lso reference o another work where the result is presetrted. Ou lai m is to be as plecise as possiblel on the other hand, these lemalks are not intended tobe a complete historical source and they serve only for orientation. This concerl ls rnainlyremarlcs on old lesults. Th e reader interested in deepel investigation of origins of thernetamathematics of alithmetic is refemecl to source books: G


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393 Bibliographica l Rema,rlcsancl !-nlbher Reacling

Ivlatiyasevid's theorem is a final step of a long tesearch inio l{ilbert's tenth problem inr,vhichJ. Robir-rson,A{. Davis ancl I-I. Putnam played prominent roles; besiclesMatiyasevic'spapers, see [Da.ris 73 Am. Niath. Vlonihly] for history ancl a cletailed proof. Finally letus note thab lhe clecidability of tire set of all sentences of the [language of arilhmeLicwil;houL multiplication] Lrue in N mentioned in 0.49 is ptoved in [Presburgcr 30] Simila.r:ly,[S] 1); the main task - arithmetization - is donein a rather sbandard way, usual in presenting Godel's results on PA, an d it is seen thatIXr suffices bo plove all necessary things. (The auxiliary coding of o-sequences ollows[Shoenfield 67] ) Partial truth definitions are of great importancel they were studied by[lvlorlbague59 ] ([Schiitte 60] has a similar notion). [I(reisel-L6vy 6S] is an irnportant paper:dealing with par:tial truth definitions; rv e shall mention it in remarks to Chap. III.(2 ) The main start,ing point for th e study of theories I En an d B En*r is the paper [Paris-Kilby 78] tl-riswa s preceded by fParsons 70, 72] where sorrre mportant part ial results wereobtainecl. Th e paper by Paris-Itilby contains among ot,hel things the equivalence of f En,I I f n , LDn ,L i l n (Th .2 .4 ) and the i rnp l i caL ions lDn-71+ BEro+ t + IEn ,BEna t e B I In(Th. 2.5) . The fact (2. ,1) hat IEn + IEo(E*) a.ppears n fClote 85 - Caracas] (witha remarl< lhat Paris and l{ossak independently proved the same result); the paper alsocontair - rs ploof of the impl ical ion BEt,* r * LAn4-1 * Iz\n- , ry (2.5) . The impl icat ionLAn-Ft 4 BEn41 was proved by R,O. Gandy (unpublished). Note that l i. Fbi edmancirculaied a preprint (on fla,gments of Pe ano arithmebic, before 1985) in which he claimedB.Dn*r and -I4,r.'-1 to be equivalent; but it still seems o be open whether It\n-;t provesBE,r+r (or , equivalent ly, LAn)-r ) . The fact that in BX"-pi , . ! , , aud,I / , " forrnulas areclosed undel bounded cluantification is due to [Parsons 70].Th e nobion "piecewise cocled" is studied in fCloie 85 - Caracas] and was introduced byI(ossak. Theorern 2.7 appears n the sarnepaper. Concerning Lheorern2.23: The equivalenceSEn- l t Q Stn e IEn is in [Clote B5 -Caracas] , properbiesof PHP in [Dimit i 'acopoulos-Paris 86.1, roperties of the axioms of regularity in [fuIills-Ptr.r'is4]. 2.24(I) seems to be afollclore, properties of local and partial approximations (? an d P) appear in [H6.jek-Paris86]. Theolem 2.25 is due to V. Svejclar unpubl ist red) .


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Bibliographical Remarks aud f'\r l thel l leading 399

Finite axiomatizability of IEn, BDr-F1 (n > 1) is an easy consequenceof the existenceof part ial satisfactions; iL is diff icult to say rvho first observed iL'

Relativized satisfaction (for Xi(X)-formnlas, X being a set) is introduced ancl invesli-gated in [Hzi.jek-I(udera89]. lClote 85 - Oberrvolfach] works with X,"(X)-definable subsetsof a model fuI of I Et, X being a subset of M; but he does not formalize relativized satisfac-tion insicle .IXr . The development ol a theory of low 4rr-1-1 el s a,ndsets of tl"reorder type oftlre trniver:se 2.63-2.71) follows Clote's work but is macle i,nside thc thcories 8J,r-1-1, notfrom oulsicle using models. Th e strengthening Lo IEn and replacement of (low) z\'11 by(low) Xj(X") is due to [[Iijek-I(unera 89]. We shall comment more on low sets in remalksto Sec t . (3 ) .

