PartA Chapter1

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FIow precious also are bhy thoughl,,":unto me,O Godl hor,vgreat is l,he sum of thern! If Ishould count them, they are mole in numbeltha,n the sand; when I a.,iralce, am still wiLhthee.

( I 'sahn 1:J9, ' l - l8)

Part APositiveResulbs nIi'ragmenbsf Ari b rmetic


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1. Basic Developments;Partial T\'utl-rDefinit ions 291.3. Particularly important f'ragments of PA result by restricting the incluc-tion scherna o formulas tp from a prescribed class.This will be invesLigateclndetails in Sect. 2; herc we rnalceonly a few particuLar choices. Ioo",, I Eo , D twill denote the bheory Q plus the induction schemaf.or g open, Is, tr'1 re-specLively. (\\'e shall also investigate a tireory with an exlendecl language.)Note bhat in Part A we shall clevelopmainly theories containing /X1 (anclcontained in P,4). This is because n If1 we can formalize a proof of ihe factthat total y'1 fr-rnctionsare closedunder primitive recursion (a careful formu-lation is presented below). This is the mosL important feature of fragmenl,scon{,aining IEt and makes them remarkably different from weaker systems.Note also at this time that Chap. V deals r,vith fXs and related theories andelaborates their specific problems. Iop"., will play only a marginal role in thisbook.1.4. Note that by (Q3), each non-zero number r has a preclecessor, .e. a ysuch that S(y) : r. Thus we may define, in Q, a total function P by thefollowing definition:

y : P(x )= . " : 0&y : 0)v ( * #6 k S ' (y ) x)lVe shall now proveseveral ormulas in 8. Recall that for m e N,, rn is them-tln numerai (cf. 0.28).

1.5 Lemma. The tbllowing formulas are provable in Q:( 1 )(2)(3)(4\(oJ(6)(7)

r I A : 0 - t . r : 0 k A - 0 ,x * a - 0 - r . r : 0 V g : 0 ,r * T - . 9 ( r ) ,0 ( r ,

5(r) < ; +a --+ 1fi, ,. 9 ( r ) + n : t + n - l - I ,n 1 r - - - + . t r : f i V r + 1 r .

Proof.ProceednQ. Weprove(1)-(4). Talce 1). I f A +0 then y:,9(z) forsome , thus r + A : S(x + " ) #0 . t f u +6eU :6 thenc -Fy : S(z) fo rso m e . Th is p r o ve s1 ) .A d ( 2 ) :a ssu m e , A * 0 , * : S ( u ) , u : S ( u ) .Thenn * ! : 5 ' ( u ) * , 9 ( u ) : ( S ( " ) ' r u ) - F , 9 ( u ) : ( S ( u ) " ) - 1 0 . ( 3 ) i s r i v i a l .(a) is obv ious y (Q+) . 5 ) : I f z * S( r ) : n+ I then 5 ' (z+ t ) : ,9 (n ) 'thus z * x : rT.Note that (5) is a schema; or each n we have a proof. Also(6) is a schema; ,ve hall construct he desiredproofs by incluction. ObservethaL rve shall use zlo ndr,rctionwithin Lheproofs (sincewe traveno inductioni" Q) ; we shall construct the (n + l)-th proof from the n-th one. This willofbenbe tire case.For n - 0, Q proves5(r) +0 - ,S(r) : r -f T. Assuming


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30 I. Arithrnel,ic as Number Theory, Set' Tl-reory ancl Logic

(6) weserg F ,5(r) r z T - ,9(r) FS(it) - ,9(,5(r)FD)): s(x -pffiT) -z: -F n - :t'. BYnr

-t-S'(iTa ) * -l- *2.( '1) n 0. assttrnen nkn f n; t 'he 'n,or some # 0'(6), *. gel, c== P(z) t-?)+J, thrrsn'-t- S r'1.6 Theorem. Fbr eachn, m ' {, Q proves he follor'ving:( 1 )(2)\ r /(4)(5 )

rn +n : YTl'f T7f f i , . f l : f f i T rL

f f i ' + n f - o r m # ' ,: t n : . r 0 V o T V ' . ' V r - n ,

x 1 n Y n l n .P r o o f , ( 1 ) s / e p r o v e Q I m + n : y l + a b y i n - c l u c b i o n o n r z . F o r n : 0 w e h a v eto prove e F fr+ O : rn , wlich f91low1 ry 1q+;. Assume we alreadyhave aproof of (1) and proceecln Q:ffi '+i +1 *-ff i-F S(n) - S(tn+') : m + n + 1''The proof of (2 ) is simiiar'- (3; Next *" ,ho* that rn I n implies Q l- * + m. IL su'fnces o assu'ren < rn. For rn : 0 the asstrrnptio' i, ,,u,".,ous. Assttme the. assertion for rna n d l e Ln 1 f f I - f l - . T h e n e i t h e r z : 0 a n c t Q 1 ) - g i v e sQ t s " * . ^ * 1 o rn : n0-|.,1 an.clwe have Q I n0 + m by the ind'uctive assumption; henceQ | -" * nL+L bv (Qz).( ) lve .or.rt*i'proof, of the formulas in question by incluctio* on n'trbr n : 0 see r.f(f;. Assttme the assertion for n and' consider

n -l- 1' Theimplicatiol r-- is-.r"nrry provable using (1); thus proceed in Q and assume* < n*a. If.n 0wsore d'on"; herefbressume.r6 P1 1:1(5)ryJgP(*)( D, hus ; :uy. 'Yr( t l - n ' whichimpl ies: Tv" 'Vr : n * r '(5) F 0 < t 'Uv f ' f (+) 'Assume tsn( rV r 1 nandproceedn 0'If r ( n the.*"i'alft{ios ttj u"i 1r;; t " < r then' v1'5(7)' itheri l , : x , t h u s r S n * ! r o r n + I < r ' n1.? Remark. (1) Q l- * -l n : k -' xiterabed seof (Q2)'

(2) Q Pr:oves

- k - n (for n < /t); this fbllows bY

(bY 1.6('I) r 'Lsing1))'l - r * a - T n V r : i k a : Vi { j : t t

r.8 Theorem. (x1-completeness9- fQ ). !"t--?(t) l: a Xs-forrnula with theonlv freevariable ",," ir"t /f F (:c) '{r) ' Then Qt- (1t')tp(r)'

Proof. Ib is sufEcienb l'o,p (h , , . , . ,T ; ) imp i ies Q l -

V ( r r , ' , r " ) First sllow, uLsin'gEs ihal, iV F1.6( i ) ' (2), thar 'Lh.ow for ear.ch

,P (14 ' , ' ' ' ,T ; ) '


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1. Basic Developments; Pariial Tl'ut,h Definitions 31for each term 't(* t, . . . ,rn) ancleach n-ttLple kt,. . . , lcn of elementsof .M,

Q s (li,. . E;) : vat(t(6,. . En)(thus, e.g. Q F (3 -F5) i, 8 - 64). From bhis t follows,againusing 1.6, bhatour asse,rtion olds for g atomic ancl negatedai;omic. Observe hal, if l/ F. . . ( / r-


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32 I. .Ar:ithm etic s Nurnber Theory, iet Theoly and LogicSeconcl, ,veprove (Vy)(S(r) + y : .9(c * y)). Let p(y) be S(r) + y :5(r + y). The proof of ,p(0) is easy. Assttme cp(y) and prove tp(^9(y)) asf o l l o w s : 5 ( r ) + , 9 ( y ) : . 9 ( 5 ' ( r ) + u ) : , 5 ( , 9 ( z + v ) ) : S ( r + S ( v ) ) . T h u s w e g e L(VvXS(") *y : S(r +V)). Cornpare h.isproof with the proof of 1.5(6): There

we consbrurctecl,y' metamathematical incluction, infinitely many proofs (foreach n, we constructecla proof of .9(r) * n : r - l n *1i" 8); here wehave a single proof in -Ioou,.,f (Vy)(S(") -t -y : S(r f- y)). Clearly the latterformula implies each instance of the folmer scherna. blow let us prove, inI o p " r , ( V r ) ( r - l U : y - F r ) . ' L e t 9 @ ) b e r * y : y f r ; w e s h a l l a p p l yinducLion for tp. We have provecl 0 + y : lJ -l 01 assttme r -l lJ : A -l r andl'eason as follows:

,9(r) u : S(r + a) S(y f *) : a +,S(r) .Thus we haveproved vr)(cp(r) -, v(s(")); by bhe nduction axiom we get(V t)e ( r ) .(2) We prove r+v) *z : * * (y+z)by induct ion n z. F i rsb , r+a)*0 :* - f @ + 0) - n * A i s c l e a r . ssu m ex* A ) - f z : * I ( y - 1 - ) a n dco n s i d e r( r * i l + S ( z ) .W eg e t r + a ) + S(z ) , 9 ( ( r + y ) + z ) : S ( r+ (y + , ) ) - -r j- S(a + z) - , * (y + (S(z)). This completeshe proof of (2). Note thatfrom now on we may write sums ike c t V * z + u without parentheses.(3) First prove0*r : 0 by inductionon n; then prove 9(r)+!J @'ry)*aby induction on y; finally, prove r * y : A ,, r by induction on r. (Let uselaborate on the induction step for the secondproof; assumeS(t) + y :@ * u) -l-y. Then (axiom (Q7)

(inductive assumptionplus associaiivity)(axiom (a5))(commutativity 1))(axiom (a5))(axiom (Q7)plus associativity).)

(4) Prove (r -l u) >r, : (r 'r z) * (v * z) by induction ot z.(5) Prove (" * y) >t : , r,(y * z) by induction on z. Thus prodr,rcts iket: * A 'Fz ,r u (or ny zu) are meaningful.(6 ) Prove r I z - y + z -> r : y by induction on z. The induction step:assume *z : U*z --+r : a ancl *,9( z) -- v-f S(z)' Then S(t+z) : S(y*z)by (Qf) ancl 1-z : y + zbv (Qz); hus : U.(7) Prove


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34 I. Arithmetic as Number Theor.y, Set Theory and Logic1.L4 ernrnu/opu'.,).1) r lv itr (1, < u)@z y7(2)Pr i , r ne (z )t r > T&(Vu< "X" l r * . ' u TVu : r ) ..P roo f .1) Assume ly . I . f y :0 then .o r :0 we havez < ?J rz : y ; soa ssu l r r e- #6 a nd z - y . Then , + A , t h u s z ) 1 andz : z , r 1 ( z ' k r : A .(2) Observe hat f.orx ) T, u lr impliesu 1 r,. n1".15 ernma (/op",r).( 1 )T l * , r l r , " 1 0 , ( r l y a n d y l z ) - > t l z ;I l u - , t l y z ;2 ) (x( 3 ) ( x y k a " ) x : U ,y k x l " ) - , r l @ * r ) ;( 4 ) y - + 0 + ( l l , . r < r ) ( 3 ! u < y ) ( " : y u + u ) ( d i v i s i o n w i t h a r e m a i n d e r ) .Proof. (1)-(3) are easy. lVe prove (a). Let p(u) be th.e fbrmula gu I r.lVe h.aveg(0) and for some u, e.B. for u - r * 1, we have -p(u) (sincea@ * 7) : yx -l A > x *U > r). By /op"r', there is a u such thaL yu 1 rand y(u -F ) > r. Since ( is a linear order and multiplication by a non-zeronumber is monotone, ?, is unique; furthermore u ( r. Put u : x - ytr,; thenr - Au , * u and u < l J (o iherw iseu - - y - l u ' ,Au + y : y (u f 1 ) ( z , acontradiction.) ClearIy, u is unicluely determined. n1.1-6 emma (Iop"").(1) I f . ) 9 then r - a : ( r + z) - (y t r ) ;(2) i f r 2 y then (" - y) * z : (r * z) - y;(3 ) ( , - y ) , - - rz - az ;(4) z l r and z ly i rnp l ies l r - a .Proof. Exercise. n1.17Defin i t ion ndLemma /op"n). is euenif .2 lr ; r is odd f i t is not even.For each , either r ot fr -l-1 is even.Proof 0 is even;assumer + 0.Consider he open ormula2n,1r and clenoteit by p(u). Clearl y, p(O)and for someu',--,q(u) e.g. br u: x). Thus thereis an u such hat 2u, z anc l (u* 1) > , .Then e i the r u : r or 2n: r . ! tand in the latter case2(u + 1) : 2 + 1' n1"18Theoremand Deftnition (pairing, op".). For eachn,y, there s a unicluez such hat 2z: (z + i l@ -l y+ 1) *2r; th is z is clenoLedr ' ,a) 'Fot eadntlrere s a uniclue air u, y such hat z - (r ' ,y).P r o o f .i t h e r l ( r + a ) o r2 l ( z+ v + 1 ) ; h u s l ( ( " r v ) ( "+ a + 1 ) - r 2 r ) .

Therefbre lo r some 2, ,2z : (C I l- il@ -lV -F 1) * 2x; cieariy, this z is unique'Now talre any z ancl using /op",.,, incl an r surch hab r(r -1 -1) I 2,2 and


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1. Basic Developrnenbs;Partial TYuth Definibions Jb(r -F1)(r -F2) > 22. (Consiclerp(r) - r(r - l-1) < 22.) CIeaily,2z r(r -F1)is even; et u be such hat 2r :22 - r(r + 1). lVe ha''re I r (otherwisewer,vould ave2n ) 2r,21 122 - "( r + 1), 12 )- 3r 122 ancl12-F3r is even;t lrtrs r f- 1)(r -F2) : r2 -l 3r -f2 < 22, acontracliction). ut g : r -r; thenx -f U rt 2z: (r - l -y)(" - t y - l -1.) 2x, tJ.rus : (r ,y). t l1.19Definition" Let T be a theory contair:ringIoou,r. forrnultr, (x) is saidl,obe E. (il.) in T if there is a tr'," brm.trla .II,., brmula) /(x) such LhatT'F rp(x)= tp(x).Furthennore,p(x) is saicl o be Z\n in 7 if it is both Dn i n7' ancl ln tnl'.1 .20 xamples. ly, Euen(r) , Pr ime(r) , r : (n,y) areXs in /op". , .1.21 Theolerur"Let ,/ f -fopen.Fbr eachnattrral nurnber n and any fbrmulas9 , t b :(1) i : fg, t f t arc En(f ln,Ar) in ? then soare?krL andgY Lt;(2) if . p s An in 7 then so is -pi(3) if n > 0 and t! is En in ? then so is (!r)r/;(4) i:f n ) 0 and tlt is IIn in 7 then so is (Vr)),r/.Proof. tr'ullyanalogous o that of 0.34. For eachn ancleachchoiceof V,?b,a?'-proof is constructed.)

i.

