10 Subsystems of PA

7/28/2019 10 Subsystems of PA

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10Subsystemsf PA

I cannot br ing a wor ld qu i te round,A l though I patch t as I can.I s inga hero 'shead, argeeyeAnd beardedbronze,but not a man,A l though I patchh im as I canAnd reach hroughh im almost o man.If to serenade lmost to manIs to miss ,by that , th ingsas they are ,Say hat i t i s the serenadeOf a man that p laysa b lue gu i tar .

F om T te ,", *,,Y'),)r':;ir:.'i:i,2iIn this chapterwe return to consideringhe subsystems2,, 82,, of PAdefinedn Section .1. (Thereader hould onsulthe diagram n Exercise7.3asa guide o the variousmplicat ionsetweenhese heories.) he twomain results f this chaptershow hat these ubsystemsorm a hierarchy,and hereforePA is not finitelvaxiomatized.

1 0 . 1 : , - D E F I N A B L E E L E M E N T SLet MFPA- and,ul M. We shal l onsiderhe substructures"(M;A) ofM, whicharedefined n a similarway to K(M; A) of.Chapter8, except hatonly elements efinedby ),, formulasare n I((M;A).DEpt l t r ro rv . e t MEPA-, le t n>I and le t Ac :M. K"(M;A\ in thesubstructuref M consist ins f a l l b e M such hat

ME 0(b,d) AVx(0(x,a) - ->x: b)f o r s o m e (x , y )e I , , a n d s o m ee A. (S u c h a res a id to b e ) , , - d e f i n a b l e i nM overA. \

I t is c lear that K"(M; A) is a substrt tcture f NI: for example fb ,ceK"(M;A) a re de f inedby n0,a) and [ (2 , 17) espec t ive ly , here130


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2,,- lefi ab e elementsr l ,Ee- I , , ndn2I , thenb*c is c le f inec ly the^ I , , o r rnu la

ay , z ( ,? (y , ) Ag(2 ,a ) Ax : y + z ) .

l 3 l

JustasK"(NI;TheI("(M;in Chapter8 we denoteK"( lv l ; a}) and K"(M;A) moresimplybyrz)and K^(M) respect ively.next theorem s the analosue f Theorem8.1 for the structuresA ) .

T l t e o t r e u 1 0 . 1 . e t n > k > 1 a n d s u p p o s eA = M F l Z k - , . T h e nA c- ("(lvl: A)


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132 Subsystem.rf PAand so the un ic l l re z sa t is fy ing i r , , . , o l fA l= 1ry1(u ; ,)Acp(2 , )AYu{z - lcp(u ,u ) ] i s i t t K" (M; ,4 ) andMEcp(2 ,6 ; , s requ i rec l .E

Not ice hat ,even f t l :A anc l l :1 , I ( ' ( fu | :A) may be nons tandard .This r ,v i l l appen or exarnplewhenMFPA+axy@), whereX is A,, andNFVTIX(x). (Such ful exist by the incompletenessheorem.)ThenK t (A 4 ) : "N an dc o n ta i n sh e eas t eM such ha tM F y (x ) .Bu t asN { o , ,Mwe cannot ave : ne N, s ince therw ise FX(n) .Now f l ("( fu| ;A) isnonstandard,77, ancl is in i te A: {c1}ay), henthenK"(M;,4) wi l l fa i l to sat isfy ,4.This s becauseor al l cel((M;A)thereexists e N such hat

M FSarr, ,(eld, c))A Vy(Sat,, ,(efct, ) ) - , y : c)Since this formula is the conjr-rnction f a I,, formula and a I1,, ormula,Theorem 10.1givesus

I ( " (M A) FSat ' , , (, d,c])AVy(Sat;, ,(e,[a,y]) -+ y : c) .Thus or anynonstandarcle K"(M; A)

I ( ' (M; A)FVcae< d[Sat ' , , (, la ,c ] )AVy(Sar ,, , ( r , ld ,y l ) y : . ) ] .Howeverf K"(M; A)F PA therewouldbea east uch e K"(M; A), d11saysatisfy inghe above ormula.But th iswould mply d,,eN, since t n-rusteless han or equal o any nonstandard e I( ' (M;,z l) , and so I( ' (M;A)wor.r ldnly have ini te lymanyelements, hich s mpossible. husTneonevr10.2.No consistent xtension f PA is f in i te lyaxiomatized.Proof. If Zr PA is finitely axiomatized,hen or sorne e N all axiomsofT are I I , , . Considerany nonstandard FZ and take a>N in M. LetI( :K"(fu[;a)k >0 . T h e n ( " ( f u | ;A )E IZ k .Proof Let b e K"(M;A) andconsider11 ormula (x,D. rc

