11 saturation

7/28/2019 11 Saturation

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L7Saturation

The sea w:is wet as wet coulclbe .LcwisCalloll. T hc Wulru.s ntl lrc ('urpctttcr

Recall rom Chapter8 that, givena theory T, a typeouer T is a setp(t) r>fformulasn f in i te lymany ree-var iablessuch hat f Up@) sconsistent.fMFT then the typep(i) is e i ther eul ized r onr i t ted y M, and there salways omemodelof ?"that eal izes ( i ) .ClrapterB showed hat the modelsK7 (for complete ;


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r42 SaturutiortClearlyM realtzes(,r) i f f S is cocledn fu\,soquestions bout SSy(M canbe recluceclo quest ionsabout which typesare real izecl n M. I f S isrecttrsiue,hough, ly Corollary3.7 therc is a X, formula 0(x) strch hatt

n e S) PA- | 0 (n )and

n 4S) PA- r -10 (n )so hatp(x) s real izedn M just n casehe ype

ct(x): (x),+0- 'g( l);e ulis realizecln M. Now for eachn e N we have

NIFSxY< n( (x) ,+0e 0( i ) )usingLemmet .8.Thus by overspi l l

M EAxY < b((*),+ 0 N in M. Then there s a e M satisfyingM r V i < b ( ( n ) , + 0 < + 0 ( i ) )

andclear l ! ,s inceb>N, a codes he set S'Thuswe haveLpNrvrn 1.1. I f s q N is recursive hen s is coded inmodels f PA.

There s alsoan interest ing onvorseo Lemma 1t '1 'Levun 1I.2. For eachnon-recursiveqN there s

a nonstandard odelMF-PAsuch hat 5 s not coded n M.P r o o J ' .e t o o , o t , . . . , o i , . . ( l e N ) b e a n e n t t m e r a t i o n f a l l9o-sent"nces.We shal lconstruct complete xteusion7+Th(N) of PAsu ch ha t S . i s no t co d e d n K7 . T w i l i be P A U t z , , ,t , . . , r i , . ' ' ) f o rsuitablychosen9o-sentencesi.- in.'h6t sentence, 1y,SchosenSo hat PA + q, is consistent nd N Fq,'(Thiswi l l force T+Th(N), anclhenceKr+N. Such211xists y Corol lary

provedall nonstandardn

3. 0 .Supposei: PA U r,,,notof he ormSxcP(x)or . . , r i j hasbeen ounc l nc l s cons is ten t 'f o ' i ssomevariable r ancl ome ormula cp(x),henrve


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Codedsets 1.43let r ,r , be either ,or -1o;,chosen o hatT,+r,*1 sconsistent. therwlsesupposei i saxcp (x ) .f T ,+ - l o , i scons i s ten t l e t r ; *1be -1o , ,nd i f ,+ -To , i sinconsistenthen T;I o, mustbe consistent,s 7 r is. In this lastcasesuppose

T,+cp(x) ,r);:0 f o r a l l 7 { Sand

T,+ E@) - x); 0 fo r a l l i +S.then S woulclbe recursive, incewe couldcomputewhetheror not Te S bysearchingor the f i rs t proof of e i ther x) i :0 or (*) i+0 from Ti+E@),which is a recursively xiomatized heory, being a finite extensionof PA.By hypothesis,S is not recursive, o ei ther Ti+]x(cp(t)A(x)1+0) isconsistentor some G.S,or T;+ax(E@)A(r); :0) is consistentor some/S.Whichever t i s , e t r ; *1be the newsen tenceound n th isway.At the end of this constructionwe will have obtained a completecons is ten tx tens ion :PA*{ r r , r t , . . . } o f PA. (Z i s comp le te ecauseeither r i , ,1 lo i o1.r ,* 1 l-1o, or each i e N, and {o ' e ru} contains al l9o-sentences.)f a e K7 thena is definedby some9n-formula cp(x),whereT fSlxq(x). But then ,vg(x) is o, for some e N, and sinceT faxE@) wemust have: ei ther r i*r is ax@@)A(x);+0) for some 7SS or r i+t isAx@(x)A(") i :0) for some e S; and KTFr;*1.Thus a (being he uniqueelementof 1(," atisfying p(x)) annot code the set S. This applies o alla e K7,and thus S ( SSy(Kr).