Now we turn to J'X6. Since IEs and related theories will be investigatedin detail in Chap. V, we prove (or sibate) nly some few basic facts.1.22 Theorern. (1 ) .Itr's proves the leas'tnumber princi,ple f.or -!6 formr-rlas: .e.for each Xg formu)a I Eg proves

( : r ) )qa(r ) ' ( !z) (p( r )& (Vy< ") -v(y) ) .(2 ) For each X6 formula, lXs proves the following order induction:

(vr))((Vy ")v(a) , v@D , (Yr)e@)..Proof, 1) Assume lr)(e(c)&(Vr)(v@) -, ( ly < ")e@)) anclapply in-dr-rctiono the lbrmula (Vy < r)-9(y1 to obtain a contradiction. 2) Applyincltrcbiono the Xs formula (Vy < *)V(y). n1.23Del in i t ion IEo). ' I f . r ,y f 0 then gccl(r ,y) s the maxirnalu such hatu l r and u ly; otherwise cd((r ,g) :0 (greatest ornmon ivisor).Note that gcd(r,y) exisLs ince t is tLre east u { rnin(r.,y) stLch.hat(V, < min(c,y))(r t+ .1. u - ' - ' (u lx &tul t jD.

n


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36 I. Arithmetic as | lumber Theory, Set Theory au d f,ogicL . 2 4 e r n m a f t g ) . 0 < A < r - - + ! t r ( r ) ( 3 u ( r c ) ( t l cd ( r , ? / ) : xn ' - vu ) 'Proof. (0) cleai:ly we may assumey < z. obselve Lhat i.t 0 < I < y ancl-t: xtr- y,r.,hen there areu, ( y and D < r sr-rclr.haL t : rt t - YD. To seet l r isf i rst showu > y:u ) r; thus f u > g then we cart eplace t . ty u-- 'yanclu by u - r. Apply LEo.)(1) Let z be th.e eastnumbersucl: haL, or som.e ,u 1L) ILI > Au3Lz-x:LL-Au.rVe how2; gccl(r , ) . F ' i rsb rove l r .By 1.15(4),et r : zs- l t , ' t " ,z . C l la r ly ,s + 0 s ince < z < x. I f . t +0 the ' f : n - zs : r : - ( *u -yu )s - - - -r(1-F aq ,t) - y(rcr- u) whereg is the eastnumbersuch hat 1+Act > tr anclx lc l> . Then c lear ly - lAq-u 1n, tq -u { r andwegeta con l , rad icb ionwith t l re minimal i ty of. . Thus :0 ancl l r .(2) The pi:oofof. ly is simi lar noie habz ( g sincea: r?J y@ - 1) '(r) Now if r .u lr and tuly then wlz (by t. t3(a)); thus u. ' I z and z --g"i1*,y). Thi.scompleteshe proof' tr1 .25Lern rna . I Is ) gcc l (x ,A) : I -+ (Vz) (z y ' - * t l ' ) 'Proof.Assume gcd(r,u) : I and y ( r' Then, for some t'L,u t ' I : ru-Yu'

N o w i f x l A z t h e n r l r z r ' z y u z r r l ( t " - - y u ) 2 , l z . n1.26Lemrna /16). (1) Foreach ) L, here s a y ( r such hal Pri"me(y)ancl l * . (Z) I f Pr i ,me(r ) nc l la , therr ly or r lz 'proof. (1) folloi,vsby the least number principle (1.22). (2) Assume that rc loesnot c l iv ic le ; s in .ce is pr ime th is means gcd(n,y) :7 . Thr ' rs lz by1.25. L-rj-.2?Remar.lc.The reader may move now to Chap. V where "IXg is investigatedin details. He will fincl there, among other things, a proof of the followingbheorem claiming that in -IX6 exponentiation is tro definable as a possiblypartial function:

There is a tr's forrnula enp(2,r, y) sucb. ha t -IXs proves the follot'ving:( f ) e u p ( z 1 , r , A ) e r T t ( z 2 , r , A ) .> z r : z 2 t( 2 ) e n p ( 7 , r , 0 ) ,( 3 ) e r c p ( z , r , U ) e r p ( z r , * , ' 9 ( a ) ) ,(+ l e rp (z , r ,U ) k tJ ' < y ' -+ 1z t )enp(z 'r , " ! l ' ) .

lvlany results of Chaps. I-IV are inclependent of this theorem; the readermay postpone reacling ts proof. In subsecbion c) of the present secbion,weslrar,lJ.rove a wealcer (and classical) resulb saying that there is a lbrrrt:u1aenpwhich is a1 in ,Ix1 and such that J)-1 proves (1)-(4). This weaker result isbasic for Chaps. I-W.


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1. BasicDevelopments; artial T\.uthDefinitions gT(b) codingFinitesetsancl equences;heTheory .xs (unp)1'28' In this subsectionweshall nvestigatea theory stronger han -IXg a,nclhaving a richer language:we extend the languageby o n* rrnaryfunctionsymbol 2'' for the r-tir pol,verof two. The ex[ncled languug" iu clenoteclLg(ex)p). r(exp) formulas esulbrom abomicormulas f I( ,"p) bv iterateclapplication of logical conneciives nc lbor-rncleclluantifiers f ttr.e'forr11V"


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i8 I. Arithmet,ic as lrh-rmberTheory, Set Theory and Logic

ro lcls f .org lkrpZ,?;pkgZo,t t- f t2. AssLrmehab e have9(x,tL), tps(x,u.,y)rncl (x,,u) satisfying he analogue f (*) ancl nvestigaLe /(x) being (V, r.(2) fo l lows rom (1). consider(3): I f A +0 then we firsb claim that there is a largest r < y such that2t < y; then ib follows easily hat r y and z is the largest elementof y'Having (1* < D@ e a) 'wuget a least elementof y by the least numberpr-incif,le. hus-l"i ,r , prove the claim. Let z be the least number such bhatiV, a A)Q" 1.U n z u e A)(Note bhatby t.32, , ' I y is ecluivalenbo (Vu))(u c --+ r y)')


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1. Ba.sicDevelopments: Pariial Tbuth Definitious. l -.3,.[emrna (.?'Xs( rp)).(1) If u


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I. Alithmetic as Number Theory,Se t Theory and Logic

Nol,e hat, by I3a(2), u (S r) iff u 12. We furiher malce he followingd"efinition: is a res'triction f. to z (in symbols: lestrict(y,t,z)^)^if ?J 2'ancl (Vu < z)(u, r : .ttr y). Note that this notion is E[*p (enp) infto(er.p)ancl hat the latter theoryproves he following: f Restrict(U,r,z)und z ) 2:u hen (Vu)(u e n : u' A) .1.SG Theorem (/to ( erp -comprehension).For each I3" (ezp)-formula,p(u,p), Eo(ezp)pro.res

( V r ) ( l y< 2 * ) ( Y u< r ) ( u e y = v @ , P ) ) .Proof.S/e apply induction on r to the formula

( ' r) (1u


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1. BasicDevelopmenbs; artial T\.uth Definitions 4I1.39 Theorem (IDg(enp)). Fot any r)y, there exist unicluelydeterminednumbers rU?J, r )U , r -? / , r x l t , [ Jo ,P( r ) hav ing he o l low ing roperb ies :(r-Lnion)(inLersecLion)(sebd.iffer:ence)(Ca,r 'LesianroducL)(sLrmset)(power set)

(dornain)(range)

( W ) ( u r U y : . L t x V u e y )(Vu)(ue r f- l : .u ,e r ku, y)(Vu) (u r \ y : .n Q t :k" 4 y)( V u ) ( u x x y : . ( 3 u r ) ( l u e y ) ( u : ( r , r ) )(Vu)(uLJ, = . (3u r ) (ue r) )(Vu)(u P(r ) : .u,e x)

Proof. We always find a z such that ihe set in question s a E["p (r"p)-definablesubsetof seg(z).Talce he union and let x I A.Then seg@) eseg(U) and we pat z : y; there is a ,r.usuch that (Vz)(u u) : u seg(z)k (l' e r Y u y)) (by comprehension). hus u is c U gr .Similarlyfo r r AUrr - y ,Ur . For r x y obser . reha t u 1 rku < y imp l ies u , r )


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42 1. Aribl"rmetic as l\Lrrnber Theory, Se bTheoly ancl Logic

(S ) y is amc ty4 t i ng f r i n l ,oz i t .Re l ( y ) , t l om(y ) : t , r ng ( ! l ) C z anc l( V u r ) ( V u ,u e z ) ( ( t , u ) e v k ( r t , ' t u ) e v ' - ) u : t o ) '

A mapping is an in'j ection (or: is one-one) if( V T 1 1 ,2 e n ) ( Y ue z ) ( ( u 1 , ? )e v k ( u 2 , u ) e t l - > u L : n z ) ,

y is a b i ject ion of r on l ,oz i f ' add iL ional ly , ng(y) :7 '(4 ) For each r, Lhe no"htralorderi,ngof r is the unicg-reinear order y on

z s i rch hal , fo i : eac l r . ,u t : , (n ,u) e A = u 1u. (Si iow the e:c isbencef yusing comprehension.)1.41 Theorern (/xs (rrp)). (1) (cardinai i ty.) For each r, there is a u.nicluea : ccrriL(z) such th.at there is a bijection of z to (< y) '(2) (Pigeon-hole principle for fini|:e sets.) If. card(x) < card(y) th.en thereis no injection of Y ini;o r.(B) Ii y is a linear orcleron n f 0 then r has a largest and a least elementwith respect' t'o y..proof one only checks hat the usual proofs formalize in lxs (ex'p^).

(1) The clesirecl i jecbion , i t i t exists, satisfiesJ' S Z@-rv+I) ' ; bhus wemay prove the following by Efrr'r erp)-inducbion on r:

( r' ) ( l ly < rXU < 2Qt-rr) 'X/ i t a b i jecbion f r to (< v)) .(Clreck hat this is f)["P(r"P).)For n ==0 we get i:0; asstlme 'r,) or each 1 z ancl nvestigatez' LeIu : max(z) and r : z -2 t ' , then r I z , u r : .u zBzu 'u anc l hereis a y uatisiying(,r). Let / be the correspondingmapping; we extend / to abijeciion s o,f onio (< y -F1) by definineg@) = /(u) for u :t, g(u): a'ti." *npping (relation,set) exists hanks to D["p (erp)-comprehension''Io proveboth the uniqr-renessf cardinality and the pigeon-hole rinciple,we slrow he fbllowing by E["e (et:p)-induction:

-,(:, f < 2Qu*r)'X/ t" an injectionof (< " -l-1.) nto (< ")) .Tlris is clear for r :0 . Assume ,r')ancl et / be an injectionof (< x 12) into(< " * 1) . lve may alsoassumebhat (r + 1) - r (if no i we change fbr

thearg.lrrl"ntsc -F1 ancl -1(") using comprehension). ut then the restricLionot j ' to (< ,*1) is anin ject ion f (< z* 1) nto (< c) , a conlradict ion ' hiscompleteshe proof of the indLrction tep'

Ctnu"c1.r"nLly,f there is a bijectio' of (< ") onto (< y) then r - y'Thus each finite sebhas its unicluecalclinality;and we get the pigeon-hoie


18/84

1. Basic Devclopmenls; Paltial TluLh DefiniLions

principie Llsing composiiion of mappings. The fact that the composibion.ofbwo mappings thab are finite sets is a,mapping (and a .frniteseb) .tbllor,vs ycomprellensior.r.

(3) ffy in.duLction n r;; nobe Llr.aLb.[re -rniversa,l luantifi.er (Vy) ma'r bebournclecl y 2Qmax(c)l- l) ' so that )- ' [rn(erp)-incluction applies. Assr-rme heasserbion or ali z 4 x and consider z, ; et'u be bhemaximal element of z withrespecb o the orcleri .ng ancl eL u n : .u e z k, # u. Then ir < z arrclit y' -- y a (r x r) we see tl.rai:y' i: ; a linear orcler on x. By the incluc{,ionassumpt ion, has amaximal e lement 'u / 'w ibhrespecboA' . Then e ibherz oru/ is m.a;rimal in z wrt;hrespect to y.i..42 Definition (/Xo(exp)). Now we finaliy come to or-rrdefinition of finibesequences; hey are natrrrally clefineclas particular mappings.

S 'ec1Q) : (1 " < , ) ( , i s a mapp ing& dorn(z ) : (< " ) ) ;S e q ( z ) + . l h ( z ) : r i f f c l o m ( z ) : ( < " ) ;SeqQ) ku, 1 lh(z) . - - (z)" : u i f f ( t r , ,u)e z ;SeqQ) 8e ) Ih(z) . -> (z) , , : 0 ;

S e q Q ) - - + , 1 . 0 : - \ ( r ) i f f ( V , < u ) Q L , . t u : . u Q z V ' u : ( l h ( z ) , r ) ) ;f o r . . .Sec l ( z ) ,s ^ ( r l : 0 ;

0 is t lre empty sec{uence;h(A) : g;( t ) - 0 . - - t ) ; ( r t , . . . t r n t r n - F r ) ( " r . ' . , r n ) ^ ( r , r + t )

( n : 1 ,2 , . . )|ToLebhaball nobionsdefi.ned rc E["p(erp) in /to(enp).1.43 Theorem. ,[Xg(r*p) proves tlr.ebasic properties of sequences as formu-la tec l n 0.4L) :For Seg(s) ,Seq(st) ,

n

r1 )(2)(3 )( l\(5)

lh(s) ( s and (Vu < /h(s)X(s). , s) iS e c J ( O ) & / h ( 0 ) : 0 ;

( V " < / h ( s ) X ( s ) , : ( s ^ \ r ) ) " & ( u^

\ r ) ) r r , 1 , ; z kk l h ' ( s ^ ( r ) ) : / h ( s ) + 1 ;i t Lh (s) th (s t )and (Vr < /h (s) ) ( (s ) , , (u ' ) " ) then s ( s /

(Vu)(l 'u ) u")-S's6.t1r7(Consecluenbly,f. s,sthave the salrre ength and the same colresponclinge l e m e n b s b h e n s - t ' . )Proof. (2) is trivial: 0 is the ernpty mapping. Al l rnonol,onicities f (.|.)zrnd(rtr) ollow from the monotonicityof the orclereclzrir '(e-, 3 (u,y)) anclof


19/84

4, 1 I. Arithrnetic as l"lumberTheory,Set Theory anclLogicmembership (r A -->x: < y). Indeed, i t s l0,$eq(s) and clo'nz(s) (


20/84

t. BasicDevelopmenis;artialT\'Lrth efinil,ions .15Proof. lVe proceed in /X1. AssLrme Vr < u)(1y)rp(*,y) and prove.,Ly L'tinclucbionon to) the follo.ring:('r') u; { u + (:u)(Vr -Lt,la) hV tr'1 induction; thee,sistenceof a leasb sr-rchg follows by the least number principle for I'gtbrmulas.For r:0 the asserbions vacuous.Assurne Vtr < "X0 < rt, --+ trly) ancltalcey' Lt , (c f- 1); then (Vu ( r -l-1)(" > 0 ---+ lE). n1.47Definition. eo (a,z) if f (t + (t l- ?) 'r. ) | u.1.48 Lernrna (Cornpi:ehension)" or each Xs fblmula gQt ) IDl prorres hefoilowing: (Vr)( ly , ) (Yu, *)(u eo (y, ) : tp(n))(rp^uy conLain i 'ee.vtr,r ' iablesistinct from LL)y1z s paramebers.)Proof. We may assurne ) I.'Let z *- h'ull(x). lVe clairn that lbr tt l ts -- xttrenLrmber:s[ l- -L-lu)2,1 -l-(1 -lu)z are relatively rrime, .e. iheir gleatest


21/84

46 I. Alibhmetic as lrlumberTheoly, Set Theorv and Logic

comlnon clivisor s 1. ('Ihis claim is proved below.) Using this r,.re hor,vhefollowingby trr induction on f:('r.