Finally, arithmetic with a top is studied in [Dimitracopoulos-Paris 82 ] and fCegielsl


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,100 Bibliographical Remalks and trbrthel Reacling

[paris 78], [McAloon ?9]. [Ialrington conbributed by leformulahing tl-reprinciple into i lselegant form using the noiion of a (relal ively) large finite set zrnd showing tl-reprinciplebo Le equivalent to Con'(PA*Tr( I I1)) ( . f . [Par is- I - Ia l r ingtron 77.]) ; he IasL equivaiencewas ilclepenclently p.ov". i by [fuIcr\loon 80 - rappolbs] InvesLigation of instances of theParis-Harrington principle ancl their relation to fragments of P:l (as well as of instancesof th e pr. inciple of ordinal-large intervals, see below) is tl-remain conlent of Paris's Papefs[Paris eO ] lfal is 81 ] Posit ive results on fragmenl,s contained ir-r hese tw o papers form tl'r ':main Lopic of the plesenL chapler.

(1) Theorem 1. g in particular is clue to Paris. Bu t we do no t preseut it following Palis(sinc" he obtained the provabilities in cluestion by incli lect, moclel-theoretical means) V/ efollow and elaborate proofs of fClote 85 - Oberwolfach] baseclon lhe Low Basis Theoren-r. nparbicular , Theorems 1.5, 1,6, 1 .7 are due to clote. c lo te also reasonsmodel- theorei ical iytut, hi s argurnents easily yielcl cl irect proofs in fragments. Fulthelmofe) we improve CIole'sresulbsby using our stronger version of Low Basic Theorem. 1.10 is follclore'(2) HLre we elaborale the proof o f fPar is 81]; the subsect ion (b) on combinator ics rel ieson IHarr ington-Pacis77].

i ly 1.fr" hisioly of what rv e call th e Schwichtenberg-Wainer hierarchy or fasb growinghieLai.chy s as foliows (see [Buchholz-trVainer 87]): [I(reisel 52 - no n fin.] showed that theftrnctions pr.ovably recursivsin PA can al l be defined by recu'sions oll certain well ordelingsof type ( eo . Later [Schwichtenberg 71] and [lVainer 70,72] inclependently generalizecluorll* resultl of [Glzegorczyk 53] and [Robbin 65 ] to show that l(reisel's functions can becharacterized by means of the present fast glor,ving hierarchy below e6. As further referencewe mention [Liib-wainer 70], lschwichtenberg 77], fRose 84], fBuc]rholz 841'

fl{etonen ancl Solovay Bi] relatecl this hierarchy to the Paris-I-Iarlington principle anclestablished, using ptr.iy combinat'orial means, sharp uPper an d lower bounds to the{unction"(n) - rnin{o I [0," ] - ' (n * 1) l l ] ;

from this, they reproved Paris and Flallington's result. In their paper' I(etonen ancl Solovayiptroduced an d studied th e notion of an cr-large finite set (due originally to I(eLonen), cvbeing an orclinal. It follows flom their investigations that the principle(w) (V" I e,) (Vc)(3y)( [ t , y] is a- large)is (meaningful ancl) unprova,ble n Pzl . fParis 80, B1] ntroduced and investigated instances"f (Vf; anJrelatecl ihem to flagments of Ptl. We freely follow Paris's papels using fKetonen-Soioury 81]. Theorem 3.18 cloesno t occur: explicitely in [Paris 81]; i t can be found e.g. in[Takeuti 75], see also [Kurattl B6].There ar e va.riousother importanl combinatorial principles bhat can be analyzed r'vibhrespect to fragments, notably: a plinciple proposed independently by Pudlik (original pa -per unp,-rblished, ee [Hdjek-Paris B6]), I(anamori-lvlc Aloon's plinciple iI(anamori-NcAloon87], principles due to [Clote-McAloon B3 ] and possibly others'