It is worth mentioning hat every nonstandardMIPA codessomenon-recursiveet,since y nduction necaneasi ly how hatPA FVxAyV < x((y),+ 0


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144 SuturationThe fo l lowing ar iat ion f Lemma11.1wi l l occasional lye r-rsefu l .Leptua 11.3. Let f : N+N be recursive, ncl et NIFPA be nonstanclard.Then there s be I , lsuch that uIF(b), , :" f (n) o l a l l ne N. I f the rnage f fI r r r ( i ) : { f ( r t ) l i ze N} is a recurs ive e t , then the reex is ts in M so Lha tI m ( f ) : { ne N IMF x < c ( b ) , : n l .Proof. Given/:N--+N recursive, y Theorem 3.3 there s a A,, ormula0(* ,y , z ) such ha t , o r a l ln ,n e N,

N F320(n f f i ,z ) f (n ) : m.Henceby Corol lary2.9 and Lemma5.9

MEJbVx, y , z < i (0 (x ,y , z ) - , (b ) , y )fo r a l l eN. Henceby oversp i l lhe re s be M anc l eM \N such ha t

MFYx, y , z


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Coded sets 145th t r tMFVy< c( (a ) ,#0eAx


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14611.3 Showhat or al lMIPA

Saturation

S . t y ( M ){ S g N l S : { n e N l l v t F p r ( n ) l a }om e eM}11 l (a) Let 6(w,x) be the Ar(PA) formulathexth d ig i to f the b inaryexpansrono f w i s 1 'o f thep roo fo f Lemma10 .6 , ndwr i te ' xw ' for6 (w ,x ) . Show hat

S S y ( M ) : { S s N l S : { n e N l M f n e c } f o r s o m e e M }for eachmodelMFPA.(b) Let lv lFPA and let (V,e) be the model or the language f set heorywi thdomainof .V: domainof M andbinary e lat ion g ivenby (V, t )ExeyQMFxey.Show hat (V, e) satisfies l l axiomsof ZFC except he axiomof infinity.(c) Let (V,e) be a model of al l axiomsof ZFC except nfinity, which s false n(V,t). Interpreting ty as the xth digit of the binary expansion f y is 1' define0 , 1 , + , . , ( on (V , ) so ha t V ,0 ,1 , * , . , < )F PA. Check ha t heopera t ionsn(b) and (c) hereare nverseso eachother.11 5 Verify that

nt-Th(M):{ 'o t lo is a I I , sentence nd MFolis coded n al l nonstandard EPA.

L I , 2 ' , - R E C U R S I V E S A T U R A T I O NWe now eturn o consideringypes.We havealready iscussedhenotionof a typeovera theory; t will be useful o havea notionof a typeover amodel oo.DEprNrrroN.et MIPA. A typeouerM isa setp(t) of formulas @,a) of9oU {a} (wherex is a fixed finite tuple of . ree-uariables nda is a fixedfinite uple of. arameters,.e. aeM and eacha, in a is considereds aconstant ymboln the expandedanguaget,U {a}with its natural nter-pretationn M) such hatp(x) is initety atisfiedn M, i.e.,

M EAi /^!=, E,( i , a)f o r each i n i te ubse t , ( i ,A ) , . . . , q * ( i , r ? ) o f p ( i ) .p ( i ) i s a ) , , t ype(respectivly a II,,type) ff eachq(i, a)p(i) is ,, (respectivelyI,,). (*)is recursiuef f the set

{ ,EG,) ' lE(*,a)ep(f)} Nis recursive,wherey is some ixecl uple of variables rom 9o disjointwithx, and replacing he parametersA in p(i). Finzilly,at ype p(f) over M is


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2,,-recursiueaturation I41cornpleteff for all formulasE(i, ri) involving he same ree-variables asp(i) anct he samepzrrametersasp(t) ,

for some ini tesubset 0( i , a) ,

A0 k(i , a) -> cp(i ,a))A0 k(i , a) - - tcp@, )). . . ,00 ( i , a ) j o f p ( i ) .

