13 the Standard System

7/28/2019 13 the Standard System

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T3The standard YstemSo how is l i fc with yotrr r lew lr loke'/:Te l l me, is he br igh t enough o f indthat memo-PaclYou cal l a rnirrcl?Cl l i g R i t i ncFrou: 'An tt t tcmptat jcalousy' , Riclr , l9 [J4

If M isanonstapclarcloclel f PA, its rremo-pzrcl'is tsstanclarclystcmot'of setsA g N that are codecln M. In part icularwe can consider ompu-tations hat consult his memo-pzldo cleterminewhether or not a givenin teger z s in some AeSSy(M) . fn th is chap te rwe sha l l nvest iga tehestructure of these standardsystems urther: in particular we will findcertainclosureconditions hat characterizehem exactly'This will allow r-rsto constructmodelswith a given stanclarclystem,ancl n Chapter i4 wewill use these idezrs ogetherwith the embedcling heoremsof the lastchapter o construct ew nterest ing

ni t ia l segments'

1 3 . 1 S C O T TS E T STo clescr ibehe structure f SSy(M) or nonstanclarclocle lsMFPA wewil l nee


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Scott sets 173

restr ict ions. t ree, T, in th is sense orresponclso a tree in the graph-theoret iczr lensen whichal l nocles re abel lecl y sequencesf 0s ancl sin 7 and an eclges clrawn etween wo noclesf,g e 7 whenever en(/) :len (g )* 1 andg: f f len (g ) . SeeF ig . -5 or an exumple . )No te tha t tw t rcl i f ferent reesZ, Sc2'Nr may be isomorphicn the graph-theoret icense,since he samegraphmay be label led n dif ferentwzrys.A l tathP s 2'N is zr reesuch hat, whenever ' , . P with len(/) > len(g)theng: / f len(g).Thus a path s a tree wit l - r -ro rancl i ing.A tree Zc2'N is nf in i te f f 7 is inf in i teasa set.Regardingnf in i te recs,we recal l he wel l -known ncl erY mportzrntLevrvrn13.1 ( l (onig 's emma). I f f is an inf in i te ree then there is aninf in i tepath Pg T. tr

The next step s to cocleeach sequencee 2' ' r \ iby a natur i t l n t turbert , f t . N. The fo l lowing nethocls part icular ly c>nvenierrt .

I f l :

The tunct ion . l ' 2 ' :Nr+N sca l lec l yad ic oc l ing , nc l seas i l y eeu o be tbi ject ion.The f i rst ew vzrl t tesf t ' l are given n Table2.With th iscocl ing ive , clcf in i t icr lsoncerning cts -2 ' r r tcattbe appl icclto se ts RcN anc l v ice vcrsa . Thr - rsRqN is a t ree (pu th , e tc . ) i f t

( i f len ( ( / )> 0)

( i r . f O)

hFig.5 A tree.


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l l r l ThestuncLurdsy.stem' [ 'ab lc2 Dyac l i c od ing.

. l ' e 2 ' " Il ta(.)1000 1I 0t 1(XX)0010 1 00 1 1100:

t)I23tl5678910I t

{ " f .2 'n t ' l t l t .R } is a t ree (pa th ,e tc . ) ,a r - rd t ree Zc2 'N i is recurs iuer .e . ,e tc . ) f f { ' f t l f e 7 ' }gN is recurs iver .e . ,e tc . )The key notion in this chapter is that of a Scott set, a beautifr-rlcornbination f ideas rom recursion heory zrnd he theory of trees thatf irstappearedn Scott 1962).Roughly,a Scottset s a collection,%, of setsAcN, which s closedunder Booleanoperat ions,e lat ive ecursion" r ldundera form of I . ,6nig 'semmzr.More precisely:DerrNlrr


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Scott sets

Proof. We must verify (n), (b), ancl c) holcl orclelrni t ionf a Scott et.(a) If A, B e SSy(M) recodecl y u, b respectively,tryanyc e M real izinghe recursiveype

175then

B : { tc - ' ( r , ) rn er l } { F - ( n ) l ne A lis rccursiven zl , so Be .9f. hus by part (c) of theclef in i t ion bove, here sC e f such hat Cc: B uncl

P : { f e2 ' t r l tJ ' 1 C l cT i s a n i n f i n i t e a t h .Bu t then D: {FQ) l r teC le ' iL ' , be ing ec t r rs iven C, and D is an in f in i tepathof A in the sense f the cocl ingunct ionc.

The key resul tof th is sect ion s that the stanrJarcly51s6s f countablenonstanclzrrclocle ls f / ' ;1 areexi ict ly he coLrntablecott e s.One halfofth is esul t s proviclecly the fo l lowing heorem.Trruorrnrr.r3.2. Let M FP,z{ e nonstanclard.hen SSy(fu|)s a Scottset.

tr: S'Sy(M)n thethen ztUB is coded

p(x) : i ( r ) ,+ 1 )


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Scott ets(4) tvlFzt(c) dAd=-bA (r), 0.