Chapter I IX(1 ) Th e informal lotion of an interpletation of a theory in anolhel one appeal's to be la'thel'olcl; bu t lve dicl noL attempl to identify particular references. Th e first work dealing withinterprebations in connection with systerns of f irsb order arithmetic is fTarski-lvlostowski-Robiirson 53];an ol d paper is f ivlontague 57]' Feferman's fundaru'ental paper fFeferman 60]deals also with interpreiability; [Monia.gue 63] coniains a model-theoretic characterizationof interpletability (cf. also []Iei, jek66])'Simiiar.ly, we pr.esentno information about the origins o['the notion of partial col-iser-vativiLy; but the-first resr:lt concerning partial conser-vativity is [I(reisel 62 ] shorving thirt


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Bibliographica.l Rema,rks and f{rlther Rear,cling 40I

-Cono (Ptl) is I11-conselva.tiveover PA . We shall give detailed references to later wollcsbelor,v point (,1)).

Sequential bheocieswere introdr,rced by [Pucll6k 85]; a similal notion was investigahedby [Vaught 67] F\.iedman also had a similal notion but cl iclnot present an exacL definition,cf . [Smolyriski 85 - Ft'ieclman's researchi.tlCtlo is one of var.ious ntelesl,ing subsystems of second order arithrneticl we t'eler toSimpson's for : thcomingboolr on this topic [Simpson 86-90] The moclel- theoret ic prooI ofconservativity of ACrlo over PA (1.16) is a, olklore; bh e proof of finite axiomatizability isa variant of Goclel's proof of the fact ilnat GB p'r-oves he schetna of complehension [Godel40] Se e also belo.,v.The notions of a binumelation and numeration go back to [Godel 31 ] an d ale studied indebail in fFefelman 60 ] G


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Bibliographical Remarks ancl Ftrrther Reading

4.2b s clue o Guaspari and Solovay for' ? f PA): see [Guaspari6'l ] (their examplesare rnorecomplicated ha n ours)4.26 1) is classical cf . fRogersoz]); (z ) f]Injek B7l; (r) for ? = zF, | - I1 1 [l l6jek71], generalizeclor T 2 PA by [Lindsbrom a], (a ) implicit in flindstrom 84 ] for T 2 PA .4.2'l For T ==ZF,V = II t Solovay unpublishecl)or PA, I = I/ r and Consu [fl6jel


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Bibliogr:aphi cal Remalks and f,\rbher Reacling

a folrnula nA-necessarily A) and investigates theil arithmetical intelpretaLious. Such aninterpretation 'r assignsa sentence of arithmetic bo each propositional variable, comtnttteswith connectives and interprets n as provability: (trA)- is Pr"(,,i-) (for a f ixecl plovabil ibypreclicale or a fixed arithmelic ?). There is a naLr-rral xiomatic sysbem I (or G) satisfying(fot reasonable 7 and Pr') arithn-rebical cornplel,eness:G I tl iff for ea.cir ariLhrnet,icalinterpretation {, , ? l-- A*. A pioneering paper is lsolovay 761 an extensive monograph is[Smoryfiski 85 - Self].Fur-bher rnportani names (see biblioglaphy): Boolos, Sambin, de Jongh, Magari, Mon-tagna, Artemov and others. One can introduce a fultl,er moclality > of inlerpretabiliLyancl investigate moclal interpr-etability logics and logics of partial conset''rativity (Svejdar,Visser, de Jongh, Veltman, Belarducci , Savrukov, H6jek - Montagna).