Not ice hat, when tal lc ing bouta type over a model,we al low f in i tetymanyparameters from the model o appear n formulas n the type. (Inabstractmodel heory, nf in i te lymanypzrrametersre sometimes l lowed,but for the purposes f this book we shallrestictourselveso only finitelymanyparameters.)It is easy o check (using he compactnessheorem) that p(x) is a typeover M (with parameters e M) accordingo this newdefinition ff p(x) is atype over Th(M, a) (the theory of the structureM expandedby addingconstantsor eachelementof r i ) accordingo the def in i t ion n Sect ion .1.

Den'rNrrroN: model MFPA- is 2,,-recttrsiuel yaturatedff every recut-sive ,, type over M is realized n M. The model M is recursiuey saturatediff every recursive ype over M is realized n M.This definitionof a model M being ecursively aturated eallyhas itt leto do wit h the language9,6 all that is required s that the languageof M

hassomesuitableGodel-numbering.husbeforeproving he next resul t ,that everymodelM hasa recursively aturated lementary xtension,weshalldigress lightlyand discuss ecursiveanguages.Let I be a language or the predicatecalculus.Then -7 contains hesymbolsA y - t : , O l V a n d v ; ( l eN )

togetherwith extra constant ymbols 1 ie1), re lat ionsymbols\( ieJ),and funct ionsymbolsF1(kK), where ,J, K are (possib ly mpy) indexsets.Suppose lso hat, for each eJ and each e /(, the symbolR, hasarityrnreN ancl he symbolFe hasarity noe N.DeprNrrror.r. he first-order anguaget above s ecursiueff there s a 1-1functionz from the set of all symbols of I to N such hat eachof

{ r (c , )l e1}s N{( r (n) ,n) l e / } s N2{0( f* ) , n t ) lke ( } g N2, ancl{r(r) ie u} ru

e i t l re r MEVI(O( i , a )Aor MFVt(0( i , ct )


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148 Sutttrutionelre ecursive ets. Note hatsince vis 1-1, hismeans hat.1, . / , ( rnust1 l lbe countable.)Given suchzi unctiony, we canclefine heGodel-nttmlteofa str ingof symbols r0.rr. . . !1 f L asbeing

t . t r , , t r ' . s i . l I r ( r , , ) , y (s , ) , . . . 77( , rk ) ] ,wheref*u, , , . . . , . r7. ]s the funct ionclef inedn Sect ion .1Notice hat f I ,J, K are a l l f in i tesets hen t is easy o construct uchafunct ion v. Thus any language with only f in i te ly many nonlogicalsymbolss a recursiveanguage.f I is a recursiveanguaget is easy oconstructecursive lgor i t l imshat decidewhether 'x s the Godel-numbero f an 9- te rm ' , ' . . . o f an 9- fo rmu la ' , e tc . For examp le he se t S ofGodel-numbers f 9-terms s defined o be the leastset such hat:

ififif

x e { v ( u , ) l i e N } h e nx e { v ( c , ) l i e l } r h e n(x,y) e { (y(F^)n1,) lk[* , r ( ( ) ] nr ' n [ r ( , ) ] .

[x] e S;[x] e S;I q a n d s r , . , s u S t h e n. " [ r ( , ) ]" ' ^ [ r ( ) ) ]

S

Note too that (by relabeling ariables f necessary) e may alsoassumethat the functionz above or a recursiveanguage54satisfies(v,,) y(vr)