Now et P be heset

111

1 ' {ne N |&1 az< d(p ( r , z ) : r t ) ) ,so P is coc lec ln M, by Theorem11.5 .We c la im hat PqT is an in f in i tep a t h .T o see ha tP cT n o t e ha t f M F a z< d ( p ( r , z ) : n ) t h e nM r ( r ) , , * 0by (3 ) , thus ne T. To check ha t P is a pa th no te tha t r f t t1 , r? rN,MFp(c , z r ) : n tAp( r , zz) : nz fo r some 21,2 . ,1 (1 ,hen wi thou t loss o fgenera l i t y e mayassume 1 {2 . t .B r - r then 1) mp l ies ha t l12z t , so z , isstancletrcl ,nct 3) i rnpl ies hat M Fp(nr,z1) - n1,so N Fp(n2" ,) -- n, (sincet l re ormtia p(x,y):z represents in PA-), thusrz, s a restr ict ion ) f n, ,and so P is a path. t l

We now aim towards proving zr converse o Theorem 13.2 that allcountable cottsetsare real ized s he stanclardystem f somemodel ofPA.The next heorem s he mainstepping-stoneowarclhisconverse, utit is of interest n its own right too; t shows hat Scottsets orm a useful inl


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180 The standarclsystertfor each unct ionsymbolF o f 9, and earch ,, , f t t r+ t N , and e t l so

n : c Q f ( n , ) : c e Sfor each constant ymbol c of ,9. These are a l l wel l -def ined, ince ore xa m p l e i f n t , . . . , r ?1N , , S F Jv1 y( f ( r , ) , . . , , ( n ) ) : o u soF(f("')But i f c i - c,p then c1: c,e S, so SF (/(n,) , . . . , (n, ,)) : , , henceF( f ( " t ) f (n ) ) :c ieS s inceS is comp le te . h is shows ha t the re sn t ,+ rNsuch hatF( f (n , ) , . . . , f (n ) ) : f (no* , ) i s n S ; bu t suchna11 rL rS tbe unique since t F(f(n)f( l ) - f ("0*,) so :ro*, . A simi larargurnent hows hat or everyconstantsymbo l o f I the re s a un iquene N such ha tMFn: c .We complete heproofby showing hat or any ormulacp(vu,. . vo)ancla n yn o ) . , / ? p N ,

MFE(n , , , . . , nn )Crp ( f@i(This suffices ince his relation s clearly ecursive n S, so the set

{ ( tE (v) t , d ] ) l , c teN, q g , MrE@)]is recursiven S hence s rn f, and asSF.f i t a lsoshows hat MFT.) This sproved by induct ionon the complexi tyof E.For atomic ormulas, tfollowsdirectly rom the definitionsaboveand the equalityaxiomsof thepredicate alculus.We prove the inductionstep or the existential uanti-fier. Clearly

MEAxE@0, , nr, , )CMFcp(n1y, , txk, r * t ) for ' some t7,11NcE ( f @ i , . . . , f @ ) , f ( n o , ' ) ) . S f o r so m e , . n , N .

B u t f E U @ ) , . . . , f @ o * , ) ) . S , t h e nw e h a v eaxcp(f(ns),. ., ffu), x)eS,

and converselyflxcp(f(n()),. . 1 @o), ) eS,

then3v1vE(f 1), ,fQr),vu) s,

SOcP(fnr), . f (n , cu,)s,


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I t iz Thc s unrlu rcLsy.surtisa cornpletey[)cover vl ,ancl t rvi l l o l lor,vhat17zrrtclcnce ) is r-eal izeclin lVI.sirrce learlyq(i, d) e !,t ' .$/c neet l rp lyshr-rr .vhat ry s f in i te lvsal isf lccln M. sir . rcct i : ;obviou:; lyco m p l e t o . u t f cp l i , ( 1 ) , . . , cp r ( i . t l ) r e i n 1 $ . d ) , h e n x l A f = , q , , ( i . d )is in r7 , - ry omp le teness,rnc l o MFAIA l - ,cp , ( . t ,d ) ,s ince the rw ise " 'wou lc l c incons is ten t .