(f ) In this book we pay vely little attention to advanced methods of .proof llveory; inpalt,icular, cut elimination is used only in Chap. V, Secb, 5, Proof theoly is an extlemelyIarge domain and we shall not tly to sketch its aims in a few lvords; insteacl, we refer bobasic monographs; [Schiit te 60], [Takeuti 71], lPohlels 89]. See also [Schwichtenbelg 77].An application of prool th.eory Io fragments of arithrnetic is fSieg 85]; we obtain some ofthe results presented by him using model theoretical ra.ther than proof theoretical methods( in Chap, IV.)

Chapter fV(1 ) The construction of a.non-standard model of PA using definable ultrapower goes bacl


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.t04 Biblioglaphical Remarl


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Bibliographical Remarks an d Further Reading 40 5

Chapter VBoulded A.rithmetic rvas intloduced by Parikh [Parikh 71]. I Ie introduced th e systemwhich is nowadays clenoteclby /Xo or .[z\0. Sorne weak fragmenl,s hacl been consider.eclbefor.e.Shephelcison [S]rep]reldson 64, 65, 67] considered Iopun Goodstein [Gooclstein 541an d Cleave and Rose [Cleave-Rose 67] studied so-ca lleclEt-arilhmetics. These systetns

ar eecluational systems corresponding to Gt'zegorczyk clarsses ', [Glzegolczyk 53 ] Anol.herecluational sysiem was introcluced by Cook [Coolc?5]. tlis system PI / is closely relatecl lo52 1 this rela.tionship is described in [Buss 86 - Bounded Arith.]). After Par:ikh, most ofth e research on l lo an d fXo * J-lr rvas done by Paris an d trVilkie. Th e most importantpapers are [Par is-Wilk ie 81- As Sets]and f lVi lk ie-Par is 87]. Let us note thai [ lV i lk ie-Par isSZ 1 m publishecl several years afbel the resulbs of the paper were obtained. LaLer Buss"rrler.d this areawith hi s book [Buss 86 - Bounded AriLh.] . Several other rnabhemaLiciansconlr ibuled to this field and tl-re esearch is going on.(f) .1ne result on /op"n is due to Shepherdson lShepherdson 64]. Fo r further indepen-clence esults on f6pqn se e ffuIacintyre-Ivlarker 89 ] [Adamowicz 86] Theolems 1. 1 and 1.2ar e due to lvlcAloon [NIcAloon 82 ] (w e present Kaye's proof of 1.1). Paris improved The-orem 1.2 to cover -od"lr of IEt fParis 84 ] Theorem 1.4 (only for.IX6) wa s proved in[Parikh 71]. The hierarchy of theories IDo * Q;, IEg t Er p and lX s ! Superexp appearsin tl-repapers of Pa.ris and lVilkie. A systern ecluivalent to IEs * Ex p is ment,ioned alsoin [Fr.iedrnan B0] The Lheory -[-Er and relat ed systems were investigated in [Wilmers 85]and fKaye 92 - Open] . PIND an d LIND axioms were introduced in lBuss 86 - BoundedArith.l .We mention only very briefly EsPlIP, a very intelesting subject with several nice resultsand open problems. lVoods showed ihat IsP.FIP proves Beltrand's Posbulate, -rence hatthere are infinitely many primes. (Note that by Parilch's theorern an y ptoof of the infiniLudeof primes in Bounded Aritl-rmetic must give a piece of information about the distlibutionof primes.) Wilkie proved a weaker version of EoPIIP in /Xo * J-2twhich is sufficientfor Woods'proof. The resulls are presented in [Paris-Wilkie-Woods 88 ] A weak form ofindependence of PHP was shown by Ajtai fAjtai 83]. FIe proved that if we extencl f-Dswit lr a new lelation symbol R(n,y) (and do not add any special axioms about .R except for'th e inducbion for the new formulae), then PIlP(R(r,y)) is not provable in such a tl'reory(which is denoted by lXs(n)). I le proved a similar result for th e parity principle whichsays that an interval lO,2n-F t) cannoi be partitioned into two-element blocks [Ajtai 90].(Z ) Computational complexity is a very broad subjeci. In Secl. 2 we have mentionedonly a few resr-rlts. here are several books about this subject; we recommend the followingones [Aho-Hopcroft-Ulman 7,1], [Garey-Johnson 79], [savage 76], fWagner-Wechsung 86],[Balcazztr-Diaz-Gabarr6 BB , 90]. The vely recent llandbook of Theoretical Computer Sci-ence fvan Leeuwen 90] is also a very good source. lVe have quite neglected an irnporLantpart of complexity theory whic h is the complexity of boolean functionl the best referenceis lWegener 871.IIere we shall only state the authols of the theorems of Sect. 2. Theorems 2.2(a) ancl(b ) are due to llartmanis and Stearns [Hartmanis-Stearns 65] and ]Iartmanis, Lewis andSbearns []Iartrnanis-Lewis-Stearns 65], r'espectively.Theorem 2.4 is due to ]Iopcloft, Pauland Valiant fFlopcloft-Paul-Valiant 75] Theolem 2.5 r,vasproved by Savitch [Savitch'/0] .Tlreorern 2.6 was ploved independently by hnmerman and Szelepcz'6nyi fhnrnerman BB]an d lszelepcz6nyi 87]. Theorem 2. 6 is du e to Cook [Cook 71]; in this farnous paper heformulatecl the P L ntp problem. Let us note in passing thaL alretr,clyn L956 Goclel haclasl