(t < * --' (ly)(V, < r)l(u 1 t 8z @) --, tr, o (t1,r))k ('" > t v -r("))) -, sccl(y, -F 1 )- rt);:) 1)l

For t :0 ta i re?J:7. . A.ssu.me'r ) anc lconsic ter - i -1 . Let y be as in (* ' ) .c nse l : -.,p(t). .I'.[len(,r.)holcls or f repLzrcecly t -l - 1. nd fo r 3ras it s .ands.Ca,ce : ,pU) Pui yt : v 't (t -f- t + f )) ' r '2. Then clearly t eo (.r l ',2) anclt - t t k rp (u ) imp l iesn Qo ( 'U tz ) . I f u > ' t y - ,p (z t ) thengcd (y . , I - l ( I { - u )z ) : ' Iand,gccL(y, -F (1.+t)z) - 1. Moreover, by ou.r claim, gcd(I -F (1 + tr ')2,'L{( ' t+t)z),=' -| ..Now if c diviclesboth 1.+ (,11-u)z an.dyt : Lt t ' t -t- t -f t),)

t hen , by '1 .25 ,c l y o r c l 1 - l - ( 1 .+ t ) z ; thus c : 1 anc l ve a re d-one .L remainsto prove the claim. n

(Proof of th.ecLairn.)Assume u 1 u ( r; then 0 < u - t-t r ancl 'u- ulz.

wribe u7,u..t"or 1*u,1+u and iet c be sr-rch hat c | 1-l-T.r1, cl ' I{u12. Then forsomec t . ,b r , vehave-u1z - ac , I1 ' u . t z b r , ( I+u tz ) l c tbc : c t ( Iau1z ) . S incetrivially (' I -f u1z) | a( I * rr,1z) e get (by subtracting) (1.+ tt'1 ) | o,(u1 - zt z) :cr (u- . , ) r . t r , r ident ly , cd( I l t r ,1z, t ) - - \ thus, by 1 .25, ' I *u tz)1"( r -u) lc tz .By the same reasoning, (1 ,- tr,1z)la,which togeLherwith 1 ltt lz: crcgive1 . r t l z : c l a n d c : 1 .

-Remurk. ote that Lh.eormttlau eo (A,z) is Xg in ID1.1.4g Def in i t ion ( IEt) . (1) (y , r ) o-codesa seq,rencef length r i f for eachn 1. t: there is a u ( g such. haL (u,u) eo (y,r). If ihis is the case then1y,z) , is the least u ( y such hat u,r ) eo (y, , r ) .' -

fd l @,") is an eaponent i t t l equ 'encef length r (Enseq(Y,z,o)) f (y , ' )o - coc les seq r l ence f l eng th r , r ) ! , (A ,z )g :1 and , fo r eachu 1: t -7 ,(y , z) t t - r l : 2 * ( t l , , ) " .( 3 ) e rp ( t : , u ) i t ( 1y ,2 ) (E rseq ( IJ , z , r 1 ) & ( t l , t ) * - u ) '1 .50Theorem. (1) I E1 | (Vr) (3 !u)(erp(r ' ,u) ) '(2) l t 'h.e ormttla exp(r,u) is d1 in tE1'

(S) ff we clefi.ne n IL'1 the function 2n hy erp(r,u) then al l axioms ofIEo(erp) arc provable in ID1.ProoJ.1) F i rs t showin - I t r '1 hab f Etseq(y,2 ,rc) , Erseq( 'y tz ' , / , ) andrc r t

then, for'each u < r" (y,,2)tr, (y' ,z/),, (uniclueness).This is proved byinduction on ?.i,he formulain cluestionbeing f6 in -IIr. Similarly we provet l -n t Et :seQ(U,z,c) nd u 1n impl - iesL1( !J ,z) t1 ; furLhermore) tseq(y,zr t )and u ( u ( r impl ies (y,z) t t 1 ( 'A,r ) r .

Then prove the fbllowing by tr'1 inclLtctionon r:r ) - ! * ( l y , z ) E r s e q ( y , z , r ) .

rl


22/84

1. Basic Developments; Partial Truth Definit,iot'is 4'l( ir lote th.at the formula ()y,z)Enseg(Lt,.z,r) is X1 in IEl by conbraction ofc i l rant i f ie rs . ) he asser t ion s ev ident or r : 1 . Assume r ) 7 , . [ lnseq(y,z , r )and (y, z) r - . t : q; put q ' : (x ,2q) .By 1.46, Lhereare yt ,z t s t tch ,h .a t

( V u< q ' ' ) l u o y ' r ' ) = . ( , o y , t ) & ( l i < r ) ( ! u < u ) ( u ( i , u ) )y u =, *,2r0).WegeL trx.se.q(yt, t ,:r 1).(2) Clearly, rp(r,u) is 11 asdefinecl;ut sincewe have 'I), "-"rp(r, u) iseqr,rivalenLn IEl to (ftl;)(* l, kexp(rc,tr.')).hus he result ollows.(3) We proceed n. I . l1 . Clear ly ,20:1. ' I f .zE : u and 2.0+1 tr . r ,henweh a v eE n s e g ( y , z , r F t ) , E x s e q ( y tz t , r 1 - 2 ) , A , " ) * : u , ( U ' , z t ) r + r : t u ; b u t , ,by the claim in the proof of (1), (It '," ')* : u and thtts tr : 2u. Thus theaxioms 'f.or2t are provabie. It remains to prove that induction fbr Xs ("*p)formttlas is provable. But this follows from the fact that each Ds(erp) formulais /\1 in .tD1. Let us prove this faci.To prove that atomic Xg( exp) f.ornulas are A1 it suffices to show bhat foreach. erm t of Lg("rp) and each variabie r not occuring in /', the formula' t: t is 41 rn.t81. This is clear or f atomic and the induction step fbr,9,*and * is easy. For example, : ' r+s is ec l t i i ' va lento lu ,u < *)(u : ' tku:skx - LL u) ; i f z t : ' t and u : s are 7 l1 i1 f I1 then so is z : f * s by1.45.) Consider r : 2t and let the fo:rmula u : t be A1. IL is suffi.cient toremember that u :2u is d1 in IE: (see (2) above); r:2t is ecFrivalent o(1 , < x ) (n = 2 t tku - t ) .

Ibr the rest of the proof it sufHces o recall that fbrmulas /1 in -IX1 areclosed under connecbi'ves nd bounded quantifi"ers. n,li

1.51 Discussion and Definition. (1) Recall that we have I' formulas and IInformulasl these are particular formulas of the language of arithmetic. Then wehave tr',, and IIn sets of natural numbersl i.e. sel,sdefi.ned (in the standardmodel l/ by X' formulas and I/r, formulas respeciively). A' sets are sebstlrat are both \)n and I1'" (c,f. Sect. 0). In 1.19 we defined a forrnula to beEn (I f n, A") in a theory 7'. Now we turn our atbention to sets of naturalnumbers defined by such a tbrmula. Instead of saying that a set X is defi.nedby a forrnula that is 8,, in 7 we say.that X is ?-provably X' , (similarly forIf n, An). This general izes o X g l/o ,k : 2,3, . . ..Clearly, if ? is sound, i.e. // is a model of 7' ancl X is T'-provably E" (Anetc.) then X is Xn (etc.). The converse eed not be trtre, cf. CJ.rap.V, Secb.3.(2) A formr-rla p(r,A) rlefinesa t,o'tulu,nction n T i I7 F (Vr)(1ly)tp(r,y).We may then extencl ? by defining a nelv fr-rncbion syrnbol 'F and the axiomtp(x,F'(r)). We m.ay againcall the resulting theory ? but care is necess&r'ywhen clealing with hielarchies of forrnulas) e.g. r,vedisbinguish tr'g-fbrmuias


23/84

4B L AtiLhmetic as Slurnber Theory, Set Theory an d Logic

zr.ncl.0(er.p)-.[ormu[as..l .f he brrnr-rla cteftn.ing i ' in 7 is D"-\n ?'l .hen wesay th.at I' is In tn T', et;c.

c.l.ear:ly,f. ,p(x,y) clefinesz-r,ol.al uncl,jon in 7 a,nc[T' is sound bhen ,pclefi.nes total. funcbion. n M. A fr-rncbioir : il/ --> IY is T-'prouably total if -iL lras a ctefinit ion,p(r,y) which clefin-es botzr.l .utncbion n 1-'.The frrnctionf is T .proaubly En (eLc.) f i1 ;has a cLefi.rlitionwh.ich s E, in . 7. (This is alrar:l , icrr lar ase ol. ( i).) The furncbion is a 7.'-p'rouabLyotul L'n fttnction tliL lras 3,clefir1iLion,pwl-1ic.hs En in 7'ancl clefinesa l,otai function in 7' InpalLicttlar i,vecall f T'-prouo,bLyecursi,ue f. t is ?i-provably total )li; sincewe slr.allo.fben. e inl,erestectn.[tr'1-provarbly recr-irsive unction.s we shaLicalithem ius| pro'uably ecursiae.1"52 Lernrna. A.ssr-rmef ) /op"r'. (1) If F is a function sy'mirol X1 in if tlr-enF i s A l i n T .

(2)If.$ is A-1in.? and.F'is afr-rnctionsyrnbol ,41 in 7 then. he fbrmula,()" : : r(a))rb is A1 rn T'.Proo.f. f ) I f ,p@,A) is a,sabo're hen the formtt)a ()yt * y)'p(*,a') - i" X1 in7 ancl defines the complement o, fF; thr-rsts negation is .tr1 and defines -F.

(2) Let $t be (rr S r@Dv. cleariy, t l :t is ' l1(by contraction of guanti-fiers). Br-rt un T', rh'is al.soecltLilralento (Vz)( " : f (A) -t (V"


24/84

i. Ba.sicDeveloprnenbs;artia,l r!rrbhDeftnil,ions ,[9Tlrtrs .f tbe lhncl;ionscle{-inecly p.,$,y ate clcnotecl y F,ti,.H respecLrvel-yLheir l - t r(r ,0) : G(x) ancl? F F(:r , - l - '1 .) :Jy'(r , , f (x,z)) .Prooi. X jusl; clesci:ibes.hecotLrseof 'values(Ersecl rr/aszr , ar.tictLl.ar i,rse):

y ( u , z , y ) - , ( ! s ) ( , 5 ' e q ( s ) k l h ( s ) : . . t & ( s ) s : r - J ( ; r ) 8 z ( , r ) , ,y B t(V i < - r ) ( (n ) ; r t ,= FJ x , , ( r ) ; ) ) .' . l ' ranscribingthis'wiLh. the help of g,$ is Lrivrzi l . uL l , i resom.e;Lear:ly,1J1 in T. .t.\t.was usec] o prove (v:r.,2)(],y)x(*, z,tJ); un.icluenesss ezr.syprove ancl a peclanbic elaborai;ion o[ c]etails of the proof of .tr(x, z -F'1.\H(* . ,2 , . [ r ( : r , .2) ) s le l l to the.reacler .

1.56 Remar:ir. Tire letntnasays (in contraclistin.cbion o 1.54) thab insicle E.1we may defi.ne total /1 functions from. olh.er d1 frrncLio:nsby primitiverecu.rsion.. ote that ll:.i,'; eneral.izes asiiy to 1:rirnil,ive ecursion on L,he otrrseof valr-res, :f.0.


25/84

I. Arithrnebic as Number'll 'heory, Set' Theory ancl Logic

Proof (1) followsclireclly rom 1.55.To prove (2) work in I D1 ancl ake anyx) ;wes l row hab here s apr ime p> r .L ,e l z : l ru l l (x ) , .e . (Vu < r ) ( rL lz )anclbalce". l - ' I .By .26, bheres apl(r)- 1), but p is dist inct rom al l rz a.(This is the classicEuclid'sproof.) For ezrch , let Ip(*) be the least primenumber greater han r1 by what we have ust provecl, p s a toLal \1 function.Thus the fLrnctionP o 2 t

Px*\ : lP(P*)is 41 and total - both provably in I D1. This is ihe desired increasingenumeration of al l primes.

(3) A. secluence is a prime decomqtositionf all members of .s are primesancl the sequence s non-decreasing, .e. (s); < (r)lr-r for al l i < lh(s) - I .We claim that for each r ) 0 there is a uniclue prim.e decomposition s suchthat LIs : r. Exisbence s provecl by induction: bhe prime decompositionof 1 is the empty sequence0. Let r ) "I ancl asstrm" (Vy < *)(y >' 0 -+y has a prime clecomposition.).'Let,p be the largest prime dividing r (it existsby the least number principle fbr Xs formuias) and take the y such thatr : p , ry (d iv ide rby p) . Now U 4 rc,so le t s be a pr ime decomposi t ionof y.Then u ^ (p) is a prime decomposition of r. n1.59 Rernark. (1) Prove the unicluenessof the prime decomposition of r asan exercise.