A par t ia lconverseo Lemma 13 .5 " ha t any coun ta t r le ecurs ive lys;aturatecltructure s ,9f'-satr-rrateclor sorneScott sct (, rvi l l be provecl nClrapter15. Not ice that the proof of Lemnra 13.5shows htt t uny typep(i .A)e !/ . ' (and ot urst complete ype)overan .9I-saturatecltnrctureMis real izecln M.ln part icur larf ,4uls an l l -saturatecl rodelof P,4 thenSSy(M) 9(, since t r l =N is in 2'( hen the typep(x) { ( * ) , , *0 n e A \1U (x) , , \ ln 4 A l

is recurs ive in , so s n l l ' . anc l ences ea l i zedn M. Converse lyl 'ue V Icocleshe setA c N, then4 : { n e N l & / F ( u ) , , * 0 1

ant l he ype eal izecly u in M , tp,r ,( t ) : { tp(x) lMr p(u)} is n .? 'sinceM is#-saturatecl. hrrsA is recursive n tp,r,(cr),o.zl 2I'.This lneans hat ourpromiseclheorem, hat anycountable cottset l is SSv(M) or somenon-stzrndarclF PA fol lows rorn he fo l lowing heorem.Trteonp,vr 3.6 Wilmers, 1975).Let 2f,be a countableScott set, T ' t tconsistentheory n a recursiveanguage , ancl ttpposcTe 2'{.'.Thenhereis a countable f-strturatecltructureMET.Proof The icieas to nr imic he proofof Proposi t ion 1.4,Lrsingheorem1-1.3ncl he arggmentn Theorem13.4 n placeof the cl i rect ppeal o thecompletenessheorem. Wilmer 's or ig inalprooi was much shorter,btr tusesmachineryhatwi l i not be clevelopeclereunt i l Chaptcr 5 ')We shal lusenotat ionsirn i larro t l - ratn the proof of Thcorem 13.4 ' r- rparticular t -V is zrny ecLlrsivearrgr:,irge9"' is the recursive anguageiesulting rom aclclingufficiently nanyconstzttttsrl, o that c,, 4"' for al l9"'-forriulas p(v,,)ith onlyv0 ree. i/e regarclhe G0del-nuntbers.riformulas f I AS recLlrsiveubset f thr:Gocle l-nutrbersf forrnLr lasfy* .

UsipgTheorem13.3 ,we wi l l f inc l sequencef cons is ten theor ics , , ,T 1 , T z " . . i n r e cu t s i vea n g r r a g e qu . t , . 9 " , " ' su ch ha t e X f o r e a chgiQTi*, for each (in f ict s: , . * t l l [ ' rea recursive xtensiop f Tl , asclef ineclr t Exercise 3.B)ancl7,n ' l -T' f t l r each . In the cas;e f 9 lr ' Tr ' we

fl


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It\2 The .rtunclttrtl ;y,tferttsa contpletey1-rctverv l ,ancl t r ,v i l lo l low fhat17zrnclence1-r)s re l l izeclin ful sirrce learlyc1(i,a) e I' .We neecl n ly shor,vhat 1 :; in i tc lysat isf iccln iVI.sincc t i r ;obvioLrslyco n r p l e t e .u t f e ( r , ( i ) , . . . , q t , ( i , r 7 ) a r e n q ( * " r 7 ) ,h en= l i l A r 1 = , q , ( i . r t r )i s n r7 ,by comp le teness,nc l o MFSxla f - rc f , ( i " t l ) . : ;hceo thcrw ise . ,wou lc l e ncons is ten t . tr

A par t ia l converse c l Lemma 13.5" hat any coun tab lc ecLr rs ive lysaturatec' lt ructure s /f-saturateclur someScott set f , iv i l l be provecl nchapter 15. Not ice that the proof of Lernnra13.5shows that arty rypep(i . u) e9( (ancl ot just a completeype)over zrn 9l i -satLrratecltructr ,ve 4is real izedn M. Irr part icular f M is an Pl. :saturatecltof zrs recl l rs ive ubset f the C(icle l-nunbcrs f formulasof92't:.UsingTheorem 13.3,we wil l f ind zrsccluencef consistentheories , , ,7 , , T r . , . . . i n r e c u r s i v e l a n g L r a g e s Y r , - 9 , , 5 - ) t , . . . s L r c h t h a t I , e X f o r e a9;=5|1r, for e i tch ( in fact -9,. , wl l l be a recursive xtension f S) iasdef inecln Exercise13.8)anclT,*1Ff for eacir . In the case f ! l r , T 'u, e


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,9c.oft ct.tptrt /1,,: ' f a,cl . l f , ,ezt 'be nyconrpreteonsistentxtensionf r,us i r rgheorem3.3 .