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406 Biblioglaphical Rematlcs ancl F\rthel Reading

bo Laclner [Ladner 7b ] Theolern 2.9 is due to Ba]ter, Gill and Solovay [Baker-Gill-SolovayTb]. Th e polynomial Time llierarchy wa s intlocluced in fstockmeyer 76]' The Lineal TimeFlierarchy wa s studiecl in [Wrathall 78]. Th e reference fo r NepomnjaSiij 's Theolem 2'14is fNepomnjabiij Z0]. Wrathall proved that, Linl-t s eclual to the class of rudimenla,ry seLsi ' trocluced"in firnullyan 61 ] It is not cl i f f i .cult to show that luclirnentary sets ar e just4o definable sets (,-,uitg th e natural cocling of secluences).Thus Theorem 2'16 fbllowsfrrn he r resuh;. Theorerlr-rs .1 7 (i) and (ii) *"ru ptou"d in fFlaltmanis-Lewis-Stearns 65 ]an d [Hartmanis-Stearns 65] respectively. fh"or.rn 2.18 is clue to. Za U 17,eU eS].Theorem2.19 wa s possibly ,,"u", "*plicitly statecl, but it is a typical application of z6'k's meLhocl'Tlreorem 2.2I is clue to puatat. f{irther r.esu}ts n time hieralc}ries cau be lbund in lPalis'WilkieB1l.