(2) lvlany theorems of elementary number theory formalize easily in I E1together with their proofsl fbr example, the proof of. Bertrand's postulate(saying thaL for eachr ) 0 thereis a prime number p such that u < p


26/84

1. BasicDevelopments; artial T\'uih Definit ions b1develop them furLher; we are going to show Lhat logical notions (lihe formu-las, berms etc.) are d1 in /X1 and that "IX1 proves their basic properties. Weshall detail careful forrnr-rlations; roofs consisbmore or less n checking thatinformal proofs presented in Sect. 0 can be read as proofs in IE1. Our gainwill be twofbid: We shali see that some reasonableparts of logic fbrmatze inI D 1 a'nd secondly, we shal] be able to e,rpand exp::essivepossibilities of -[X1by introclucing variables f.or En (nn,, An) seLsof numbers. This will ire verytrseful.1.60 Ttreorem.Let T ) IEL,Iet A'tn,Op",,Arnbe formulasd1 in 7 andassurne hat ? proves Lta,Op' to be clisjoint, f" non-empty and Ar" toc1efine Lotal unction, i.e.

"F (Vr) (Op' ( r ) > (1!y)Ar" (* ,a) ) . , wr i te : Ar ' ( r ) for .z l r ' ( r , i l )

?' l- (3r)( At" r) k (Vy)(At" a) -, .-(Sect'@)& Op'((y)o))cf . (0.50)Then there are fbrrnulas Expr' , AppI" that are d1 in 7 and such bhab 7proves (.Erpr', Appl ') to be a free algebra of type (Op" ,Ar') generated byA t ' , i . e .

T I A t " ( r ) - -+ npr " ( ( r ) ) ,T | .App l ' (o , ,U)

: (Op ' (o ) ,Seq" (s)lh (s ) : A r ' (o ) ,a : b lT t App l " o , ,a )& (V i < Ih (s) ) (Ezpr " ( ( (s ) ; ) - 'n C o n c s e q ( r ) )E r p r " ( y ) ,

and for each Xl formula g(z) (possibly with parameters),7 F (Vc)(At ' (r) -- e@\ k& ((Vo, ,y)(Appln(o,, y) 8z Vi < lh(s))g((")r) - , ,p(a)) ,- , (Vr)(Expr '(r) - ' ,p(r))

'llhus atomic expressions re expressions; pplying an operation to a se-quenceof e,rpressions f the appropriate length givesalr expressions; achnon-atomic expression rnicluelycleterminests componentsland Expr is theleast )'1 set containingal l atomic expressions rrd closecl nder applicationof ooeraLions.Co'naention,We shall idenbifyatornic e,rpressionse) with atoms r if thereis no danger of misunclerstancling.This comespondso th.eusual conventionof omitting superfluor-rsraclceLs.


27/84

52 L Arilhrneticas rlumber heory,Se tThcoryanclLogicProof. Define ApytLu(o,s) (o) - ' Concsea(s); ,ve iefine u to be a clerivationfbcmalizing bhe clefiniLion n 0.5.1;defi.ne

ExTtr' (u ) = (:qxq is a clerivatioir and s is il;s la-r.stlernenL)The rest of tl:.eproof consisbs r.i ch.eclcing;he proof of 0.51. t-l

lrlol,r vre cou.Lcldefine a At presentation of berms e'r.nd ormulas of airarbitrary language; nsteaci,we restrict ourse.i .veso the languageof ari th.metic.leaving ,hegeneral case bo th-e reader as an exercise.1-"6.1 heorern. In IE1 vre can defi ire consLantsSt, - l-t, '1.o, o, (t, 0n, -",* ' , V', A1 precl icaLes ar', Terrnt, A'tJormu, Fornf an.d 41 functionsAppl'term", Applforrnn such. hat basic properLies of terms a.n.clormttlas areprovable. ly'lore preciselv, "Itr'1 proves the fbllowing:

(L) (Term", Applterrn") is a free algebra over variablesn ancl the consLani;0' as atorns wibh tire operations 5'' (r-rnary), Fo , r. ' (binerr:y);(2 ) An aLomic formula" consists of :" or (o l,ogetherwith two term.s:.t l tform" (t:) : ( lu, t < r)(Term" (s) k1' 'erm" (t)

k . x : ( : t , u , . L ) r : ( < n , " , f ) ) .(3) (Form',A.7tplform,o) is a f-r:ee lgeb::aover atornic forinulaso as aboms

vrith the fbliowing operations: -" (unary), --+' (binary), and.each variable".(4) There are infiniteiy many variableso:

(Vr)( ly 2 r)Varo v))(5) Terms' are disjoint from formulas":

(Vr) - . (? 'e m'(x) k From" (x)) .In a still more transparent vray' his may be formulateil as follows:Write

r - l * l t insLead , f .lpp l ter -u(-F" , (* ,U)) ,r , tn A insbeacl , f .lpp l ter rn ' ( * ,o , ( r ,y) ) ,S ' ( " ) instead of Appl ternz"(5" , t ) ) ,t :" s insteaclof (:t,-f , s) ,I 1 : " s insLeac l f (< ' , ' t , s ) ,-u( r ) insteado, fApTtLfor-u(- " , ( t ) ) ,

r - r * y ins l ,eac l f Appl forn l 'u ( -+"(* ,y) ) ,(V'u)z instead o,f. I 'ppfform'(u,(*,y)) where Vctr"(tt)

'l 'hen /I1 ploves 'Lhetbl.lowing:


28/84

[. Basic Developments; Partial Truth Definitions 53'k bhere are infinitely many rariableso,'l ' each variablet is a termn,'k i f t ,s are berms"bhen -Ft s, ' t ' { ,e .s ,5 ' " ( f ) re termsu,'r 'atomic formnlaso have the form t:t s or'/


29/84

54 I. Arithmeiic as Number Theory, Set Theory an d Logic

(2) A completely anal.ogous ,heoremon the construcbion of a Z\1 tunctionby inclucbion on formulas is now e'viclent; or example, tve may define the sebof all its free variables' of a formnla' as follot'vs:

o / e \- l lTceaurt -L : u) :.Freeuar"(zr -" u) :Freeuaru - ' r ) -

tr't'eeuar"r -r t y) :Freeaar'((V'r.o)r) :

Var-of (u)tt Var.-of(r) ,sirnilarly,

Freeuar"n) ,,Irreeuur'r) u Freeuar"(y)Freeuarnz) \ {u} .

Furthermore, n 1'81we may clefineotal d1 functions Suhstu substitu-tion), Val" (evaluationof terms) such Lhat IX1 proves he following:(3) (Substitution nto terms.)Subst ' (r , , t ) : t i f V ar '( t ) k1 'erm'(t) ,

Su,bst ' ( t1 t tz,n, t) : Subs' t ' '' t , n, f ) - l - ' Subst ' ( t2, , t ) ,itt1,t2,f are berrnsu nd r is a variable"; imilzr,rlyor,F',,9'.(a ) (Substitution nto formulas.)

,9nbst '\ - ' t2 , , t ) - (Subst "t1 , , t ) :n Subst ' t '2 , , t ) ) ,Subst ' -" z x ,t ) -n Subst"z r , t ) ,

Subst "21n ' 22 , , t ) : (Su"bst "q , , ) - " Subst "'2 , ' , t ) ) ,Subst ' ( (Y 'u ) r ,n , ' t ) (V 'u , )z f t t : x ,Srubsl"((V't))2,, ' t ) - (V'r) Subst ', , ' , t ) i l u I r '

(5 ) (Value of a Lerm.) If f is a terrn' and. is an evaluation of its variables",i.e. a finite mapping whose domain consists of some variables'' among themal l variables' of J then:

Vul"(t,z) : z(t) i f t is a variable' ,Va l " ( t1+o ' t z , z ) : Va l " ( t t , " ) * Va l ' ( t z ' , " )

similarly for 'l'o,56 and. r,,9.1.6b ltemark, We have constructed a clefinition of secluences)erms, formulas,etc. Lhat an:e lin IE l ancl such Lh.atbasic properties (in particular, closureproperties) are provable in /L-1. or-rr definitions clefine he corresponding,]otior,, (r"tr, lLnctions, etc.) in //; fbr example, p \s a formtLla tr A/ FForm"fg] (this was cliscussecln Secl.0) . Br-rtsinceour definitionsare d1 inID1 we can use x1-completeness f. El (see1.9) to get the lbllowing:(i) For each ff,f is a berm ff N F Term'(t) i f f . l 'D1l- ' I 'erm"(T),


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56 I. r\rithmetic as Number Theory, Set Theory ancl Logic

1.69Lemma.There are ormulasXi(r) ,nl@) with two freevariables read:r is a Xf formula', similarly or //) such hat (1.)boLh Xfi(r) and ,lfi(r) ar:ed1 in IE1, and (2) IDr proves he fbllowing:(i) fory - 0,r3(r) = rli@);(ii) nil-rt(r) it r there s a variable"'ua - (Y'u)z;(ii i) similarly or Ei,*,Proof Exercise.

r ancl " D; formula z such b.hat

We are now ready for a definition of szr.tisf'actionor Xg1.?0 Theorem. There is a formula Sats(z,e) which is 41that -IX1 proves Tarski's sabisfaction conditions (cf. 0.6)i.e. I El proves the following:(i) Sats(z,e) * zisa Xf formula' and e is an evaluation'fot: z' ,( i i ) i f z is X i and " : (u : ' u) then

S ats(2, ) : VaI" u , " ) : VaIo r , " )and similarly for z - (u , (' ,) ;

( i i i ) i f z is Di and. : ( - "u) then Safs(r , " ) i f f -Sato(u," ) and simi lar ly forz : (u - t , ) ) ;

(iv) if z'is Ei iia , - ((V'ru1evaluationof u coincicling',vithe on Freeaa," ') \ {tr } and such that, '(*i is clefineclncl" '(*) l et(u2) we have Suts(u,,et)'The proof s in 1.7'I-I.73.

1.?LDefinition (/tr1). (1) g is a partial satisfuctionor xfi formulas' ( p andtheir evaluations'by numbers< r ( in symbols:Psat6(r1,p,r')) f q is a finitemapping whose domain consistsof al l pairs (t, ") '',vherez is Dfi, z S P, eis u.nevaluat ion ' for 2,,e C (S p) x (S r) ,rct 'nge(q) {0,1} and Tarski 'sconclitionsholcl or q whenever hose things in cluestionare clefinecl,.e . foreaclr (z,e) e clorn((t),( i i ) i i r : ( r : ' , j t t " t t q ( z ,e ) : T i f f Va l " (u ,e ) : Va l " ( ' , " ) , s im i i a r i y o r, : ( u ( ' r ) ;( i i i ) i f , ) 1 - , ' u1 ' then (2 , " ) :1 i f f q ( r ' , e ) :0 , s im i l a r l y o r z : (u * ' ' ) ;i i " ; i f s : ' ( (Y " i u r < " .2 ) , r ) then c1 ( .2 ,e ) :T i f f f o r each e / q (< p) x (5 q)as n ( i v ) abovewe haveq ( ( " , " ' ) :T .

(Note tltai e is assumecl o be clefi.necl nly fo r (some) variableso y suchthaL g ( p and evaluates them by numbers ( r. )

(2 i S" i1( r , " ) i f f there ^re q ip , r such tha l PSato(c l ,p , r ) and c1(z,e)1'

nformulas.in IE1 and such

tb r X6 formulaso,


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1. Ba.sicDevelopme.Ls; pa't ial Tr'*th Defirriiicrr,,; :).1I.72 Lernma" 1) PSats s d1 in /tr'1.(2) Ih proves that if QJ..Q2re prr,rtiaisatisfactions bl I'fr bhe' l,lrr:ycoincidezr,the intersectionof their dornain.s.(3) .ftt proves thal; fo l ezrch. , r, bhere s a rJ sr

an.d also recal l 1.52(2). l loiroof. (1) Recall that d1 includes J["?(er7t)example,dom(cl) s characi;erizeds follows:(Vr dom(q))(12, e 1. r)(r : (t, ") & e i s an evaluizr,Lion'ot o r k z i p k , e ! ' r ) k

( ' lz, e < II(p,r))(r is an evalu.al ionu or z 8r,3 t , z p k e 1 r . - > ( r , " ) e d o m ( q ) )

r ,v l rere (p,r ) is a termmaior iz ing al l sr - tchr ," ) ; ta lcee.g. (p- r r+:02. Therest is lefL as an exe.rcise.(2) In ID1 we prove tire formLrlaPSats(r f i p , r r ) U PSats(q2, 2, 12) >- -+(Ve < q lXbo ih q t ( r ,e ) and c l z ( z ,e ) e f ined+ Qt ( r , " ) : qz (z ,e ) )

by incluction on z (in the form 1.a5(3)): f cn(z,e) and q2(z,e) are defi iredandz is atonric bhen the conclu.sion ollows by the definition o{ .PSatglit z is -,'uare u -->nu then by the incluctiveassumption,c11(u,,e)nd ez(u,e) are definedand eclua l and bhesal r ]e foru) ; thus q1(" , " ) : Qz(z,e) . f z is (V" , r .u1" u;2)u,and q;(z,e) is defined then e(zo2) s clefined and e(tr.,2) r; tl:.us if e/ c (< < r). 'Ihereforee .maybe defined or u but the valtLes irrelevant br q'(t.,") since'u cannol,occu.rn z. Thus pr-LL'(r ,")" : q(z,e |) wheree.L rneans he resbrict ion fe to argurne:rtscliffer:entrom u. $how tLsingcomprehension ;hat g/ exisbs;PSatg(Q'u,r) is e ' ' i ic1ent.If ti is " tr'6 forrnula Lhen v,rehatre ,o cliscLrssevere-rlases;buLt obe ,hal,no\,v?., s uot a varizr,bieoo .r/ehave only to in.restigateonly everlr-rabions


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53e gfor

(((Vr ) ( r e a ,n e : t D,nc : r 11 , , , l )b-urthermore,ut,n cis En anclr e l,n d is IIn in" E1.1".8?rterrrarh. eL Lrs u,nLrna,rizehat we haveclone n the plesenl,section.Wefirst introdtLceclRobinson'srithmeticQ and pro''redt to be Ii1-complete.Wefurrihel nL.roclucedop"n and proved n it somehigh-schoolaws for numbers:associabivity nd co.mrnnlativity f aclcliLion nclmr,rltiplication, istribubiv-ity, canceilation,monotonicil,y' , tc. Furthelrnore, we exhibibed he pairingfr-rncbionn i'oo"rr.T'hen we showecl n /16 sorle ploperties of clivisibiliby(anct eti; llorough iuvestigaLion f fXs bo Chap. V) . In /to( erp) we clevel-opecta coclingof frni te $ecluencesnd finite sets and proved sornebasic acts


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2. firagmentsof First-Orderr\r' i thmetic 61abotrt finite sets and their cardinalities. Then .',veturned to I Ey we showedthat I Dyprovably recursive funcLions are closecluncler primitive recursionand then we developed arithmebization of metamathematics in this theory.We defined termsn and formulas' and proved their basic properties. Our finalclevelopmenthas been a definition of partial satisfactionsl for each n we havetlre formulas Saty,, and Satp,,, wibh provable Tarski's properties. This en-ables us to introcluce variables for Ef, sebs,etc. (for any fixed n). This basicapparatus will be used throughout Chaps. I-ry.