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2,,- ecttrsue stturutrcn t49ProoJ.For each t r ,neN there z t reco t tn tz rb ly any se tsp ( .xu , .. ,x , , , "I n , . . . , 1 , , )o f f c l r t l t t l a sl ( x 1 , ,. . , x , , , , 1 0 , . . . , 1 , , , )i t h at m o s t h e r e e -var iables hown,anclwhose etof Gocle l-numbers'rp(t , l t l rp ep(i ,y) i isrecr r rs ive .he co l lec t ion f a l l suchse tsover a l l t t , rneN is a coun tab leul io lr of coupt lb lesets, o s a lsocountable. numeratehiscol lect ion yp u ( i , ) . p , ( i , t ) 1 . . , p i ( i , t ) , . . . so h a t he r e e - va r i a b l e sn p , ( i , y ) a r eJ g , . . . , X , , , i , o , . . . " 1 , , 1 .Now let M,,--M. SinceM is infrn i te, zr rc lM'" ' ) :carcl( lVl) , whereM' '" 'deno teshe seto f tup les : (u1 , , . , ( t , , , )eM >f ome in i te eng thm* L .Associate i th eachpairp,(x, l ) and c7: (u11. , a, , , )m;* I new constantsymbols i . , i . r , i . , , . t , , c i . , i . , , , ; ,f ld in aclcl i t iono these onstztntymbols)acld to .9 further constzrnts aming each element u e Mo. The resultingla lguagehascarcl inal i tyarrd(M11).e nowconsiderhe ol lowing heoryZin th isexpandedanguage: hasaxioms(a) 0(a) for everyg-fomrul a 0(i) ancleveryu e M, sLrchhat MrF 0@);(b) E( . , . r i .0 r ,c i . , i . , , , , ,7 ) o l ' eve ryEG, t )p ; , eYQrye N a t td everyct11,1 a,,e M such hat p(i, ar, . , a,,,)s a type overMu'This theory is consistentby the definition of 'type over M11' nc' l hecomplctn"r, h"or"m, so t hasa modelNI ,of cardinal i ty t mostcard(M).By i ient i fy ingeachaeM{tancl he element f M, real iz inghe constantorc we may assumeM 12M,,ancl learlyM, reahzes very ecursive ype overM, , .Con t inu ingn th iswaywe ob ta ina cha in u lu


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n * 1 coniuncts(EG,Q /\cp( i ,a) )A ' ' ' A ,p( t . ) )

150Theorem .11.

q @ : { (

SaturationNow consider he type

' rp(t,t) 'n e

q(x) is clearlyequivalent in the predicate alculus or 9) to the typep(i);ho*"u . r , q ( ; ) i s recurs iveanc l o t us t r .e . ) s ince 'V( i ,a )eq(* ) ' can beclecided y the following algorithm:

compute he largest nteger n e N such that' r ! ( i ,y) ' ' ( . . . (q( t , )n E@,))n ' ' ' Acp(t , ) )1(n+I conjuncts) or some 9-formula 'q( i , l ) ; compute (n), and if'E ( i , i l ' : f (n ) then p( i ,a )eq( i ) - i f no t , then rp (x , )QqG) '

(This observationhat any r.e. setof formulas s equivalent o a recursiveset is known as craig's tr ick.) similarly 5: {tcp(*, )1lrp(*, ci)ep(,r)} isequivalent o a primitive recursive etof formulas,by considering,(r):kfor somenaturalnotionof the numberof steps computationor f(m)takes.Thus 'recursivesaturation' urns out to be equivalent o 'r'e'saturation'nd'primit iveecursiveaturation' 'Propositioni.q provicles s with many ecursivelyaturatedmodels,but nbt all modelsare recursively aturated'For example, f T is acompleteonsistentxtension f PA, K7 s not recursivelyaturated inceit fails o realizehe following ecursiveype:

p(x) : {a Iy0 y) -->10(x) |0 *) an t o'for mu zt}

:/(")1N )

f ( m ) : ' E ( i . y ) ' );:'o' rot (m) J

However,we haveTuEoneu 1.5. et MFPA be nonstandarcl.hen&/saturateclor al l n e N'Proof Suppose (;) is a recursive I,,-type ver M withSince

is l,,-recursivelYparametes c1 M .

n* | conjuncts(rp(t, ) Acp(*,d))n . . A E@,a))

fo r somem{nand the computhalts n


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2,,-rec:ttrsiuectturcLtion i5 Ii s rcc t r rs ivc , . ts coc lec ln M l - ty ome :e lv l . y l -e rn rnul . l . S incep( r ) sf i r r i tc ly at isf iedn lVI,