An interesting area, which we clo not cover, is count'ing problems. A typical cluestionis the following: .1 o fo definable sets have -D o definable counting functions? (F(u) is acounting funcblon for A, if lr(o) is the number of elements of A smaller than c.) 1'his isclosely ielated to the questions about PH P in IEo.If Xo sets ha d Xo counting functionsancl their properties *e.u prouuble in.Ixo, then .Ixo would prove x6 PHP. Th e relation ofcounting to Bounded Aritirmetic and approximations of counting functions wele studies in[earis-filkie Bb a.nd BZ]. Recently Toda proved a result from which it follows that the set'sin PH do not hu,v" .ounting functions in PH, provided t'hat PH does not collapse fTodag9] . llence, assuming that FH does not collapse, the answer to the above question is alsonegative.(3) Bennett was the first to show a !o formula for the telation z = E! fBennett 621Paris found another such for.,mulaan d Dimitracopoulos [Dimit|acopoulos 80] ver: i f ied thatth e inductive clauses ((c.1) an d (c.2)) ar e provable in f.Do fo r Paris's fotmula' A diffelentxo definition of "*ponlrtiation is in [Pudli,k 83 - A definition]. In Sect. 3 we use th e idea of[N.t*n 86] : fi rst to build a weak form of coding of sequences (based on bina'ry expansions)and then to define exponentiation. Flowever Nelson does not worlc in -f 0, he works intheories interpretable in Q. (Also he uses base four expansion of a single numbel insteadof the binar-y expansions of t*o numbels.) A folmalization of syntax in lxo is consideredhere fol the first tirne, though the icleas on which it is based have been around for sometime. A for:malization of uyniax in /xo l- J-2rwas made in [wilkie--Paris 87]. Theolem 3'37is new. The corollary that context-free languages belong to xfl follows also from the resultof [ !Vrathal l7B]thatconLext- f r 'eelangtragesarerudimentary.(.t) The definition of the theories ,Sr, , an d their fragments Si,:f; is-d]e to Buss [Bussg6 - Bouncled Ariih.] . The theorems aboul the relationship between different axioms ofinduction and related principles ar e provecl n that book or in fBuss 90 - Axiomatizations];(Theorems 4.b,4, ' ( ,+.S, +.10,4.13), Our model- theolet ical approach o witnessing theoremsL inspired by Wilkie's'proof of Buss's theorem fWilkie 85]. Theorern 4.29 wa s proved byI(rajiiek an j Takeuti, th e proof here is due to Pucll ik. Theolem 4.32 first appeared in [Bussg0 - Axiomatizations], i t is a strengthening of Buss's theolem from fBuss 86 , BoundedA'ith.]. We have u.lrit"d Wilkie's ploof to this strengl,heuing. Th e r:esult about modelsof fr.airnents of 52 wirich we have extracted from hi s proof (Theorem 4.31) might be ofa., inJepunclent interest. Theorem 4.38 is clue to l(rajicel


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Bibliography

This bibliography grew out of or-rrworking database of papers related to ourresearch in foundabions of ariLhmetic. Several papers have been adcled whenworking on the booic, in par:bicular we aclclecleferences o some papers andbooks on complexity theory. The choi.ce f items has been considera.blybiaseclby our interests and the sorlrcesavailable to tts. Still we belie.'re ha t it canbe a valuable source lbr mabhematicians working in this fi.elcl.

Our sources have been the following: First of all, we used the excellenl,Omegu-bibtiography f muth,ema'ticnlogic.This is the main sourceunbil .1984.It was impossible to copy any section from the bibliograplry as a wholel eachitem has been selected (for criteria see below). In particula::, section F30in th.e Omega bibliography contains many papers not included here. Ourfurther soLrrceswere: r\ bibliograplry by Srnoryrislci,circulated some yearsago, our o'wn works and their lists of references,most impor:tant journalsand proceeding volumes of relevant conferences,our collections of preprinLsancl reprints and some f'ew in:formation rebrieval sessionswith mabhemabicaldaLabases.Finally, some colleagueswere sent listings of their papers containedin Lhe bibliography and asked, o send completions.

Our criteria for inclr.rsionof a work into the bibliography have been, un-fortunately, rather vague: we included papers about which we were sure or atleast suspecbeclha i they'were someho,,v elevanL or topics elaborabecl n thebook ancl for or-rr uture research. Since the boolc s clevobedo the metarnath-emabicsof first-order arithrneLic wibh special emphasison fragments of Pea.noariLhmetic, incluiling wealc ragments and, on the other hand, to interpretabil-ity, parbial conservaiivity and some parts of model theory of lragments, thesetopics are ernphzrsizedalso in the bibliography. Little attenbion is paicl totopics as aclvancecl roof theory or seconclorder systems. The fz'Lcthab someparticular paper is not inclucleclcloesnot mean t,hat we hold ilb or ilrelevant:lh" ,"rroon lrray be ihzrt the paper has been unknown to us or at ieasi; r'vehzrveno t lcnor,vnts content.