2. Fragments f First-OrderArithmeticRecall that in Sect. 1 we already nvestigatedsome ragmentsof first-orderarithmetic, notably /to(erp) and IJ1. Now we are going o investigatesys-tematically fragmentsobtainedby postulating a number-theoreticprincipleas a scheme or all formulasof a certain class. n subsection a) we shall dealwith fragments based on induction, the ieast number principle and collec-tion; in subsection b) we shall study various other principles.Recall alsothat in Sect. 1 we exhibiieclsatisfaction or fr,-formulas (["-formulas) forany fixed n. In subsection c) we shall use this device o show that most ofour fragments are finitely axiomatizable; hen we shall generalizeand showihat under someassumptionswe can exhibit in /X1 a reasonable atisfactionfbr the relativized arithmetical hierarchy; namely for formulas X, in a setX. In subsectio" (d) we apply this to particular fragments; his will give ustechniques ery useful n ihe foliowing section.Subsection e) is an appendixpresentingan alternativeapproach o fragments n the logic without functionsymbols.Results of this sectionwill be used throughout the book.(a) Induction and Collection2.1. Here we shall investigate the following four axioms that we met alreadyin Sect . 1 :(Iv) ,p(0)&(Vr)(e(r)-+ (S(r)) * (Vz)e(r)Q'p) (Vr)i(Vy< dvfuD --. p(r)) -+ Vue(r)(Le) Qr)e@) , (lr)(tp(")& Vv< ")-'v(y))(Be) (Vu) l(Vr ")(1a)p@,a)) (3u)(Vrcu)(1yS r)p(* , ,a) l

They are cal led the successor ,nduction axiom given by ,p, the ordet' in-duc't ion ari,om by V, the leas'tnumber ariom given by V ancl the colLectionaxiorngiven by V respectively.Boih the axioms of induction and bhebe sufficientlyclear. Let us give a verballeast number principle appear tolormulation of collection: hink of


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E2 I. Arithmebic as Number Theory, Set Theory ancl Logic,p(r,y) as defining a.mr-rl t i -valuedunction Q: ,p(x,y) says hat y is a possiblevalue of @ or r. Call 0 t,otalbeneath r.r f each x 1u has at leasl,one possiblevaltte. Call rF cofinal on u if for each u there is an c ( t such thal. erllpossible


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2. ft 'agmentsof First-OrderArithmeticlbr each pair o En, T e nn (thus Lhe axiom says: if o' and 7r'ar-eequivalenLbhen nduction holcls fbr o - and obviously also for JT). Similaily LAn.In 2.4-2.b and 2.'l we formulate the principal facts about these theories.2.4 Theorerm"Fbr each n, the tbllo'wing nine theories are mutually ecluivalenL:

I Xn,, I 11'n,I' Dn, I'IIn,LEn, LfIn,,

I L's(I]"),I'Es(t),),LEo(Dn).

2.5 Theorem. For eachn,(1,) E"+1 1 BE1-F1+ IEni(2) BEn-F1 + BIIn Q LAn'r1 4 IAn11,(3) For each Dn-'formulag,, the formula (V" 5 y)tp is En in BEn.(Here =* means "contains" (i.". "proves all axiomsof") and g means "isequivalent o".)Rernallr. It is unlcnown whebher IAn and LAn are ecluivalent; all the oth.erarlows will be shown to be strict in Chap. IV (the theories in question areno t ecluivalent).2 . 6 Def ln i i i on . Le t p ( rg , . . . , xk ) be a fo rmu la . De f i ne n . IX r : g is a z -p i eceof p i f q is af in i te mapping, dom(q) : (< )h-mrrange(q) g {0, . [ ] anct

(V to< z) . . (V r r < r ) (q ( ( "0 , ..u , ) ) 1 : V (q . . r r ' ) )g is pieceu'iseodedn T' 2 If)t if ./ proves Vzxlq)(q is a z-piece f V).We often write "p..." for t'piecewisecoded".2.7 Theorem. For eachn ) 1,(1) each Eo(D") formulap is p.c. \n IE";(2) if g is An-1 in B Eng then g is p.c. n B L'na1.

In particular, each r, lbrmula s p.c. n I En lrlote hai by 2.5 (3), lormulasAn+r in B E,,41 are closedunder bounclecl uantifiers.This completesour list of facts. In the secluelwe shall present a seriesoflemmas hat provesall the above heorems.2.8 f,erru:nu.") For eachg. ,Lrp : It -'g is pro''rablen predicate ogic (trivial).T'h.trs[t ln e Lnn and It fln e LEn.(b) / to e LEo e I I Es.ProoJ. T'he second eqtLivalences obvious from (a); for IEs + LEg see7.22.We plove LEg -- /,D0. AssLrme? e Xo and let us lvork in LEg. Assume


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64 I. A|ithmetic as Number Theory, Sei Theory an d Logic

.p(0), (Vr)(p(r) -* p(S'(r))) ancl (3r)-tp(r) ' Then there is an r such that-o6 l . l r (V; < ' " ) ,[email protected] r f 0 s ince 'P(0) ,hus : '5(v) or some ;bvQl ' 'y'< ;, ifr"r ,p(y) and therefore 9( r ) - a contracliction' n

2.g Lemma. In B Dn, Xrr-formulas as well as -Ilrr-formulas are closed under'bounded c;-rantifi abion.Proof. Eviclenb .or n:0. Assume the lemma fot n and considerBEnal anda En+t formula (1A)V@,Y). Then( * ) B E n + t l - ( V r < " ) ( 1 v ) v@ , a ) : ( l u ) ( V r < ' X l v ! u ) t p ( r ' v )ancl he forrnula bllowing (lu) on the right hand sideof (+) is IIn in B E,by the incluctionhYPobhesis. n

2.10Lemtna.BDn-r1


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2. F\'a,gmenis fl FirsL-Orcler Ar.i[hrnetic(cf . 1.13-1.16;n greaber leta i lwecouldwri te (Yz)(z 1 cr + (1" < c)(z- l rL:ak-, o. (Use 2.13.) By IE1, let f be the least numbersLrchhat (q)i : 1; then obviously q)l ir the leastelement such bhat p(r).lVe have proveclLEy(Dr\ the proof of /Xs(Irr) fro- LDy(Er) is bh.e ameas the proof of 2.8 (b).2.15 Lemma. Bf)r'r1 ) IE,x.Proof. Trivial for n: 0; thus assrlmen ) 0 and worlt in BEnq1. Note thatwe may assum.e En-t (induction on n). Let tp(r) be Err, p(x) : (12)1,,(r,z).In BEnl1 assllme ,p(0), (Vr)(


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66 I. AriLhmetic as Number Theoly, Set Theoly and Logic

P roof Work in B D r.41.Assume (Vr ) (]y)p @,'u) : -(1 z)t[ (*, ")], where 9', f;are flni furthermore) assume (1y)p(a,y). We have(Vr ) (=y ) ( ,p (n ,y )Y [ ( r , y ) ) , he i rce or some t r ,

(V" < ") (1y < u)( tp( r ,y) ,h@,y)) (by En-pt) , hurs(Vt < a)(Qy)rp( r ,u)- , ( :v S w)e@,Y)) .T'lrenwe have(1A { u),p(ct,y)and we m.ayuse En (prececling.l.er--na) , bhrrsLII^ (2.'II). Thus there s a leastcl 1 cr, uch hah(1A 3*)9(o"y); bub hen.,, i , t i- , ."eastnumbersuch hat (!y)pGt' A). This completeshe proof. n2|L7Lenvna"LAr-rt t BEy141.Proof.By incluctio.n n n ' Worlc n LArr11, let (Vr ! o')(1y)0(r,A),d beingIIr. Put cp(x) : $r,a)(1y < u)O(z,g)]; hus rp(r) means hat the leastwibnessor n m.ajorizesthe least witnessfor al l z such that o 1 z 1cl. Observe hat g is Dnaf ifor n: 0 this is eviclent,and for n > 0 it followsby BEn. But 9(r) iseqr-rivalento the following formula ,b@), h@) = a 1 ctk(vu)10(t,u) -- (Yzbet'ween,c)(1y < u)0(2,,y)], hus: each,wilnessbr r majorizeshe witnessesin questiorr.No* T/(r) is iln*1. (for analogouseasons)and thereforeLA,+Iapplies: et rs be the least elemenb atisfyingV and d(re, ys)'Clai,m. V, < ")(1y < yo)0(x,y) . Indeed, the statement holds for r be-tween16 anclo by definition; f there s an r ( rs such hat (Vy < Ao)-0(*,A)then take the largesb uch r possible r is the leasteiementsuchthatr 1 ns k(Ytt < trgxr l x t - -+ ly < yo)0(" ' ,y)) ' ,which is Arra1. But ther: we hwe g(rt) k *' 1 u - a contradiction. tr2.18Reinalk. (1) The proof of-LAn-p1+ IAn+t is easyand ef t to the reader(cf . 2.8).(2) The reader may ch.eckhat Theotem 2.4 {bllows from 2.8, 2-L2 and2.I '4 ; urthermore, heorem2.5 ol lows rom 2.9,2.I0,2.17,2.75,2.16,2.77and 2.18 1). In aclclition, heorem2.T 1) is proved n 2.13.Thr.rst remainsLoprove2.-((2); this is done n the following emma'2.1gLemma.'I:fn 2 1 anclp is AnH1 n tsEn-vt then I is piecewise oded nB En+7'Proof This is provecl imilarly to 2.13(a): givenz, Lhedesiredz-piece f p(r)is the least q suchthat q i, a finite *u,ppir,.g, lom(q) - (< z)tt rang"(q) Q{ 0 , 1 } ' nd ( V z6< z ) . . . ( V " t < 4 ( v@ 0 " ' r f t ) - q ( ( zo ' 3 l , ) ) - 1 ) ' I } r e.orr.litio1 in cluestions easily shown to be An+r in BEn-'r1, thus LAnl-tapplies.This completesbheproof'sof our theorems' ll


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2. F\'agmeuts I Filst-Orcler Arithmetic(b) nmther Frinciplesanrnhcts Abou{,Fragrneids2.20 Principles - continued. Let r-tsntrodu.ce bh.ree ew principles:S'trongcollectionStp:

(Vu)( lu)(Vc u)l(1y)tp(r,y)> (=)y r)p@,a)lRegu,LarityRtp:

(Cr ) (1y I u ) tp ( r ,y ) ( l y < u) (Cx)e@,y)where(C'*) is (Vtu)(lr > ro) - the cluanbifierthere are unbouncledlym.any"Pigeonholeprincipl,ePHP(


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63 L r\rithrnetic as Number Theory, Set Theoly and LogicThe principlesare zr.sollolvs:

(P*) PF UN(e) -* (Vz)(3s)Approrr( ' )& Lh(s) z) ,( r ) TFTIN( tp )' (Vz) ( l s, \AT tp ronr (u )/h (s) - z ) .hlote that bobh (Pr) and (7*) are meaningfui n /X1 (even n fXs(erpt); '1eqis u.sed).Ihis completesour list of principles;obviously, br each ,p all theaboveprinciplesare true in l/.2.22'Iheories. (1) Theoriesbasedon 5, -Rand PHP are assrlmedbo containI Es: S l : I E s U { 5 e | , p T } ,R r ' : / X s U { R p l . p e l } ,

PHPQ) IEou {PHP(e) p e J-} .Here f stands fotr Dn, IIn and possibly Zo(D").(2) Theories baseclon.FtlC,T, P ate assumed o contain 1tr '1:FAC(r) I Etu {/ '/ C(v) v e r}Tg) : IE t r {Tv e e I }P g ) - I E t u { P p l v e r } .

Tb.uswe may trse he notion of finite secluences.ote that in Chap. V atheory of finite secluencesn /Ip will be elaborated;having this we coulclcliscuss lso he principlesFAC,T, P over ftr's. But we shall not investigatebhisnow. Our resultsare contained n the following three theorems:2.23 Theorem.For eachnatural n:(1) S'En+r e SIIn e IDnyl(2) PIIP(Ey(D")) + .[Dn+t(3) for n> L, PHP(Dn+t) + BEny lfurtherrrrore ' HP(Et) + BEt-(4) RXn-rr e Rnn e Bfln+r # BEn'r2.2.24 Theorern.(1) For n > -1., Dn+1 +) FAC(8.+1); and /Xr


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2. f,bagments [ First-Order Arithmetic 69:

\ /I D s

" / \PEz BEs\ " /

IEz,./ \

PEo : PEt BEz\ /\ , 1

IErJ

BEt : BEo.t

IDoFig . L .

The subsequent eriesof lemmasprovesour three theorems.Overview.2.23 (1) proved n 2.27-2.29, 2-3) in 2.30-2.33, 4) in 2.36-2.44and uses2.2a$); 2.24 (I) proved n 2.35, 2-3) in 2.45-2.48and uses2.37;2.25 s proved n 2.49-2.50and uses2.23 1).2.27 Lemrna. IEn'4 ) SIIn.Proof. LeI gbe IIn and work in IEna1. Let u be given; et q be the (u -F1) -p ieceof (!y)p(r,y) (ct .2.13). We prove :u)(Vr < ")((q)*, - J" > ( ly


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I. r\r' ithmetic as Number Theory, Set Theory ancl Logic

2"28Lenrw:e-. fIn * S'Dn41.Proof. Conbracbing 1-ranbifierscf. 2.10).2.29 Lelrn:ma. lln 1 Ifin-tr.