(/ ) / ./) (/ . i . ,| : 1 . / . ;U \C, . , i .o ,

Le t t p , ( x , ) ! , x , , , , , 1 o , . . .! , , , ) l ; eN ) e n r . r m e r a t el l s i : t s f f c l r m u l a s7 r ,n thc l i ' i te ly many ree-, ,rar iabt. ,hn*n, such hat {rpt rpup,,} ; t r . Sin. .9[ s ccrrrntir lr lee canenumerate ll thescsets n this r,vay.) e crefine,;inclLrct ively;ach l /1,, , wi l l be oi the form, C1. , r ,, , , I ) ' t '

t 8 3founcl

(@)Ior sonlencw cot. tstant i i) . r i .0 , . , c1.t i . , , r ,lotarreacryn.g,. increxecry some.rN, sonreconstants le 11,, nc lan integer between0 anclnt, . g_ isu ,,rr '9 , .we arrangehisso rrat or everyTeN ancr very et a,, , . ' . , , , , } ofco n s t a t r t sn g " - t h e r e s i e N su ch h ^ i ( r u " . . . , u , , e ' 3 1a i c r * , i s t heexpansiotof -9,asshown n (o). we may als. assurnerrat eacl- ti* , is arecursiveanguagen which he fornrLr lasf Sl , formsa recursive;r-rbset.Thisclone,we f incl heseque ceT,of the heories s ol lows.As alreaclymentionecl, ,,e 2{ s anyconlplete onsisentg,,-theoryextencling . GivenT,e9{",lo, is any complctec.nsistcnt g,*,- theory n ff thatJxtencrs ]togetherwith

{cp(c , . , i .o . , C 1 . , i . , , , , , 0 6 , ,0 , , , ) l cp ( t ,) ep)ancl{3v,,0(v, ,)- 0(c,)10 9, , , }

i f th is s consistent" r Z, togetherwithi3v,y (v1,) O(c,,) l e g, * , j

o the rw ise . ga in 7 , . r , x is ts y Theorem13.3 .Having found ?] for al l i in th is way, let 7.. : tJ, .^ , | , , let t (FT t:ea rb i t ra ry , nc l e t M be hesubst ruc tu refK cons is t ingt 'a l le lements f Kthat real ize olne onstant f 9." . Thenby an easy ndLrct ionn cornplexi tyo f fo r rnu l i t sim i la r o tha tg iven n Thco ie rn13 . . i ,M < K,anc l n par t i cu la rMFTifor each . we mustshow hatM is z'-saturatecr.I t c1eM andp( t_ , t t )eg r . ' i stype overM,we mustshow hatp is rea l i zedin fu | . Bu t p (_ r , ) :p i7 " i l fo r so r 'e /N, anc l ts MFTI (where theconst i t t r ts. :c j . , .0 r . ,c1 . , i . , , r ,, J le c lc lec lt the th s tege f thL copst r r rc -t ion as n (e) atrovc) herr

T,+ p,(a,a) * {Iv,,g(v,,)-u0(r,,)10 5,.. ),is consistcn, henceee fu lFEG"r7) or ezrchp(i y) ep( i .y). on the otherharrcfi f p( i , 17)s a conrprctc/ .- type l ' r^ t s r: .u i i r . . l ^ wi,by c:eM, say.t l terr i ,c=beinga f in i tesetof constarrts)ccur n Li for sornc e N. so

1t(i d): {rf t , rt) lcp s an ...4orrnLrla ncl T,Fqt(c.,t)\1.


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184 ThestunclarclystemSince?] is cornplete he set of Gocle l-numbersf formLrlasn p(x, rz) srecursiven T,e 2.(, ntcl enccp(t, a) e2X"since 'is closecl ncler elativerec i l ' s lon ,

Cc-rmpzr r ingropos i t ions 1 .4ar rc l 3 . ( r ne rn ightexpectevery non-standardmocle l VIF A to havean SSy(M-satr-rratecllementary xte sionM'> M. This turnsoLltnot to be the case, ince any 2l l -saturatcr locle lMF PA must have its complete tl-reory Th( IVI) {ol IVI o. o an9o-sentencele9(, y the def in i t ionof f f -satu lat ion.E,xercise 3.4showsthat Z(SSy(I( .) for any complete xtension of PA, thusno elernentaryextension f l {7canbeSSy(1(r)-saturated.xercise 3.5 uppl ies ecessaryand suff ic ient ondit ionsor suchM' to exist .The mater ia l n th is sect ionhas a sl ight lycompl icerteclistory. Thenotion of a Scottset s due to Scott (1962)who proveclTheorem 13 2 an


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Sc:ott eis13.4 Let T bea completeonsistentxtensionf Pr l , +Th (N) , anc l e r

,9: {tor ' l ' f o, o i tr-r!.[,1,r-sentence].Show hat .9dS.l,y(1(r).(l-lint: For eachg/,r-fornrulari(x) firrcl ,,such hut

P,4 f o, t , 1x(ry(x)A (;r),,,,,,0) .Given hat IFl lxry$) , s l tow hat heurr ique e I( .szr t is fy ing7(a) oesnot c6cle ,gby consider ing, , . )13.5 Let lv lEPA be cot t t t tab le ,or ts tanclarc l ,e t ) ) ( ' : ,S$)( lv l ) ,r r rc lupposel ia tiVI'> IV Iwith M' Z.'-saturatecl.how that for e,acl-re IVI