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4 1 0 Bil>liograplryAcxnniue.i'rN W. : Zur Wiclerspluchsfreiheit der Zahlentheorie, Ivlath. Ann. Vol' 11' l , 1940pp . 162-194Acznt p.: Tlvo notes on th e Paris inclepenclence esult. In : Niodel Theoly an d A|iLhmet'ic(Leci , . lotes NIath. 890), Spr inger-Ver lag,1981 pp' 21-31

Aonl,totvIcZ 2... Ope-a nj,.,ct1oi 'r"a,nd Lhe brue lheory of ra,tionals, Preplint 289, Poli: ;hAcerc l . c i , , 1984Aorrntowtcz Z. : A;riomatization of t l- ie orcing lelation.,vii 'h an a.pplicabion o PA, f'\u'rcl'NIath. Vol . 120, 1984 PP. .16' f -186A.oavtovrtcz Z,: .Nlgebraic opproach bo lr-incluctio n. In : Proceeclingsof th e tl-r ird Easbelconfelepce oir rnoclel ;heory (Gloss I(olis,'1985), I-Iumboldl Univ. Berlin, 1985 pp' 5-'l-5Ao.q.rurowlcz . ' . Open inclucLion abstract) , Jouln. Symb. Log. Vol. 50, '19B5p,p' 260Anrrlrloi,vtcz Z.'.lome results on open and cliophanbine ncluction. In: .Logic Colloqttiuil'84, blorbh lol land, 1986 PP. 1-20AOAptOWtCZ Z. : Open inch-rctior-rnc l true theoly of lationals, Jottrn. Symb' Log' Vol' 52,1.987 p. 793-801Aorrl,towrcz Z. : Encl-e;ctenclingmoclels of IA o + enp+ BEr, preplini; 431, lnsl ' lr lath'Pol is l ' rAcad. Sci . , 1988, P. 38ADAMoWICz Z. : A rec.,rsior, theoretic characterization of insLanceso{ BE n provable in

II^11QY), Fund. Nlath. Vol. 129, 1938 pp' 237-236ADAMo\,vIcz Z.'. Patarneter free inducl, ion, the lviat ' iyasevicTheolem an d B-f1 . In: LogicCol loquium '86, North Hol land, 1-988 p' 1-B

Aoa.ltowtcz 2. , BT11HAJSI(r\T,: ffunctions provably total in -Is, Fund. Ivlail'r ', Vol. 132,1989 ,pp . 189-194

Aolrurowliz 2. , I{ossl,I( R. : Irlote on au intermecliate induction scheme. In : Seminar-berichte 2L, Proceedingsof the Fourth Easter Conferenceon Ilodel Theory, Gross-I(ciris,1986 pp. 1-5

Annttowlcz 2,, I{ossa,tc }1.: A note on BEn ancl an intelmediate induction schemzr,Zeitschr. IVIath. Log. Gluncll. Ivlabh.Vol. 34, 1988 pp' 261-264Ao6rvlowrcz 2. , Nlonl,LES-LuNA G.: A reculsive rnodel fo r arithrnetic with weak induc-t ion, Journ. Symb. Log. Vol. 50, '1985pp' 49-54Aontsolt J. W. : Supu,ru.ttn principles in hierarchies of classical an d effective descriptiveseb heory, Fund. Math. Vol . 46, 1958 pp' L23-135Aolrn A. : Extensions of non-standard models of number bheory, Zeitschr' Math. Log'Clrundl . Math. Vol . 15, 1969 pp. 289-290f\Ho A. V. , FIoecnoFT J. E., Ulr . ivr r \ l \ J . I . : The Design and Analysisr\lgorithms, Addison- \r\bsleY, 97 4r\.lr*1i M. : Xr 1 forrnulae on finite structures, Annals Pure Appl. Logic, Vol.1-48A:r^lt M. : 'Ihe complexibY of theComp. Sci . , 19BBpp' 346-355

pigeonhole principle, 29-th Symp. on For-rndations of.AJTAI M. : Pariiy an d th e pigeonhole principle. In :Birlchiiuser 1990 pp. 1-24

Feasible VIat,hematics, uss, ScotL eds.,

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complete modalSSSR, Vsesojuz.Soc. Transl . (2)

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