K"( tu | ;)F0Q,6)AV.v (O(x ,) , g (x 1 ,6) )the t r , ince h is s equ iva len to a I{ *1 fo rmu la rnc l " (M;A)


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E,,- efinub e e emen ,s

lvlF0(0,1i) vx(0( , 6)-.- 0(x+ I , 6)),

1 1 aI J - )

T h e o re m0 . I ,

lretrceMrvx7(x, b_)sincc lvlF z:t,, zrnclyxT(x, b) is also Ir,,*, henceK"(fu| ;A)EVx7(x,6), by Theorem10. again,as required. t rThe next result sholvs hat in general propositon10.3 cannot beimproveci.t has heconsecluencehat1r, ,- t Bz,, foral lnz r, a result lueto Parsons1970). he proofwe givehere s due o Paris ncl ( i rby (1978).

T r r E o r r g v r0 . 4 . e t M F P A , A c . M a f i n i t e s u b s e tf M . a n d l e tn > I w i t hK"(M;,4) nonstandard.hen I((A4; A)V B2,, .Proof. Using he pairing unction,everyb e I("(M; A) is clefinablen fulbya fonnulaof the form

ayy(* ,y , a) ,where , s [r , , - t ,A: {a} ancl is a single ar iable. For f b is c lef ined yaZd(x,2 u) thenb is alsoclefinecly Ay(VZ(y : --d(x, Z, c1)) hich soft l re appropriate orm.) Thus for al l beK"(M;,4) there s ee N (narnelye : 'y (uu , u , ,D) , o r a su i tab le ) sLrchhat

M lauf3ur, u,(u: (Lto,,) Aux: bA Satn,,,(e fu,, , , , af)AV z I uYz 11,,< z z : (Z ,z) --->1Satn,,_,(r, t u,2,, al)Dl

SinceSatr,,,_,s I1,,-,, lSatr7,,_,is -I,,_1, Dd ,,_, is closedunder boncleclquanti f icat ionn PA, the ormula n square rackets bove sequivalentinPA) to a ),, formulaA(a,b, e,u). Thus, f t e I((M; ,4) is nonstandard,ora l l b e K"(M; A) we haveMFJe < tatt) ,(a, , e, u)

andhence since his ormula s ,, ) by Theorem10.1we haveI ( " ( fu ( ;A )E le 1 tAuA(a , b , e , u ) .

But b here was arb i t rarry as ong as t was a member of I ( " (M;A)) so wecleduce hatK"(M A)FVb < t * IAe< tSuA(a , , e ,u) .

But the formula on the r ight-hand ide of (" ' ) is a ) , , formula with abor.rncleclniversalquanti f ierprececl ingt, so i f I?(M; A)F BZ,, th isfornrulawoulclLreeqLrivalentn both K"(&1;A ) anclM to a .I,, fonnula

( ' ' )


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134 Snbsystems f PAXQ, ), by Proposit ion 7.L But t l -ren weK"(M; A)Fy( t ,a ) ) n tEXQ, ) by Theorem 0 .1 , ence

lv lFVb< I * 1 fe < tSuA(a , , e ,u) .

would have

Thiswouldcontradicthe pigeonhole r inciplc n M (Exercise .12), incec lear ly isun ique ly e te rminedy ^nye sa t is fy inguL@,b,e ,u ) ,b be ingu, where u : (uu ,u , ) and u is the leas t e lemento f M sa t is fy ingSat;7,,,( t , luu,ur, a l) . nExercisesor Section0. I10.1 LetMFPA and >I. ShowhatN< t,,M ff K"(M) N.1,0.2Let n>-L.(a) Show hat both1.X,,nd 82,,canbe axiomatizedbyets f II,,+:SentencesfQ(Uj Snow hatneither .X,nor 82,, areaxiomatized y a setof t,+2 sentences.(Hint: Take MFTh(N) nonstandard with a) N in lvl and showK" lv l a)F2u 2- Th(N).10.3 Let 82" denote he theoryaxiomatized y 1A1yogetherwith