1 8 5

( " )> M .stage

{,,p(a),M r cp(tt)\ %.Now suppose ' ') holds or each TeM. Show hzrt here s an P.&satutatecl'(H in t : For the second zrr t , epeat he proof of Theorern13.6, u t a t eachalsoerclcl ncw constant arning onre lementof &/.)13.6 Let ' f be a complere xrensionf pA, T+' rh(N) . Show har

.SSy (1 ( , ) : {ScN IS . : { r re lT l ? (n ) l f o r sorne e g t , r l13.7 Let MF PA be nonstanclarc l .how hat l re o l lowing reeclu iva lent :(a) I I1 - Th(N) e .s,5y(&1);(b) there s a nonstanclarc lc : , , IV at is fy ing1, Th(N);(c) Aue IVIN sLrchhat , or a l l b e A4Fb u, b i .sA,r -c leinablen M // 'b e N.13.U Let 9.12ea f i rs t -orc le t ' langui tgenvolv ing: onstantynrbo ls' ; ( i I2) ; e la t ionsyrnbols ,of ar i ty n1( ie- l : ) ; nc l unct ion yrnbols *of ar i ty n1(keK2). e t I ,c . l , ,J re J : . , (1= K2ancletglbe thesutr languagef ,5 t2 invo lv i r - tgc,( ie) , R,(7e /1) nc lF1,(keKt). 9r. is a recursiueexten:;ictnf' "l f iff there is a l-l flrnction y fron.r-/. i2-sylnbolso N such hat

{v(c , )i e 11i{ ( r ( R i ) , m , ) l j e l }{ ( r ( P ^ ) , n ) l k e t ( , }{ " (u , )i e N}

$,(c) l i e I7 li ( r (R i ) m1) l jeJ2){ (u ( r ^ ) n ) l ke I (21

area l l recurs ive, r rc lr (vr* , ) y(v , ) or each e N.(a) Asstrrrrcl ztncl"Vlareasabove. henclearly totb 1, ancl.!1are ecursive.f.911akeac lyasasu i tab leGdde l -nunrber ingde f inec ln i t , useExe rc i se1 .9 toshowtha t o r a l l se ts o f s t r i ngs f symbo l sron t , f1 , hese t t t r l r e7 ' } Goc le l -nunrbcr ing .i n the newsense)s recu rs i ven { r t r l t e f } (Goc le l -nunrber i r rgn theo lc l ense) .(b) Show hat anyextensionf a recurs ivearrguageby at rnost in i te lynranyr tew -tonIog ica lynrbo lss a recurs ive x tcns i t ln .(c) Given thtrt. 'y ' | isrecLrrsiveanguzrge,how hirt l ie luneuage/1"' t tThcolcrl13.4 s a lecurs ive xtension f 9 ' .


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186 Thestuncktrclsystem1 3 . 2 T H E A I T I T F I M E T I ZE DC O I v I P L E ' f E , N E S S' t - { 1O E, MThe sta r t ing o in t o r th issect i ( )ns the fo l lo lv i r rgbserva t ion .Prroposrr lou 3. l . The set of ur i thmeticat.s.

{5 g N lS: { rz N lN | 0(n)) for some J'o-fc-rrmula(x) lis a Scottset.Proof. By Theorem8.8 there s a proper conservat ive xtension V/>N.S'Sy(NI)s a Scottset,by Theorem 13.2.ButSSy(M) is exact ly he set ofa r i t h m e t i c e t s , i n ce f S : { n e N l N f 0 ( n ) } t h e r rN F 0 ( zz ) C M F g ( n ) , st1>N, henceSe SSy(M) by Theorem 11 .5 .Converse tyf S eSSy(M) ,,S : {ne N l& /F (o ) , , ;a0 }ay , henS is he n te rsect ioni th N of a c le f i r rab lesubsetof M, hence is def inable n N since the extensionM > N isconservat ive.husS is ar i thmetic.

We shallsay hat a theory 7 in a recursive atrguage tsarithrneticff Tlras n axiomatizat ionr, l i e N} such hat {rr , r l e N} is an ar i thmetic et.An9-structtrreM is ar i thntet ic : f f A/1=(N,R, f ,e l for some relat ionsR,functions , anclconstants on N such hiit the satisfzrct ionelation

tr

s - {{' .r{u,,, , vp )1 , [41 , , , . ?a ' ,1< N, r t11 , , t /4 N , I' " " 1 ) l andN R. . c )Fcp(u , , ,. . , , , , ) J

is an arithmeticset. (This notion can be extencleclo finite rnoclelsM too,by saying hzrtM is ar i thmeticf f lv l= (A, R, .C) for some ini tesetA c N,bgt these ases rre n interest ingecause very in i temocie ls of th is ornrwith the sat isfact ionelat ionbeing ecursive.)