IVI aiV y < k((.).,* 0 --->Sat:;,( /, *, rZ])fo r a l l ke N. By oversp i l lhe re s 11>N n &1sr . rchhat

tu lFaiV y < tl((c),, 0 ---Satr.,,(y, f* . a111.Lc t 6 e MFvy< d( (c) , , *0 >sar ; , (y , fh , r t ] ) ) rnc lsL rppose( t , t i ) p ( i ) .Ther r pu t t ing [ : [cp ( t , r ) ]


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152 Suturtttion- l ' l reorem11.6 o cLef ine.5.S/(&1),o hese eople 'uvott lc lle f ine ^5)(N)o be

{ {n e N | u O(n;1 | an -7 in- formula} .Th is , potent ia l ly ,cot t lc l eac l o coufus ion,bt t t in pract iceSSy(N) i : ;harc l lyever usecl .

Exercises or Section I 1.2t I .6 Show hateverynonstandaclMF PA is 2 , , - tu l l for l l n , i .e .wheneveri e Mis a tupleof finite ength' thertK"(M;r7) s not col lnal n M'l l . l* A nonstanclarclroclelMF PA is tctl l i f f or al l d e M, a tupleof finite ength,I(NI;r7) snotcofinal n M. otherwiseM isshor(.M isshort-rccur:; iuelyutttrutecl i f l ievery ecursiveypeover M of the form

p G ) : { * , < b l < i < k \ u { c p 1 G , r 7 ) l iN } ,w h e r et , b e & I a n c l: ( r , , . . . , x r ) , i s e a l i z e c ln l v l '(a) Show tl iat every recursively aturatedMFPA is tal l ancl short-rectrrsivelysaturated.(b) Sl row hat, f M N '1l.g', (aj let $ l-;ea first-orcleranguagewith cr-rnstantyrnbols ;(ie 1)' i trrelation ymbolsRiof ari tyrnlQ eJ) ancl trnction ymbols 1. f ari tyn1,(ke )' L'etv1ancl , bothsat i ; fy heconcl i t ionsl l p .147. .e.bothare 1- l f t tnct ionsrorn hesynrbols f l l into N s.t.

{u , ( . ' )i e } { u , ( . ' ) l i e l it ( u , (R , )m) l i e \ { ( r , (R , )m) l j e l \{ ( r , ( f * )n) lke I( l { ( , ,2(F*) ,r7 , ) lk( }{u ' (u , )l eN} { r , (u , )i eN}


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Tenne bau n' ; theo enxareal l recursive,ndalso hat

r53v ' ( v i ) < y r ( , r i ,r ) z , ( v1 ) y : ( v i r r )

for each e N. Def ineGoclc l -nunrbcr ingsl anclNr on str ings.r r . r :. . , r1of symLro lsof .9 byN,( , r1 . r2. . .sr ) lyr ( .sr ) ,r ( . r : ) , . . , y r ( , r / ) lN2( . r '1s2. . s) : [yr (s , ) , : (s : ) , . . , y2( , r / )1 .

Giverr hat Z is anysetof strings f symbols rorn9, show hatNr(7') {Nr(r)l e T)is recurs ive n N,(7) : {N,( r ) l reT} and henceN,(7) is rectr rs ivef f N2(?") srecr t rs ive.husN; anclN2 'z lgree 'boutwhich ets f formulas,erms, tc.of L arerecursive.(b) Given hatS (as n par t a)above)sa recurs iveanguage,how hat here s anin ject ion frornsyrnbols f - f to N such hatz(v) 1v(v i *1) or a l l r , and hat

and

{ r , (c)i e 1}{(r(R,,) , , ) leJ}{ ( r ( r^ ) , 1 ) lke }

{ t , (u,)l e N}aleal lpr imit iveecursive.I I . 3 T E N N E N B A U I V I ' SH E O R E MThe central esultof thissection s that nonstanclarcl oclels f PA are notrecursive, a resul t f i rs t provecl by Stanley Tennenbaurn (1959).Tennenbaum'sheoremmay be regarcled s the model-theoret ic lnalogLleof the Godel-Rosserncompletenessheorem,aswe shal lsee.First we must saywhat t means or an 5Ja-strtrctur eo be recursive. etM be an J4o-structure:VI s recttrsiueff there are recursive unctions@,O:N2- - -N , a b iner ryecurs ivee le r t ion -N2 anc lna tu ra lnu tnbers , , ,rz ,eN such l ' ra t