YaV ay?@, , d) --->YaY azV < tay< z9(x,y, a)for all 0(r, y , a) in 2,,.(a) Show hat BI, | 12,,-1.(b) Show hat BE^ is axiomatizedy a setof I,,n2sentences.(c) Let MFPA-. Show hatK"(M)FB>,, t t K^(M)FBI; .(d) Deduce hat 1I , , ' VB>; for al l n>1 ' .I0,4 Assume hat al l the properties f Sats*(x,y) escribedn Chapter9 areprovable n I.Xy (see Exercise9.10). Show that 1.X,, nd B),,*r are finitelyaxiomatizableor al ln>7.(Hint: Consider he sentence

Va,b[Sat;,,(n,0,b])AV.r(Sat2,,Qt,lx,))--+Sat:,,(4,[x * 1,b]))-+VxSat2,,(a,t, b])]

andsomething imilar or BI,,*,. It is unknownwhether1A11r B)1 are l initelyaxiomatizable.)

I O . 2 ' , , - E L E M E N T A R Y N I T I A L S E G M E N T SWe now turn our attent ion o the impl icat ion12,,+8t, , . Our aim is toshow hat this doesnot reverse, hat is we will construct modelof BE,,that does not sat isfy I2,, .The idea is to construct a suitable.X,,-,-elementarynitial segment f a model of PA.


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2,,-eLemertturynitial segnrcnts 1 3 5Pnoposr r roN 0.5 .Le t n7 l , N IF IZ , ,_ ; ,z t rc l e t Ic , .M be el p roperI, ,-1-elementarynit ia lsegment f M closecl ncler anc' l. Then EBZ,,.Proof.Note hat fF1A0 incc --o,, l l lF1A,, nd 1A,,s f I , -axiornat izccl .SeeExerc ise.5 b) ) .SLrpposere anc l FYx laaSlQ( ; ,y )w i th0 e I , , - , .Thenfor every eM wi thx {u ,x is an e lement f ,I s incec"M) anc l o here sye l such hat0(x, l ) is true n / , ancl ence (x,y) is also rt re n lz1 since0(* ,y ) i s I , , - ,and ( r ,_ , M) .There fo re o r a l l be lv l 1 , MFV-r


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2,,-eertentary initial segmen .sThis ' last formr"r las a I, , c lef i rr i t ion f d in fu| , so deMFax l , MF BZ,,ancl 4q &1.Then

131l ( " (N l ;A) . Bu tNIF0(x , ) , asT

is alsouseful .

I" (fuI A) En,, | - Th(tvl,0) uu,t,thesetof a l l 11,,n,ormulas (r i )with parameterseA tl ' tat re rue n M.Proo f . Le t 0 (a) be V iAy tp ( i , r , r7 ) w i th tp f1 , ,_ , , e t be l ( ' (M;A) bearbitrary, ancl c lef inecl y the 2, , formula e@,A). We show thatI"(M; A)FVx < baytp(x, , a), whichsr.r f f icesince K"(M; ) s,y l"(M; A).SinceMFBZ,, there s ce 11 such hat IVI Vi< bat


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138 StLbsystemsf PAin the languagegoexpandedby addingconstants for eachbe M anclzlfurther new constant . The reduct of.M' to 9a clearlyhas the requireclproperty, s he element f A4' eal iz inghe constant is n M' but not in' 1 " * l 7 M ' , a 7 .1Then by Theorem 1 0 .7 , d e l " * t (M ' ; d )1 r , ,M ' , s o byProposit ion10.5 and our choice of Iv l ' , "* t( lv l ' ; c l )FBZ,,*1. ButM' Fv t -1 ! ) (c l ,) , so M' F-1q(a , ) fo r a l l b e I " * ' (M ' ; n ) , soI ' * ' (M ' ; a )FVy1V@,y) as I "n ' (M ' ; a) 12 , ,M ' anc l - l4 t is 8 , , ' Thus, "+r (M' ;a ) sa t is f iesX, ,n , lo anc l ence \ , ,n1Vo. n