Conol lnny 13.8. f Z is a consistept r i thmetic heoryover alangtrage , thenT hiis an arithmetic -nodel.Proof By Proposition13.7anclTheorem l3.zl.

The arithmetizeclornpletenessheorem s the assertion hat Corollary13.8 anbe fornral izeclnc l rovecl i th in PA. We shal l ndici t te proofofth is heorcmby re-proving roposi ton13. l wi thoutreferenceo Theorem8.8. An energet ic ezrcler i l l then be able to constr t tct proof of theari thrnet izeclornpletenessheoremby formal izing hese rgLtmelrtsrrsiclepA. (The formal izat ions stra ightforwarclut lengthy,arrcl ot istra i t . t tsnthesizeof th isbookmean hat mustorni t t here I wri te hispare rthet icalremarkwith manycrocodi le ears!)i f SgN therrS is A)r f f thereare / (x)e II , ,ancl0(x)e), , s t tch hat or al l

recurslveu


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The uritlurtetized contpletertes,s l.teoren.t t 87k e N ,/ce S()N Fyr(/c) N t= (t).

I r l .pa r t i cL r la r ,y Coro l la ry .5 ,a set s A l ' i f f t i s ecurs ivc ,nc l l so r e t sar i thmetic f i i t is Al ; , ,o, somc u N.Lr 'rra lvr.r3.9.For eztch 2l, zl f )r is losed nc' lerhe Boolean perat ions fcomp len len t ,n te rsect ion ,nc l n ion ,anda lsounc le r e la t i ve ecurs ion .ProoJ'.ali is closecl ncler omplement, ince he negationof a r,, formulzris 11,, nclvice versa.n) is closeduncrer n , u sinceboth ),, ancl r,, arec losed nc le rA, y .I f B is recursiven,4 e df\ there ttreA,, ormulase,4t in the language9, ' , (A) ( :9^ togetherwith a new unary erat ion ymbol or A) sucS hat

k e BN F3yp(rz, t) (+N FVzr2(n, ) .By replacing (u) in cpandy;by ei ther he r, , ormulaequivalen ro,ue A,o r the r r , , o rmu laec lu iva len to 'ueA 'we ob ta in f ie j , and r /e f1 , , uchtha t

k e B < a N r l y p ( n , t ) ( + N F V z A ( n , z ) . t rTt tso t


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188 Thestund.arclystemWe c la im ha t he ormu la p (z) :

t1t(z) V;, < zQ"Q): l(z) - ' I ,p(,y))def ines n nf in i tepathP gT. cp(z)s cle i i r lyAlr*,since p s E,, ,ancl lso fN F cp ( k )he nN F t p ( k ) , o ( by co n s i c l e r i n g: ) . ( k ) ) N F 0 ( k ) ,h e n ceP c , T .A lsoP is n f in i te , ince is , andcon ta ins r tmostonee lement f anyg ivenlength.Thus t suff iceso show hat P is closeci nder restr ict ions. ut i fx e P has eng th >0 thenp(x , l - 1 )e 7 , and f y is in P w i th leng th - Ithen he treeT is nf in i teabovenode ,, so n part icular here s z e 7 with7 ( z ) : / , f i n f i n i t e b o v e , a n d7 : ' y 0 ' o r ' y 1 ' . S o z : 2 y * 1 o r 2 y * 2 ,w h e r e a s : 2 ' p ( x , - 1 )+ I o r x : 2 ' p ( x , l - l . ) + 2 ' , a n c l a s s u m i n gp(x , l - L )>y th isg ives 1x , con t rad ic t inghe asser t ionhatx i s the easte lement uch ha t 7 (x) : / and tP@).F lence n fac tp (x , l - 1 ) :y , so p i sclosed nder estr icton.

Lemma13.9andTheorem13.10 e -p rove ropos i t ion 3 . l In fac tweget extra information or our labours.Say a theory T in a recursivelanguages A)r f f i t hasan axiomatizat ionr, l ieN} such hat {Ir , r l i e N} isA), anclsay a mocle lM is A) i t t M=(N,R,. f ,e) where the sat isfact ionrelat ionS on (N R,f , c) is A). Then we get:CoRor- lnnv 3 .11 . f n>1 and T is a cons is ten t ) r heory n a recurs ivelanguage ,then Z hasa complete onsistent )* ' extension nd a A)*,model.Proof. The proofsof Theorems13.3and 13.4only nvolveone appeal othe closure f the Scott set9( under Konig 's emma, hus here s only anincreasen level n the 4) h ierarchy f one. n