T

Proof. By ilclr-rction n rz .Recall that;Sfl', containsJtr'g.Le{,9be ff n. lVorktn SII7, ;assumely)p(O,y) and(Vr)(( :y),p(n,y) , ( l lv)q(r- l - t ,v) ' Takeanvu and lrseSI f n to geb u such hat (Vr < "X(:y),p(x,y) ' ( )a ! u)cp(t ,v)) 'Then ttse trr, to prove V*


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2. f,\'agmenLsf First-Order.\riLhrnel,ic 7Iifllre form.uia "q codes rn on (S ") >< " is /^\n-r7, Lhe seconclconjr-rnct s41; {,husbhewhoLe s X,r.1_1.(l) p is l,otal on (( zr )and there is a,n s 0 it is .BErn.Proof. (+) In IZt -l BfIm assume V" S u)()y)p@,?J), \o e fln; thus forsolrreu we have (Vr '-- u,)(2y I u)g(r,y). Prove the following formulzr, yindr-Lction n z:

(V" < tr ) ( ls < e@*u*t) ' ;1 la(s) - :u ! -182 Vi 1 r) tp( i , ( " ) ; ) )

n

r-lLI

(cf. the remark inm : 0 a;nd is L\r,or,u: isposal.

c is eviilenL.

1.113).Observe that the lbrrnnla in quesLion s d1 forfo r m ) 0; thus the corresponding indLrction axiom is zr,t

flTtenrir' le. bser"rehat ?.:lC(fr*) + FAC(E*-rt) (.f. 2.10); hus 1.2.11) isproved.


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I. Arithneebic s Number Theory,Set Theory and Logic2"36Definition. Givena tbrmula tp,,he ollowing ormula (that canbe denoteclIvI.ttUNS(p))says that,


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2. Fra,gments f First-OrclerArithmetic 732.39 ,ernma.Leb bsbe tIr.; (ID1-F BIIm) proves he tbllowing: f Ty'6defi.nesan increasing unction Jl on an initial segmentbhen(a) (Vr < ")(1y)rh0-,( l t > a,)C.PRL',,p,(t)(b) LeL K(ct) be bheconstanb ecltlencef length a r,'/hose ach member iso. Then (tf > /r(a)) cp.RI,to(t) , (Vr < a)(1y)tbs.Proof. (a) Note tha| C'PRI,t,,(t) s II* and we haveLIIrn(LfIr for rn: 0.)Let s be the maximal code of the primitive recursive beration of iq' suchthat s ( a; 1.eb : lh(s) - 1 a,nd',7 (s),,. Clea-r,rly,< a ancl f tcr s suchthat r/g(u,u) then let I be the concatenation f o with the elemenbu, i.e.t : s^ (ur).Then f ) s, thus > cr .(b) Assurns -(Vr < ")GA)rhs. Then evidently each primitive recursiveiteration t of F is iess han 1((a). n2.,70Lernma"Let {o b" A,"; (I E1 + B lTm)proves he following: if Tpsclefi.nesan increasing unction on an inibial segrnent hen

(Vr)(1y)t!o : (Ct) C"PR.t,b,(t).Immediabe rom the preceding.

2 .4LLernma.J 'o r ach0( r ,A )e I I * bhere isaT / ( t ) f l ,n such tha t ( I Ih - fB n m) l- (Vr)( :y)o : (Ct)$ .Immediabe r:om he preceding.

2.42 Lemrna.B E*+2 + REm+l"Proof. Assume (Iy < a)(Cr)0(0 e E^+r).Using Bilm+1. show (lt)(Vy


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74 I. Arithmetic as Nlumber Theory, Set Theory ancl Logic

I{ence(Vr < ct) l (12)rp(r ,z) (1 , < t) t l t ( r ,z) l

Tirurs -IXg gives Lhe ilesirecl rninirnum.2.44Lewtuna. .RIf* =* BIIpr4-.'1Prool by indr-tctionon rn; thus we may asstlmeBfl'nl. (For nz : 0 thisfollows rom 2.43.) Let O(r,y,z) be E* and consider Yz)O(r,U,z). Workin (RII,,, + Bn"m); asslrme

(V" < a)(ly)('l z)0(*, y, z) .By 2.3"(, Iet . ,g(x,A,z)e [I*be such hatT/s def ines n increasing'unct ionolr an in.il,ial seqtnenl, an d

(VqXVy .ci(12)-0(t,Y,z)'Ihen.we have the following:(Vr < ct)(1y)(Yz)0(*,u,2) ,(V" < a) - ' (Vy) ( l ) rho (* ,a ,2 )(V" < a)--(C' l ; )CPl l I { , ( t , f ) - '- (CfX:r < u.)CPRI4,o(r, t ) -+

= (Vy< ,t)Gr)rbo(r,,z) . ( , ,)

(lqXVt > q)(V" ( a)-,CPRI,po(r,) -+(:q)(Vz< a)(V/> s)- CPRI,,p"(r,l)-+( lq) (Vr< ") (1a< e)(Vz)- - rbo(* ,u ,2)(:r /)(Vxi ")(1a < (t)(V)0(r,a, ).T'his completes the proof of.BIIm+t.

n

bv (*))(by 2.40)

(by RII,.)(definition of C)

(1ogic)(cf.2.3e)(bv (*))

nRernark.Note that the proof of.2.23a) is complete.We turn to 2.24 2)-(3).2.45'[.'emrr:ra.En-r1+'1' En+tProof.Lel ,gbe Dn' 1and r,vor lcn IEn+t- l TFUN(p), i .e.assume hat pdefinesa tol,aLunctioir -F. Then yt ts An-,r1i1 our tlleoly' Define

G(" ) : (m iny > x) (Yu , r ) ( r (u ) S y ) .G is total, r:nonoLonend Anal; showby the usual echnicl-rehat its primitiverectrrsive telat'ion f/(0) : 0, fr(r * 1) : G(H(T)) is toLal and A"-p1'Vloreover, Fl s piece'wiseocled;r-om his it is easy o conclude hat for eacho, the lestriction of H'to (< z) is a finibeset h; but /r. s an approximationof F of lengbh + 1. n


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2. Fra,gmenLsf Filsi;-Or:cler ri[hnretic 'rc

2.46 T'emrwa,T Dn-F1=+ I En+L.Proof. Triyial for n : 0 by our clefinil,ion of -TEf thr-Lsassur.rre??) 0 zr,nclworlc in Tf,'n-1-1 f IE". First Drove BAn. I'et tl.,he fln ernclassurne (Vr !u)Qy)r l : ( r ,y) . De{ lne l i ' ( " ) : (min A)rbkt . ,y) or x { u, f ( r ) :0 ot ,herwise;F is 1\n-p1 n our theory. l ,eb s be an approximation o.f F', lh(s) ) r.r .| ' .Th.en r)" > u, thus (Vr -< u)(1y S (.s)"+r) ,p(* ,y) . Bn" {b l lows.' Ib prove IEnl - t take g En+1, V@) = ( :y)X(r ,y) and assLr in t?(0)ancl (Vr)(,p(r) -r ,p(r -F 1)). For each r, let G(z) : y if y is a secluence flength (r -l- 1) an.d.for each i < *, (y); is the Ieasl z strch that y(t,z). C| sa funcLion (possibly parLial) and is An-t-L n our theor:y. f G is bounclecl, .e.a: G(r) i rnp l iesa S us, hen we get (Vr) ( iy < w)(y: G(") ) by [ t \n . r1 ,wlrich is aL onr disposal (thanks to BEn-1-r); thus asslune G u.nbounded.Thus (Vu ) ( l r , a ) (y - G( r )& ( " , y ) > , ) .De f i ne H (u ) to be the l easLpa i r(*,y) such thaL A: G(*); then -I1 s An'r1i1. our Lheorya1d -Iy' s total. l ,ets be an approximation of .F/ of length z -lI; then s is a seclu.encef incleasingpai rs (* ,y) such l ,hat y - G(r) . Thus i f ( r , 11 - Q)" then r ) r , A : G(r)and tlrerefore y gives witnesses for each i 1 r; thus (Vf < *)(1y)X(i, y) anclp(r) follows.2.47 Lemrna.T(IIr,)


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76 I. Alithmetic as hltrmberTheorv, Sel,Thec-,ry ncl Logic2,49 lLenmna."rbr eacb. 'g(p)-formula tZ l,here s a formulatUl of l,he for.m

) ' ) (Q U " ' - z r ) . . (Q*yn a ) . . ( Q z " z ) . . . V 0 ,i.e. z is both the bound in one cluantifier and the cpantified variable in aquanbifier frn'ther ouL. It remains Lo show tirat f1 can be rnade clash.-free.Tb this end let V2 be a subforrnula of F1 having a clash and s'r-rchhat eachproper subfbrmula oli V2 is clash-free. Thus V2 is( * ) ( Q a t z r ) . . . ( Y a ;a l ) . . . va.nclsimilarly for f insteacl of V. But ('r,) is ecluiva-r.lentn Ioo.n to

( Q u tS z ) . . . ( Y a ; .z ) . . . @ ;S y t - - +V a )and sirnilarly fbr :1.

Iberated use of this procedure gi'ves he result. n2.50 Lemsm. If tp 8. Iln and f is tr6(g) ihen V is L\n-p1 n IZn.

0; bhtisassume napply propositionalthe torm

@ y S s ) ( ( o ; vcalcuLlusoUg; hen we get an

7 (i ) )( l y ) o 6 ; ( x ,y ) .v,rheleo'i ate En a,ncL'n';dre ll'n,. Write oi as


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2. Fragmenl,s of Filst-Orclel Aril,hmeLic 7 , /

Usirrg gDn, bounclzrl l ly) 's:

'.llhen,I/'t is eclu.ivalentn. Ioo",l to( f q ) (Qr r t 3 zr ) . . (Qnrn < 27 , ) (n ( ly c t )oo ; r r ; )

the conjunction is IInin IEn ancLh.ewhol.e ast formutla s En-rr in IE".But this su'ffi.cesince -f is also trs(Xr) ancl thereforeLn+r in 1X".. ThiscompLeteshe proof of the lemma (anclof 2.25). tr

(c) Finite A.xiumatizability; .Fartial Ilruth IJefrn-ifiiomsfor R elativizeclArithmetical For mulasThe aim of the present subsection is two-fold: first, to show that for n )0, IEn, BEn+r and P.f,'n. are finiLely axiomatizable; we show this usingpartiaL truLh definitions elaboraied in SecL. I (d). Second, we shall extendor-rrpossibility of clealingwith infinibe sets insicte ragmenbso.[ a.ribhmeticbyshowing Lhat, under some conditions, we may speak in -ftr1 of sebs -rr-definedfrom ct giuen sel (and cluantify over such sets). This will give us very usefuimeansof expression.

2.51 Discussion.Let us survey our sebsand membersir ip in aribhmebic.Firsh,we defined in /Iis(erp) the memebership predicate with respect Lo i,vhichnumbers behave like h.ereditarily finite setsl in particular, lXo(ecp) provescomprelrension f.or E[e (ecp) {brmulas. Furtherrrrore, given auy formula p(r)with just one variable, we may introcluce a consbanb A, say) for the se t of allnurmbers satisfying p together with an acl lr.oc nembership predicate suchthat r A jusb means p(*)..Anaiogously for forrnulas with more variables- v/e may inLroduce a constant for ihe relation defrnedby ,P.In particular,assuming tlrab r,a(r,y) cleflnesa LoLal r-Lnction which we denoted' TI'AN(p))we ffray w.ribe : F(x) fbr (r, a) e F, which in tr:rn jr-rst neans,p(r,y). Nloregenerally, oulrV may contain free vai:iablesdistincb from. those clisplayed; heyact as pararnebers. In this case \,ve lIay consider the parameters l;o be iustcoclesor the colresponclingclasses;f g is tp(t,put') bh.en X,ar (or evenx e pdr) mean$ simply g(r,par). r\.parti .crLlar aseof this is given by partialtrutlr clefi"nit ions:t Sat(2,e) is a formula which is a partial saLisfacbionor-l-fbrrnulas (".g. Ii,.,-formulas) th.en we may taiie I'-formulas with extrcily onefi.eevariable for coclesof -f-sets as we clid in Sect. 1 (.1). In this last case ',vehave clouble profii;: firsi:, we may cltLzrnl,ifyv.er J--seLsand second, rve helve"it's sn-owing"-iL'ssnowing lernrna saying, roughly, for each J--fornrulzr p(r;)bherl,:he .l-set coded by rp consisLs xactLyof al l nrLrnbers satislying 9(r).


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73 I. .Arilhrnetic as Nurnber 'Iheoly, Se t Theory and Logic2,52'-fltreorern.:'brn ) 0, each. f the theories Inn, BE'n-lr, -Pt'n is finite-tvzuciomatizaltle.Proof (stcetch).Lel, n be given..We slr.ow 'ha| IEn is finitelv a,riom.atizable.Obser-re hat the asserLion",5aty,r. saLisfi.esl'ar:si.ci::uth condibions lo r X7,-formtrlas" is a conjunction of fi.n.ilel.ymany conclitions (one for atomic .[or-mulas, one for each connective, bwo for: bh.e ourr.decl.luLz-r,ntiliersnd two.[ortLnbotrndeilones, say). 'lhLt-siL is a single formr-Liarrovable in IE1; leb I-n beth.e iniLe subtheory o, f .'D rnalsing his forrnula meaningful and provable. InJ, we can express he single sentencesaying

(i.) ryX E'l-set')\ " / [ (0 1, , , 'Y 8z(Vn)( iD D,n x --+S(rc)Qr,r , X)) - ' (Vr) ( r e r ' , " x) ) ](each .!fi-set satisfi.esncltLcLion).Observe tlrat this is in facL one particttlzri:instance of .I',"-indr-rction since 5,2 is X,r, tlr.us (+,) s prova.ble t IEn; onthe other hand, each instance of X,r-induction follows from l" + (*). Indeed.,take a Xrr-formula rs(r,a) @ being a paralneter); then lF' : Subst"( ,9',y)is a Xi-set - a formula' with one free variable. A.dcling possibly one newax iom lve may p rove Sa tn (g , [ ( " , t ) , ( g , i ) ] ) : Sa t ' | ( / , { ( 2 , ; ) } ) = 'pQ: ,a ) .Now the classX codeclby V' is indtr,ctiveby (*); this gives -Ir. n?.53 Deftnit ion (/X1). A set X is pieceu'ise oded p.".), i f Ibr eachu there is asecluerce of zerosand onesof.le:ngth'rr r-rchhat (Vi < ")((r); : I =i e )Y).Il.ernark. This is in fact a scheme of definitions, depending on the chosennotion of a set and membership (cf. 2.51). Recall also Def. 2.6 (a formula ispiece',risecocled n a theory T ) l'Et). Eviclently, th.e relation is as follows:V ( x 1 , . . . , $ r ) i s p . c . n ? i f f ? p l o v e s h a t t h e s e to f a l l n - t u p l e s r s . , . . ' , r n )s u c h h a t V ( r 0 , . . . , n n ) i s p . c .2.5.-1 efinition. (1 ) Let X be a new variable anil let new atomic formulas betlre old ones plus t X where f is any tertn. tg(,.f ) formulas restilb from newatomic formr:las r-Lsing onnectives and bourncled uantifiers.