( N , @ , e , e , n r , n , ) = M .In part icular, ote that only countable a-struct lu 'esanbe recursive.Trrronevr 11,.7 Tennenbaum'sheorem).Let MFPtI be nonstztnclarcl .T lren&1 s not recursive.n fact f M = (N, @, O, @ , f i11,,) thenneither@nor e is recursive.


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154 SuturutiortProof. By the rematr l


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1 5 5sLrpp l ies ls

2inclg : {rct)|o is an .9 o-sentene and pA- F-1o}

'4 , B are c lea r ly .e .anc l i s jo in t , ; ince A- i s f in i te ly x iomzi t i zec lnc lcons is ten t . I f : , z l i s recLr rs ivenc lC l t B :A , then we mely lssume is aset of Gocle l-numbet 'sf J4o-sentencesby ieplacingC with the recursiveset C': c f - l to l lo is an J/o-sentence)f necessary). ow consider he set

Tannenbcttt t't'; the t entw i thsucha pa i r ,name ly he r .e . se ts

A : { to l l n is an .Zo-sen tencenc lPA- | o l

I I l o has ro ead ing s , i sC ' : 1 |L t t - t - - ' l o t | t - t o l eC a l c l t , s t q Cu -i:-Lt l r " ' t o r

uoo' ]/c is even, anclht is no leacl ing ls,

r - l o r + Co r ( o t e C .ei tner)

we leave it to the rezrcler s an exercise o check that (a) C' is simplyconsistent,.e. , or no J4o-sentenceis both [r ] ancl[-1r] n C'; (b) for a l l9n-sentences , either lr le C" or [-tr] C"; (.) C' :,4 anclC" n B : c/; ancl(d) i f C is recursivehenso s C' . But then by Theorem3.9, C'cannot berecursive, o C is not recursive i ther.The existence f ci is jo int , .e. , recursivelynseparableetsA, B can beprovedby moredirect ecursion-theoret icrrgurnentsoo. See or exarnple(Cu t land ,1980 , . 133) r (Rogers , I96J, . 170) .We nowshowhow heseicleas anbe useclo givea more directprooofof Theorem l.T, appl icablein a wider sett ing.TIIponev 11.8. Let MEPA- be nonstanclard nclsuppose&1 sat isf iesoverspill for d,, f ormulas zrncl he stanclarcl ut N. (For exarnple anyA,y-recursivelyaturatecl odelMF PA-, or e lnymocle l f IA,,wi l l have hisproperty.)Then &1 s not rectrrsive.Proof. Let A, Bc.N be cl is. joint,.e., and recursivelynseparable.incethcyare r .e . hereare4, , onnulas r ( r ,y)andp(x,2) such hat , or a l ln e N .

n e A ( ) NFA ya @ ,y )n e B+NFaZ{}Qt,).

itnrl


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156Thus or a l l ceN,

Satnrution

MFYx,,2< lc-1(a(x,) Af i (x" ) ) .since his sentences A,, ancl rue in N, usingTheorem2.7- Applyingoversp i l lo ( l , ) , heresnlN in M such hat

MEYx,, z< u- l (a (x ,) AB@, ) ) ' ( + )Le t C be t hese t ne N l M F f t < a a ( n , y ) } .Then CaA, f o r i f n e A t h e nN F a ( n ,m ) f o rs o m em e N , h e n c er y F a t < c t u ( n , y ) ; n dC n B : Q ' f o r fne B ' then N f p (n , ) fo r some I e N, henceMFaZN andseM such hat