There is a lso a useful var iat ion of Corol lary 13. 1 for completetheories .PrroposrrroN3.12.Let n> 1 ancl et T be a complete onsistent ) theoryin a recursiveanguage . Then T hasa A)r model.Proof. I i the language9+ is constructed fron I as in the proof ofTheorem 13.4,we constructa complete9"'-theory T"' extendingT withT* e A). The construction f the moclel ncl erification hat ts sttt isfactionrelationsa) is henachievedxactlys n Theorem 3.4.The dea s to const ruc t t reeas n Theorem13 '3 ' f { 'p , , 'f in i tesetof sentences f . ' r , we szlyQr. . . , cp1,s " ' -consistent

T * cpr " ' ' l cpr* 3v,,9(u,,)t 0(t , , ) lcp ccurs n (p\,

u

. . , c p , , ) i s aif f, (PrI

isconsistent.hepointhere s thztt, sing he act hat7-iscornplete,his


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7.lLe ritht.netizec[.ornpletertessheo enx r 8 9notion srecursiven I. Ttr is s becausefc l t ( c u , , . . ? c o t ) , . . , g t ( c t , , , . , c u , )

are 9*-sentences nvolving new constants 0r, . . , c : / i /ls showr-r,hcrr{ rp , , . . , e t } i s " ' -co r - rs is ten tf fT I - - l4x1 , . . , " , [A l_ ,cp , (x . r , )AA j=1( Iv , ,O , (u , , )0 ie iD ] .

whichholclsf anclonly fTlax, , , [A,;=,rp,(*, x,)A A j= Avr71(u,,), {) , (x,)) ] .

I t i s a lsoobv ious hat i f (p r , . . , (pk , s ' , ' - cons is ten tanc lo i s any _g . , . -sentence,heneither cpt" . , cpk, } is , , ,_co'siste ' t ,or {cp, . . , Er,- lo} is" ' -consistent.hus zlny " ' -consistentet czrnbe extencledo a completeextension * ofZ + {3v,, (v, ,)+ (t(c/)10e L*}.

Bttt I* canbe taken o be A)' by takingsome ixecl ecursive nurnerzrtionor,or, . . . of 9 'r ' -sentencesnd clef in ingr _ f o,*, i f { r , , , . . , T i ,T i ,o i+,} is ".-consistentc i + r - l r o , . 1

o t h e r w i s e .So r, can be complttecl singan oracle or' (an axiornatization f ) Z, thusT":{r i l i e N} is recursive n T, ancl a completeg' , ' -extensionof therequired heory,so Z ' , 'eA) by Lernrna 3.9. t r

This cl iscussiontr ings us bacl< o consicler inghe incompletenesstheorem (Chapter 3) ancl Tennenbaum's theorem (Sect io ir I I .3).Proposi t ion 3.12 ogetherwith Tennenbaum'sheoremcan be usecl oreprove heGocle l-Rosserncompletencssheorem r A-. (This ust i f iesthe ret-utrrkn Sect ion11.3 hart ennenbzunr 'sheoremcan be consiclereclas the moc le l - theore t i c anzr logue of t l ' re Goc le l -Rosser theorem.) Theargument oes ike th is. I f T were ar ecursivc ornplete onsistent xten-sion of PA- , then z woulclhave a recursive ocle l fu | , by proposi t ion13.12.SinceMFPA- we may regarcl he stanciarcl ocle las an in i t ia lscgment 9. p1.But N cannot reclef inccly a ) , formul ay7@,y , u) in ful(where0 e A,, , \ ,or i fYn e tu lQteN ) M Fay)Qt, , t t ) ) .

then he truth of any ), sentence zV(z) (whereVe Ao) oulclbe recr.rrsi-


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190velycleciclecly usinghe

The s nnclurcl,ss entrecrrrsiveat isfact iorre lat ion


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The urithrnetized contpleteness theorent T9I(c ) i f MFt : 'F ( . t , ,. . . , s r ) r where . i r , , s1 , M 'F ,s , s theGocle l-nLrmberf an g-ternt ' each , anclF is a /-ary unct ionsymbo l o f _y , then lV IEva l r ( t , 1 : ze -Jb r , . . , b1eN s. t .ri,Ii- ?(r- ; val,y.r,,u) : D;arrdN l= (rr3 . For a l lp , u M, i f ME 'p is theGoc le l -nJmbero fn 54- fo rmu la ,hen:( r ) i f MFp: rR( . r , , . . . 7s , ) t he re r , , ,s11r_ te rmszrnc l i sa re la t ionsynrbo lo f 9 (poss ib ly : ; t l - tenMFSat , p ,n )gl b , , . . , b , eN \ V I F4 1 : ,v ' l 1 y ( s ; ,) : 6 , a n c l FR ( b , , " . , ' h , ) ;(b) i f fu lFp:t (p,r ,r ' ) l or sorne 4-formulas p, Qe fu| , where r Aor V t l ten VI Sat" p, o)e M FSat, r l ,n),r ,Sat,v.r , ) ;( . ) i f MFp- ( - lq l fo r some _54- fo rmu laqeM rhenrizl Sat,yp, n)e MF I Sat, q, n);( .1 ) i f fu lFp : tQu ,cr l o r some efu r and someg- fo rmuraqeM,whereQ is V or I , then

MEsat (p,a)e M rea(crom"ta)(;) Satly(q,tb iD)(here, ake -> i f q:V, and take A otherwise).