(2) copy the definit ion in 181,i .e..,define fi(x")folmulas in the obviousway. Clearly,,p is a X6(X)-fbrmula iff IXr l- [


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2. Flagments of First-Order Alithmetic Tg2.56 Remarh. Clearly this is again a scheme clependenbon which notion ofsets ancl mernbership is nseit. We can summarize the theorem by saying bhatin .I1 we can define sabisfacbionbr sets tr6 definable from a set X TtrouidedX is p.c. Th.us or a p.c. set X,we have cocl ing br )7o(X) sets; they are al lp.c. and hence sal,isfy he least nu.mberprinciple. The enigmaLic formuiationsauyingLhaL Sols is /11(rV) unclerLhe asstunption of X being p.c. means,pecla.nticaily, h.e following: we have a flt(X) formula Sat,s and a nt\)formula Satt and IEl l [X p... -> Sa'tg,y ==Sattg,y).2.57 Corollary (Saiisfaction for X[(X") formulas and IIE(.X') formulas.) F'oreacir fu ) 0, there is a l7r(X)-forrnula Sat2,1-,u(z,e) sr-rchhaL /X1 provesthc followin.g:

If X is p.c. then 5ol5',&,X obeys Tarslci truth conditions for Ei@')-formulas, Xo being interpreted as X.

Similarly 'for IIp(X).(Obviously, Satr,k,;g is constructed. rorn ,9atg,y - or more precisely, fromtlre Lwo fbrms of.Sa'tg,y, exactly as Saty,lr was constrrlcted from Satg.)Caution. Nobhing is claimed on X[(,(")-sets being p.c.!2.58 Corol lary ("It 's snowing"-i t 's snowing emma). I:tg@s, . . . ,r&) is X"(X)then -[X1 proves the following:

I f X i s p .c . t hen lp ( "g , . . . , r k ) : Sa ty ,n ,x@@o(ag , . . ) ,0 ] .The resL of the subsection elaborates the proof of 2.55 and contains someadditional technical devices.2.58 Definition (/Xr). q is a partial satisfaction fo t Xfi(X")-formulas' ( p,their evaluations by numbers ( r and partial inLerpretation of X by a sbrings of zeros and ones (in symbols: PSattg(q,p,u,s) i f g is a finite mappingwhose domain consistsof al l pairs (r,") 'wherez is E'f,(X"), z 1p, e is anevalua l . ion or z , r e ( rP anclTarsk iconclitions hold f.or q whenever defined, i.e. we have the same conditions asin 1 .71 ancl , n adcl i t ion, u) i f z is ' t X" 'where t is a term' theu g(z,e) : 1ift (s) Va\t,e) : 1.Il.ernark. (1 ) Noie that general power rI J is clefinable n [))1 by the rtsualinductive conditions, see 1.58.

(2) Prove in 1'))1 ihat i f I is a term",.t 1p and e its evaluationu bynumbers S r then Val(t,e) S rp; this fbllows fi'om the tact th.at Lrndel ourassumption on e, VaL(t,e) < rlh(t), which is provecleasi ly by induction onlh(t). Thus the concl ibion h(s) ) rP guarantees hat (s) Va\t,e) s clefined.2.59 Lemma. (1) PSat i is 41 in - I f1 .


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B0 I. Alibhmel,ic as hlurnber Theoly, Set'Iheory ancl Logrc

(2) ttr proves he follor,ving:f PSat$(q,p,r,.s),PSat'i(tt ''p ', ,1"',s') irclscoincides rith s/ on. he inbersecLionf i:heirdomains bhenbhenq coinciileswith q/ on the intersecLionf their cloma,ins.(3) Fr-rrth.er:m.ore)II1 proves: or eacirp, r, s such hal, s is a.sl,rin.g f zerosa.ncl nesancl h(s) ) rp there s a q such ba,t' t 'Sat"s(c1,p,r,s).Proo! is fully analogotts to the proof o' t I.72. To ge t (3 ) prove Lhe followin.gby in-cluLctionn p (r zi.ncl being parameters):

l h ( s ) ) rP - -+ 1q )PSa t " ( c1 ,p , r , s )

2 .60De f i n i t i o r r / I 1 ) . Le t ,K be p .c .Def ineSa t ' s ,aQ, r ) f f t he re a teq )P , r , s ,such h.a t s is a piece of.X, .PSa' t$( r l ,p , r ,s) anclc lkQ,e) : 1.

Floie th.at Sats.y(z,e) implies that z is a X[(X')-formuj.a and e is i tsevalr-Labiono.2.6L Proof of the Iu-[ai,nf'heorem2.55.We prove that the formula Sats,y(2,,e)has tlre properties stated. Work in -IX1. First observe thab tbr z e Df,(X"), eits evaluationt, and assuming X to be p.c. (so that we have arbitr:arily longpiecesof X) Le.m.ma .59 irnplies Lhe ollowing:( t ' ) -Satl,xQ., ") iff t here ane .P,r, s such hat sis a p iece f X, PSat i (q ,p , r ,s ) nd q( - " z ,e ) : 1

Tlris showsthal, Satg,y obeys Tarski condition for negation. Looking from.otrtside.II l observe hat the definit ion of Sct'ts,xs tr1(X) in /tr1 and ( 'r)gives a II{X) definition of Sa.ts,*under the assumpbion that X is p.c. Theproof of other Tarski conclitions is similar and is lefb to the reacler.

lVork again in -IX1 ancl asstLmeX p.c. Then we m.ay speak on Xfr(X")sets, Xo being interprebed as X (briefly, speak on Xi(X) sets). It remains tobe shown thab each I '$(X) set is p.c. Bub this is now tri 'vial: i l z t6(X')is a formula' wibh exactly one free variable' and r.u s arbih'ary, Lhen the'r.u-pieceof z is easi ly obtained from any q such that PSatfi(c1,P,r,s), wherep ) z, r ) ru and s is a satisfactori ly long piece of X. This completes theproof. fl

VVeclose this subsecbionwil,h a lernma on Xi(,Y) forin:ulas which willbe r-rseful aier. Recall ihat having proved 2.55 we al.sohave Saty,n,y ancl.Sutn ,,-r,x .f. 2,.57).2.6? .[,ernmu.There is a forrnula'WSat y,1 which is d1 in .[E1 ancl such thab.l'IJ1proves the following: for each X p..., each z ti(,Y") with exactly onefree variable and eac.h , the following erreecluiva,lent:

n


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2. F\'agmenl;soI f i irst-Olcler. ' AriLhmet;ic B]( i l . Saty,1,y(r,l* l) (where [c] is the e',ralu.aLionf the free -rariableo{. by e;( i i ) (3s p ieceof )Y.) WSat ,1(2, , s) ;( i i i ) (: l to)(Vs piece of x longeL: ha:n to) WSaty,1(r,*,s). FtLrt|.er:more r/1pr :ovesWScr, t2 ; (2 , :v ,s)&.s C st - i lVSct t2 , .1(z , t : , ,s / )i r ron.oton ic i t ;v iW, l n ' t2 ,1 ( r , * , s ) s reac t : u i -h tesses - thea t i s fac t i on f . Ly , .Proof.! l : : j .p,r(r,*,s) says: h.ere n t,zrz'L t'6(X") anclLheres a, yPSat i ( rJ . ,p , , ' ,s)ndg.(zr , ln ,V. l ) :1wiLhheobvio1_r ,. "ur ingof [ r ,y ] ) .Clearly, bhis s eclui',ralenl,osaf ng tha,Lhere are?,t,)zI z as above ancla A < s such hat for al l , 1, tr , r"'ch that .p, jnt \kt ,p,r ,s) ancl l (zt , r ,y) tsclefined, e haveq(zt,r,A) ,='.1..[ 'he res6s evicle.r:t n(d) R.elativizettHierarchy in Eragmentsllere we shall nvestigaLe eLs .c. n I En and n .BDn_t-.t" Z 1). In particu.lar,we sho.,v'ha,tBEnl-1 prove$aLIEnyl sets o be p.c.; we ntrocl:uceow An+Isets n BEn+l (in 2.69) and provetheir basicproper:ties2.'7'.1).rhe conceptof a low An4-t set playsa very prorninent ole n t.henexbsecbionn connecl,ionwitb. the analysisof provai:le fbrm.sof l(onig's lemma. F'inally we e.rhi|it zr,cLass f sets calledP^8*o(x.) sebsmeaningfr-rr.tnX'n (z.rc); -weshow tiratIE,7 proveseachE;'o (xrr) cla.sso be p.c. ancl An_rr.Tihiswiil be lseful fbrgeneralizing esultson Konig's lemma provecln BE,,,]ll to resulLs n IEn.2.63.Lernma"1) IEyl proves achXfi set to be p.".(2) I.Dn proves hat /fitotal functions are closedunder p:r:imitiveecurrsion.B) BDn-l-1 proveseachAir-t seb o be p.c.Proo!. Imitate the proof of.2.rB (a) (but now working i,nsi,d,ex1: we jLrstprove one theorem'with a universalquantifier over aII Ef, sets): Le i X be aIfi set ancl et z be given: the z-pieceof X is the leasr sequence of lengthz + L such Lhat (Vt < z)(i e .X| + (r); : 1). (Alternatively, yor_r ould cleriveour assertiondi.rectly rom 2.13 (a) using bhe act that the forimrla Sat2,1is11 and thereforepiecewise ocled n .[I'1.)(2) Routine.(3) Imita,te he proof of 2,.Ig nsid.eBL'n_t-.1. n2.64Dc{inil,i 'n (/x1) (1) A set x is unboun.d,etlf (vr)( ly > i(a x), i .e.( C n ) ( x X) .(2) A set-,( has Jreorder-type J'the tniaerseo.t.Lr.)f for ea,ch there sa seqLlence of length it entlmeraLingncr:easinglyhe fir:sL elementss, i.e.

S e q ( s ) r , t h ( s ) : x k ( V t < r X ( u ) ; X ) & ( V i < j < ' ) ( ( r ) ; < ( r ) i ) &( V i < " ) ( V y < ( r ) ; X y X - , ( l " r < i ) ( a : ( u ) ; ) .


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82 I. Alithmetic as l{umber Theory, SeL Theory and Logic

2"65Lernrna.For nZ'1, IDn'provesbhateachAfl unbouncled et haso.L.u..ProofIf X is Afl unbouncledhen the function F(*) : (leasty)(A e X k y >r) is Afr toLal; hus the function G definedb.yplirnitive recursion o be

G(o) minXG(r 1-1 ) : r (G( r ) )

is Afr, total und G is th.e ncreasingennmerationof all elernents f X. Gis p.c.; thu.s or eachr, the resbriction f G to (S ") is a set - the ciesireclsequence. n2.66Corollaly (of 2.63).(1) In I f)r, wehavesatisfactionor Xfr Di l formulasanclblrereforemay cluantifyover Xfr(tri) sets. 1'Er'proves hat eachXf i@n)set s p.c.(2) In BDrr_y1,w have satisfaction or lfr(aT+) and therefore mayqnantify over Xfr(A[*r) sets.BEn-rt proves hat eachD|@i-r) set is p'c.Rernark.Compare 1) with 2.7 (I) or 2.13 b): there we had a schema, erea single statement. A schemaanalogous o our present (2) is possiblebutcttmbersome.2.6?Theorem. Ixt- l -BEm,"n 21).(1) trachE|@"r") set s a a k set; hus,3@?*) sebs oincidewith dfi sets.(2) Ei@D setscoinciclewith Xft seis.Proof. A cluichway to provethis is to fi.nd actual) forrnulaso(x,par) G Ernand rr(r, pur) e II * and to show n ID1 I BErn that if x is xfr(dfi) thenfor somepar) r .x is ecluivalent oth to o(r,par) and to r(r,par). Theresult followsby the "ib'ssnowing"-it'ssnowing emma. Thus assumeX to bet6(y) where Y is Afi;Iet z be the xi(x/)-formula defi.ning lrom Y andlet u G Ek, u Llil, both defi.ning'. observethat the fbrmula "s is a pieceof Y" can be'wri t ten both as at(s, LL)u), 1 e Zvn n BEn, and zr1(s, ,u),11 fly, in BDrn, e.g. cr1 s

(V i < /h (s) ) ( (s ) - I -> Sa t2 , * ( ( " , l t l ) & (u ) ; 0 - -+ Sa t 'n , , , ( ' ' ) ) 'I\or,v

r e X = ( l g , p , r , s ) ( sp i e ce f Y k P S a t ( c1 , P , r , s ) kq ( , , [ " ] ) 1 )and

r ce lX = ( 1 q , p , r , s ) ( sp i e ce f Y & - P S a t ( q , P , r , s ) k ' l Q , f r l ) 0 ) 'T'hiscompleteshe proof of (1).


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2. Fragrnents f Firsi-Orcler Alithmet,ic BJ(2) is easy:by (1), a Ei@fi) set can be clefined y a formula of the form(1'u)2, where z is afu; in Lheusuaiman.ner 1"u,)zmay be replacecl yXt-formula (contraction of cluantifiers).