Coro l la ry 0 .9 howshat 2 , ,and 82, ,+ , re 'c loserogether ' than nemight hink.There s no non-tr iv ia l onservat ionesult etween82,,* ,andIZ,,*rs ince or example t can be shown hat -f) , ,* lFcon(1), ,) ut 82,,* tVcon(1),,),where con(.{I,,) s a natural [I1 sentenceexpressing1),, isconsistent'.We concluclehischapterwith thepromised heorem hatBZ,,#1I,,.Thistheoremwas irst provedby Parisancl (irby (1978)and, indeperrdently,Lessan 1978).The proof here s Lessan's'THeonEvr 0 .10 . et MFPA, AcM f in i te , n7 I , and suppose hatI"(M;A) is non-standard.hen "(M; A)F BZ,,but I"(M; A)f l 8,, .Proof. We havealready een hat "(M; A)F 82,,.Let A: {A}andsupposeb e I"(M;,rt) is nonstandard. ur strategys to finclw e M with r,v b and aI, , formula0(x,w) such hat, or al l xe I"(M;A),

x e N I" (M; A)F 0(x ,w)fo r then "(M; A) t l -0 , as " (M;A) + N.We find w by usingoverspill. he following sobviously rue or allx e N:MFAw< b{ len ( )> x A lw ]n : [ ( u u :0 ) ]

nVi < len(w) [r] ,< max(l v,, 0)1, )Aly Sat,r , ,- ,( [r ] , ,y ' a]))AVI < len(w) l > 0- ' ( [ur] , lwf,- 'A ( l 3z Satr7,,- ,( i ,z , al)

yVy(Sato , - , ( [ r ] , -, \y , a ] ) >32 {y Szr t r , , ,, ( i , fz , i ] ) ) ) )V( [ r ] r : iAaz Sat , r , ,, ( i . [2 ,a ) )

AVz(Sat r , , , - , ( i , [ r ,] ) - ->ay z Satp , , - , ( [ r ] , -1 ,2 , i ] ) ) ) ) ]( In words, w codes the sequence , ] , , , t ] r : , [ ' ] ' . - | such thati r ] , , : t (uu=0)r and or each0


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Tlrus,. y overspi l l .here s n;( b in fu lsuch hat he ormula bove nsic ' lecurly brackets ) above s true for somex>N. The ic leas t l iat rv nowcocles seqLlencef I I , , -1 ormulas ,vhich re al l sat isf iedn [4, the eastsat isfy ingru]; e ingat least s argeas he least sat isfy ingr],- , , and al lstanclard1,,- , ormulas re consic leredt some stanclard)tage .We claim that, with w chosenas in the previor.rsaragraph, rnclc e I"(M; a) arbi trary, e N holds f f

I " ( l v l ;A)FAy Sat , , , ,, ( [ r ] , . , y , a ) ) .To prove his,supposeirst hat ce N. Then

M FayS a tn , , , ( [ r ] , ,y , a ) ) .But Iw] , .


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140 Subsystems of PAand induct ive lyssuminghat hemodel at is f iesZ,, - , ) onsiderhe east-numberpr inc ip le ppl ied o 3; N in M. Show hat here s no.X,,ormula (x) wi thonlyone ree-var iab leuch hat

x eN i f f l " ( tu | ; )F0(x)for a l l xe l " (M;a) . Flence lec lucehe use of the parameter in the proof ofTheorem 0.0 cannot e avoided.10.8* Prove IZ,n* tcon(1I , , ) or a l l n70, as fo l lows: n the natura l educt ionsystem f Exercise .6 eplacehe nduction xioms,,E where isan9o-formula)by the nduction ule

l , -1cp(u) , p(u 1.)f ,

- T 9 ( 0 ) , 9 ( s )

(u any variable not occurring ree in f, s any?a-term, E anyX,, ormula of 9) .(a) Show that each nduction axiom I. ,E of /X,, is provable in this system. Hencededuce hat the correspondingprovabi l i ty predicateprovableTa,(x)s a provabi l i typredicate .or 12,, s defined n Exercise3.8.(b ) The cut eliminaton theorem for this system says that if p is a proof of thesequent then there is a secondproof, r7,of I such that in an y instanceof the cutru le

A,,cp A,,1cpA

in q, the formulzr is eitheran axiomof P,{ or a formula n one of the equalityaxiomsor a 2,, formulag(s) occurringn the inductionaxiomabove.Stateandprove his heorem n 1.I1.{Hint: Modify the proof n Schwichtenberg977.)Deduce in 1X,) hat f there s a proofof 0 :1 f rom hissystem,hen here s aproof of 0: 1 such that al l the formulas occurring in it are 2,,or f1,,.(c) Let q be a proof of 0: I, al l formulas n q being ,, or 7,,. Using II,,*rinduction how hat

Vx Sat4,,,(V/ f , .r )for al l Seqlrents in q, where \(/ f is some il,+ r formula equivalent othe disjunction of al l formulas in f . Deduce that --1provable,r,,(0= l) isprovablen / I , ,n1.

10 Subsystems of PA

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