M FYu < bau< a lw < a(0 t t , t ,w)A (3 < ' l ( u ' r s )* a t < ua(u 'y ) ) ) 'T h t r s6 : { n e N l M F S t ( t + t + ' " + t : s ) ( p r ( n ) r s ) } a n c l ence f @ i s r e -cu rs ivehenC is r .e . ,s ince

C : { n e N l 3 r e N r @ r @ @ : n. (pr(n) s))

( ,)

for somesuitable ,e N.D:N \C is a lso .e . by a s im i la ra rgument ,wherea( t t ,y ) ' in p lace f 'ay


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Tennenbaurn' theo ent r57reasonable ay, th is proof of Tennenbaum'sheorernseems he mostnaturalapproach.'I 'ennerrbaum'stheclremciln be extenclecl ncl noclifiecln many w?lys,especia l ly i th the useof Result7.8, and he exercisesol lowins ake thereacfern someof these l i rect ions.n Sect ion 13.2 'we i l l reconsiderherelat ionship etween he Gocle l-RosserheoremanclTheorem l I .7: i tturns out that there is a stra ightforwarclroof of Corol lary 3.10 fromTheorem11.7usinga rnodel- theoret iconstruct ion!

Exe cise.sfo r Section I 1.311.10 Let" vlFPA be nonstandarcl nd countable.Define a recursivebinaryre la t ion@ c N2such ha tM | < : (N , e ) .1 1 . 1 1 * L e t M = ( N , ) , O , @ , n r , n ) F P A * i l , , - T h ( N ) b e n o n s t a n d a r d ,h e r en> I . Show hat ne i ther @y:z no r xOy :z i s de f i ned y a d , * , (N) fo rmu lao(x,y , z) .(Hint: Let A, BcN be definecl y ),,*,(N) formulasa(x),l l /) respectively uchthat for no A,,* , (N) ormula p( , r ) oesNFdx(a(x)- - - rp(x) )AVyl ( f r (y)Arp(y)) .A, B cttn be found by consideringExercise9.12, for example. Now use thefol lowing overspi l l principle (which is true for al l nonstandardtvtFPA * i l ,,- Th(N))

i f NF9(k) for a l l ke N, where 0 is E, ,n , thenMFYx < b1(x) for someb > N.)

Il . l2" Let MF PA- be a nonstandarduch hat or al l .X1ormulas (;) t here s an31 formrrla /(x) such hat MFVi(9(*)q,@)). For example,by Result7.8, anymodel of.PA has his property.)Show that M is not recursive.(Hint. SupposeM is recursive.(a) Show hat f I is 4 , , and ie M, then rzeNl tWf 0(n,r i ) } s recurs ive.( b ) G i v e n h a t . 4 : { x N l N F a y u ( x , l ) } a n c lB : { x e N l N F a z p @ , Z ) }a r e r . e . ,disjoint,and recursivelynseparatrlewherea anclB ared,,) considerhe set

D --{be tvll V FYx. z< b(-lct(x,)y-1|(x. z))\.Given hatb e D isnonstzrnclarcl,l iow hat

6 : {n e N IM FAv< bct(n, )}

is recursive ncl eparates4, B.


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158 Suturutirtrt(c) Given hat no b e D is nonsternclarcl,how hat or any9,y-fot'mulup(:r)herc ser ,, fornrultt p(x, ) such hat, for ttny r VI\.N,

Vr e N!(N t fQt)CA,IF ( i r ,c) ) . )I . I3 Let v lFPA be nonstanclarc l .s ingRestr l t .8and Exerc ise .10show hati lrerearepolynomials(;) , q(t) e N[f ] sLrchhat or no bi jection' : N -' M is he set

{ r ieNlMrp( l ' (n , ) ,. . ,1 ' ( r , , ) )+qU'@,) 1fu, ) ) Idefined y a fl l formula n N.(Hittt; Let S:{tot lo is I11or -I1and vl lo\ be codedby a e IVI tndlet p, q satisfy

VI Vx( (a ) , *O

11 saturation

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