Thus M strongly nterpretsN iff the truth of any 9-formula or value ofa,ny -term is given lry an ga-formula over M. Thus Corollary 13.8 saysthat, f 7 is an ari thmetic heoryandN FCon(T), hen here s a modelof Zthat is strongly nterpretecln N. As alreaclymentionecl he arithmetizeclcompletenessheorem s the formal izat ion f th is heorem'inpA.TttEotrervt3.13 The ari thmetizedompletenessheorem).Let fuIFPA,letL, 0, proot '7, , on(g) zr l l be as above.Then if MFCon(g) there is an9-structure l/ with clomaincM such that NFo for every (standarcl) 4-sentence where MF?(to1), and N is strongly nterpreted n M.* I f ,moreover,

MFYa?(t t l ; , ) )

lVlEVuJw, p (V < Ien w))lw),)A proof7(p, I A w -->rp,,1))where p,,is some canonically hosen) -sentenceexpressingthere are atlet tst e lements' ,hen he domainof. ulcanbe takenequal o the domainof //. n' "In fac t : l 'Sa t ,u ( - r , .y )s hc or rnu lan pur t o t ' thc l c l i r r i t i onl 'Ns t rong lyi r r tc rp lc tcc ln 4 ,1 'rvcw i l l h lvc MFV. r , (0 ( . r ) -Sut ,v ( . r ,) ) , rn i l t l o l k rws' r 'onrl r i s ha tNFolorevcry tanc larc l.Z:scrrtcnccr such hirtM E0(tot\ .

or


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192 Thestunclurcl.y.stemExercises'orSection13.213.9 Sl row hat here s a Scot tset ' , { ' L rchhata l l A e ,T areL\ t .( l lh i i . Lct l I ' - ,95y( / (1) or a 3 ! i cornp lc tc xtcns i r ;nf p . .1 . )13.10 (a) Let T= PA- have recurs ivex ionrat izat ion,inc lLrpposeclec ic lesl l11ancl I l ' sentences.how hat heproofof Theorent 3 .12 anbe mocl i f iec lo grvea recu rs i ve F{oe I I r l T fo } in wh ich

2, - tp ,(ct) lq(t) lM Fa@), cp Z]is recursiveor each uple rie M of finite ength.( b ) U s e ( a ) t o g i v e a n e w p r o o f o f C o r o l l a r y 3 . I l b a s e c ln t h e a r g u r n e n t f o l t o w i n gthe proof of Theorern13.12.13.11+ (a) Def inea provablyecul 's iveunct ionn PA, F(n) : (e , ,1 ,whose aluesareGodel-numbers f 9*-sentences xpressingthereare at least r elements'.(Take or rp,,he sentence

3V1y ,; , , v,, 1X\;*r-T u, vr)bracketedn somecanonicalway.)( b ) I f p r o o f T ; ( x , y ) d e n o t e s ' x i s t h e G o d e l - n u m b e r o fp r o o f o f y i n t h e p r e d i c a t ecalculus.or9n', show hat

PAFY a)p proofy,^(p,(o -- V n]),whereo is a suitable finite)conjunction f axioms f(c) Convince ourself hat,wereyour ife o depend nTheorem13.13wi th a l l deta i ls .I3 . I2* Let M,N be 96-structures,v lFPA, NFPA-interpretedn M. Definean embedding of M into N

P A - .i t , youcouldg ivea proofofand suppose is s t ronglyby

. f(0",)0";f (t + '' ltut): f(x) + N1N.

Show hat/ is c le f inab len M and ernbedsM onto an in i t ia lsegment f N. UseTarsk i 's heorem n the undef inabi l i tyf t ru th o declucehat he embedding isnot e lementary.13 .13 Def ine a fo rmu la 0 ( * , y ) exp ress ing ' x i s t he Goc le l -nu rnber o f an e tx ion ro f1),' . FIence efinea formula p(a) epresenting1.f,,is cotrsistent'. et fuIFPAbenonstanclarcl. se Exercise10.8 ancl overspi l l o f ind nonstanclat'cle .421 ithlvlFy@). Decluce hat lvl has a proper end-extension F PA that is stronglyirrterpretedn M.

13 the Standard System

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