2.68 Corollaly. Tor n,msalisfi .eshe leasbnumber principle and is p.c. L"i-r-t(A|) set2.69Defini t ionBE"-r l ,n) r).A set Y.s LouA"n-r-r.t i t is A?n-rt ndeach,i((X) set s alsoAi*.r,Retnarh. This notion comes from recursion theory. Note thab seLs 12 defin-able in N are exactly al l sebs ecursive n 1{ (the X1-complete seb), .e. setsX such tlLat dg(X) < 0' (dg is the Turing degree). A Az set X is low if(dS(@))' - 0' , i.e. the jnrnp of ihe degree o' fX is as srnall as possible. Areader not familiar with the notions invoived may disregar:cl his remark.2.70 Rernarlc. The definition of a low Air+t set is meaningful in BD,_;1(" 2 1) since this theory has sabisfaction or Ei@ir+) formulas. Obviously,in B En-,r1 we ffray cluantify over low Ai*r sets: they are just some particularA'"-t-t sets. Let us generaLize:we may speak on low AZ6) sets if we knowthaL A'r(X) sets are p.c.

atr

2.7'L T'heorem.BE{m > 2), The following equaliLies re e;Kpressil:lenil provable n

lowA\ : Ioru iQowA'*) : IowAi@"*-t): Ai(tow Ai), EiQow Ai) : Ek

Proof Worlc in B E* . We lcnowthat low dfi sets are p.c. (since Afi setsare); therefore each seb rfr-clefinedrom a low Ai, set X is p.c., thus alsoAifX) makes sense.Moreover, f. X is iow Ain and Y is Ai(X) then Y isAfi (since t is Xi in a //i(X) set, .e. Xi in a Afi-set, i.e. Xfr, anclsimilarly,Y is nfi) and thus Y is p.c. lVe haveproved AiQow Ak) S Z\fi. But bh.enloruAi(low A)r) C LouAfr. This completeshe proof. n

:ia

In tlre rest of this subsectionwe discuss En instead of.BEyr-,r1.2.72Theorem(/I,, , fr ) 1). Each t3(t;) set s a dfi_u1seL.ProoJ.Similar to 2.6'1,we find folmulas o'(x,c,z) ancl r(r,c,z) En'4 and.Un*r in I Zn such that 1'Dn prov'eshe following:


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84 I. AriLhmetic as Nurnber Theory, Set Theory ancl Logic

If ,Ks is Xfi-definedby c e Eh arrcl'Yzs Xfi(X")-defined by z from ,Y."t t r e n V c ) ( r eY : o ' ( * , c , 2 ) : r ( r , c , z ) ) .Ily the ,,i|,'ssnowing"-it'ssnowi:n.gemma we get Lhat,Y, is a /\l-,-1 sel,.Debails ollow.itiolv, -f)Jt"proves( ' , , ) r Yz = ( l ) (s is a p iece f X"8e( lq , u , ,u ) (PSat ( t1 ,n ' ,s )

k q ( r , [ r ] ) 1 ): (Vs)(ss a piece ,fX" - ' (Vg, ,u)(PSal((q,,s)kq(rfr l ) defined" q(zlr l ) 1)

Llere ,sis a pieceof.X" is Ds(Sa't ,n),, herefole o(D^) in I En; by 2.25, .l 'is z\r11 in IDn. Hence he formuia (:s)(...) i" ( 'r) is En-F7 n IEn and so(VuX...; iu "f[n+r n .[En. This compLeteshepi:oof' n2.73 Definition (cf. 2.54). (1) Let X be a nev/ variable and add t x (ta term) to bheatomic formulas. I.6"'. X) formulas result from new atornicformulas using connectivesand bounded cpr.anLifiersf the form (V" S ll),(V, < 2v) and similarlyf9r 3.(2) In IEl defineE3*'(X') formulascopying he definibion'2.?4Theorern(cf. 2.55),There s a formr-r\a at[\(2, e) such hat -[tr1provesthe follorving:If X is p.c., then Soffip obeys Tarslci bruth conclitions or Di*e QY")formulas with X" interpreteclas X, and eachE|"o (X) set is p'c' and bhussatisfieshe ieastnrmlre, principle.Under the assurnption X is p'c'" Satu|*'{'is d1(X) in /X1..proof. rVe hall define eSa-t$*e(t,p,r,s)(partial satisfaction)anaiogously o2.ll,but .wemust be sure hat if g is defineclor a formr-tla v" < 2u)9(t,. . .).anclan e evaluatingu by somer then g will be defined for the evaluation e/extending e and evaluatingu by any number


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2. Flagments of First-Order Alitl-rmetic gb- for each u,u and each sbrings sr-rchhat /h(s) ) rr,, bhere s a g such bhatPSatfi 'e rt , ,u, .

ThenSatfiY(r,") is clefinecls(1q,r t ,u , ,s) s is a piece f X, pSat f i "e(q)LL1, t ) )s)ancl (2 ," )_ 1) ;

trsstunin.g p.c.,satfi 'fr(z,e) s equivalento (vg,zt,u,s) s is a piece f xancl ,9utf i" 'o(q,u,u,,s)ncl (z,e)clefinecl Q(z,e) 1). n2.75 Corollary. We may introiluce Sat"],pn,* n Lheobvior-rsway ancl fbr eachn prove n IE1 the following: f x is p.c. then sal"J!r,* and, atuffpr,;g beythe respectiveTarski truth conclitionConsequenb[y, e have he correspondir:g it,s snowing',-it'ssnowing em-ma.2.76Theo'-em" En proves hat eachFS'o(2") set is p.c. ancl s 4p..1.The proof of th.e act that eachE[p set s p.c. s routine and uses he finiLepartial satisfactio_ns.o prove ha t each ?*o (xr) set s d,r11, first show hateaclr actuaD E6ne .!,r) formula is z\rr-1-{n I En (generalizng 2.a]g-2.b0)anclihen mitate the proof of.2.72.2.77Definition" (1) In the aboveexponenbiabion ay be replacedby any botalzl1 function H; moreover,in IEn we may define X to be a xfi(xr) set iffor some total /1 function f/ anclsome En sery, x is rfr(x;. We havesatisfaction or E'6@") setsobeying rarski truth conclitions.(2) we rnay clefineow 83"'(x") sets(or lou E'il(Dn)sets) n .Ix,r: a setx i2 !o^w 'ot@") if ir is z{ (2.) ancleachzt6\ ser s z{ (2,).This isusefr-rlor generalizations f the low basis heorem n the nexi section.(3) From here on out we shall writ e LLn insteaclof low E'ne,J (n-uerylow sets).2.78Lenrrna.Fbr n ) I, I En proves t{L.Ln) and Bx1( LLn), i.e. ncluctionanclcollecbionor E1(Ll,n) sets.Proof. This fbllor,vsi'o'rn he obviousmoclificationof 2.'76 fbr r,i(x,,) seLsrather than. z'i"p(D") sets)ancl he fact that eachEr(LLrj set s"a z|i@r)set: rvegeb ncluction. Collection ollows rom induction in the Lsual wiy. nCantion" n saying",( is D1(LL,)" we mean th.at or someLLn sety,.x isEtV).2.79Tlreorem.For n) 1, IZn provesArG,Ln): LLn, i .e. f Z is LLn andY e At(Z) LhenY is LL,.


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86 I. rh'i thmeLicas Numbel Theory,Set Theoly and Logicproof. Let, Y,,Z be a$ above and leb zv e tr(It). Then, for approprizr.teE{Z)-formulas' V, th, ',vehave

Y * {a | ( l s p iece , tZ )WSat , r1 (P ,a , t ) }Y - {A | ( ! s p iece , tZ )WSat r , r ( rh ,y , t ) }

(For WSat see .62.) UsingBE1(LLT) we get a comrnonbou.nd:(Ya - a ) ( l s p iece f Z ) (WSatr ,1 (p ,U ,s)Y WSat21(h ,u ," ) ) '

Thus, for some41 formr-rla ,f is a pieceof. 'Y ( ls piece f Z)6(t ,s) .

Now X Et(Y); i.e. or someP e E{Y),X : { r | (3 t p iece f Y) WSat (p ,x , t ) } -: { r | ( 3sp i e ce l Z ) ( = t ) ( 6 ( t , s ) W S a t r , r ( P . , n , , ) ) } '

This slrows hai X e Et(Z). Thtts tr(y) e titQ) I E6@^) and conse-qtrentlyY e LLn. We haveshownA1(LL,) e LL". U2.80Corol lary. I t") . l f . Z LL, thenA(AL(Z)) - &(Z) '

(e) AxiornaticSysterns f Arithrneticwith No 'FunctionSyrnbolsThere are various siLuabions n which it is useful to worlc with fragments ofarithmetic formafizec] in a language having no function symbols. we shallencounter sttch situations repeatedly in this book. In tire present subsectionwe prepa.t'e he necessary formalism.2.81-Definition. -L l is the language with a consbant 0 (zero), binary predi-cates :,3r,9 (equaliby, ess-thatl ,stlccessor),ernary predicabesA, IUI (ac|-clition, multiplication). Bouncled quantifi.ersare defined in the obvious wayl{brmuias og L' with al l cluantifiers bounded are called bound'edt ormttLas ot)fi-formuLas. Dl ancl II I formulas result from tr'f -formulas in the obviousway.2.82 Ftemark. We clescribea rabher weah axiom system called BAt ' T'he mainidea is that 5, A, IvI may describe partial functions ancl thal there tnay bea largest elemenb (top). The functions clefineclby S, A, IVI have to satisfythe usual incl.uctiveconclitions rvhenever he values in cluestion are definecl'


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2. ftagments of First-Olcler ,Ar.ithrnet,ic gTSince lve clo not want Lo spend much time polishing the axiom systernr,veshall make no optimization (rninimalization) of the number of a;rioms.2.83 Definition. BAt is the theory in Lthaving the following axiorn.s:(1) Axiom.ssaying h.at ( is a iliscrete inear orclerwith bhe east elemenb .(2) S(*,y) i ff y is th.eupper neighbourof r w.r.t. (.Irtirther axiomssay that .A and lz.Iclefine inary operaLions, ossiblyparLial.(3) Indtrction propertiesof A and IVI:A ( r , 0 , 2 ) : z : : t,5 ' (a ,a ' )8 t ,9 (z ,' ) - , (A (* ,y , z ) = A(x ,y t z t ) )A t I ( r , 0 , 2 ) : : 0S(A ,a ' ) A(2 ,n , z t ) . - -+( Iu I ( r ,A , ) = tV I (x , , ! t ' ,' )(4 ) CommuLativityand associativityof A anclM, disbributivity,monotonic-ity of addition, monotonicity of mr-rltiplication y a positivenu.mber,bherelat ionof ( to addit ion: 1 A - ( lu < y)A(*,u,A).Caution: equalitiesare unclerstoodo be saying "if one side s defined bhenthe second s too and both sicles re ec1u.al",.g.

A ( * , , A , 2 ) A ( y , r , z ) ,(1u ) A( r , y ,u , ) A( r ; , ,u ) ) = (1u) (A(y , ,u) k A( r , u ,u) )

(associativity);monotonicity for A reacls( t l (n ,z ,u ) k A(y , , r ) ) - ' ( " S y: u ( ,u).

(5) Schemaof induction for ffi-formr:rlas:(e(o) (Vr, )@(r)8a,9(r ,) - , v@D-- (Vt)p(r)

2.8,tr ternark" (1 ) The reader may try to gei rid of (parts of) (+) by provingsome of these axioms from the remaining ones n analogy to the coi'respondingproofs in -Ioo.rr.(2) Show that B At proves the least numbel principle in the usual wa,y.(3) Prove the fbilowing in B tLt:

( ,z l ( r , J , ) Uy0 S A) --+ 1zs < z)A(n,yo, z0)A. ( * ,A ,2 ), ( r I z ky S r )( lVI(x,t/ ,)kA0 < A)-+ (1zs< z)IuI(r,I /0, 0)( lvI(r ,y,z)k t /- 0) - ' (o 1 z ka S z)(or jusbaccepLhese orraulas s urthel axiorns).


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BB I. Arithmetic as Number Theory, Sel,Theor:y ancl Logic

2.85 Definition. (a) fffi resu-lts rom B.tltby adding the axiom ",9, 'A , Iv Idefine tobal functions" ((Vr)(fy)S'(r,y) eic.). It fol lows tha1, here is no topelemenb (Vr)(=y)(" < y)). IXf, results frorn f D|by allowing any Dtn-formttlap in th.e nducbion scherra; similarly IfIL.

(b) Ttl t is ari thmetic wiibh a top - tire exlension.o'L.BAtby the axiotr '"there is a top element" ((lz)(t/y)(y S t)).?.86 L+:nrtna,TAt proves inclucbion or each -Ll-formtLla.Proof. This is evident: each cluantifi.eran be boundedby the top. L- l2.8? Remarlc. CLearly, n "[tr's we may define S, A, ]VI in the obvious way(S(" , y) = y : S(n) , A(r ,A,z) = z : t r * y , e tc ' ) ; ihen we obviously ge i ;IEo F /I [ ancl similarly IEnl IEl. Ot the ot]rer hand, in ID's we haveaxioms stating that S', A,, NI define total functions, bhus we rnay introdtrcetheoperabionsS,. .1- , , r .To prove IE[ ts IEg,etc . , we need the fo l lowing.2.88 Lernma. For each bounded formula p(x) there is a bounded/ formurla,p'(x,y) (wiih one new lree variable) and a constant k such ilhat I D| (enricli.edby the function symbols 5' , -F , 'k ) proves

A l . / \ l / \/ \ s > n ' ; -+v \x ) = I \ x ,u ) .zProof. The only problem is wiih atomic formulas and this leads us to terms'For each .bermf and new variables y, z let z -, t l:e defined as the followin.gexample shows:

z :a (xt * rZ) ,koB s( l - r < y) (1u2 S A) (A ( r t ,12 , I -u1kX l I ( -u1 ' , : x3 ,u )2 )u , : z ) ;

then z -y t is bouncled/ and if k is the term f understood to be a naburalnumber then I E I F y 2 ( m a x * ) " ( z t : z : y t ) .(Observe that if if consists of n symbols then k ] n; ancl if x is the tuple ofvariables n t then IEt F t ( (max*)".)

LeL fu : tz be atomic and le t fobe this formula interpreted as a number.Then\-t/l J0 F (max*) / ' < U - -+ tL : tz = (1 , < y) (z :a tykz :y tz ) )

ancl similariy tor t1 1 t2. This sh.ows lte constrr-tctiouso' fg' fo'- tp atomic.The inclLrctionstep is easy. tJ


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3. Fragrnents ancl Reculsion Theoryea,ch, IDhts IEn.

8 92.89Theorerrl.ForProof'Byhave

(a) l l im

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