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7/28/2019 7 Collection

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7Collection

'A re f i ve n ightswal 'mer han one n ight , hen? 'A l i ce ventured o as l< .'F ive t i tnesas warm of cottrse. ''But they shor,r lc le f ive t imes as colcl ,by the same rule- '' Jus tso l ' c r ied he Rec[Queen. F ive t imesas warm, uncL ve t imes ascolcl- iust zls 'm ftve t inles as rich as you, unl f ive t imes as clever! '

LewisCarrol lFrom:Through he ooking glas:; nclwhatAlice ottnd there

The col lect ion axion?s are the ari thmetic analogue of Fraenkel 's colLect ionprinciple n set theory. Fraenkel'sprinciple says hat if / is a functionidenneAby a first-orderormula)whosedomain sa setx, then he rzrnge f/can be col lectecl 'together sa subset f some et . Thispr incip le whichwasa very powerfu laddit ion o Zermelo 's r ig inal et heory) 'whenputinto he contextof ar i thmetic, ecomes:

an 9o-clefinableunction/with finitedomainhas initerange. ( " )

i t is easy o see hat (" ') s true n N, and n fact thesearithmetic ollectionaxiomsare provable rom P.4. Surprisingly,hey are very powerful' anclareactuallyequivalent o the ncfi-rct ioncheme ver the weak theory1A,,.Collection laysan important ole n the model heory of.PA, and we willexplore spects f th is n both hischapter nd he next.Anothei appl icat ion f thecol lect ion xiomss to show hat he ormulaclasses, and 1, areclosed ncler ouncled 1r-rantification.ore generally,.I1and II1 form the bottom evelof a hierarchyof formulzt lasses 2, fI2,2., f l . , . . . , ancl ollection ellsus hat eachof these lassess closecl nclerbouncleclual t i f icat ion.We shal l tartby descr ib inghishierarchy nd heroleoI col lect ionhere.

7 , 1 T H E A R I T H M E T I C I _ I I E R A R C F I YRecal l hat A,r s the class f a l l bounderlormulas, .e. formulaswhoseq u a n t i f i e r sl l o ccu ras V x< t . . . o r 3 , r< t ' . . . F o r co n ve n i e n cea n c lto make he ol lowine lef in i t ion oreelegant) ezr lso enoteA, 'L ly 'ancl1B


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The urithrnetic h e nrchy1I,,. The fol lolving clefinit ion clefinesclasses ,, and 11,,inc luct ion n r?.Dr-,rrNrl toN. forrnula sforrnula s I1, ,* ' f f i t is ofThusa formula s I, , i f f

t , ,*r i f f i t is of the ornraicp( i ,tlre form Yicp(i,y) with rp.2,,.i t loo lcsike

79for a l l r re N by

y) with cp fI,,',Lr

V accorclingoIooks i l ce

f i rVtr f i r . . . Qi, ,E( i1,2, 13,wlrere all cluantifiersn (p are bor"rrrcledarrdwhethern is oclcl r even).Sirn i lar ly formula

, ,,, )Q i s f o ris I1,, ff it

V t r f i2v i r . . . Q i , ,cp ( i r ,: , i r , . . . , i , , , ) ,We al low blocksof quant i f ierso be ernpty,so that for example ny 11,,fo rmu la s zru tomat ica l l yo th | I , , *1anc l , r+ r .We sha l la lsosay ha t aformLrla (;) isZ, or 11,,f i t rsequiualento a -I,,or a If,, ormLrlap(x) n thetheoryZ or the modelM we are consicler ing,.e. i f

Trvi(0(x) E(r))or

M rv i(o(x) q(;))When i t is important o indicate he theory or mocle l n which thisequivalenceakesplacewe shal ldo th is thus: 2,,(T), i l , , (T),2, ,(M) orn,,(M). Usual ly,however, or Mlwi l l be clear rom the context.A formula s A,, tf i t isequivalento both a ),, formulaanda II,, ormula.As before, hese quivalencesakeplace n some ixeclheoryT or mocle lfu | , and, e indicate his (wherenecessary)y A,,(Z)or A,,(M).Thus, orexample,by Corol lary3.5 al l recursive ets are representecly A,(N)formulas.Not ice that every ormula s equivalent in the predicate alcul trs)osome ,, or 11,,ormula. ndeecl , y prenex ormal orm (see, or example,Bridge 1977, p.5a-6) any ormr,r la(x) is equivalento a formulaoI theform

Qr l rQ: / : " ' Q , , l , , cP(Y ,)for some l N, where psquant i f ier- freenclQr sei the V or f for each .Thr,rs is 1, ,or. I , ,or some z N. Also,sincewe al lowquant i f ier locks jor Vi to be empty n the above lef in i t ion, e have l ie obvicl t tst tc lusionsI , ,EA, ,+rc) , * ' z tnc l1 , ,cL ' , , r rg i l , ,+q .igu re4 shows hesenc lus i t lnsncliagramaticorm.The hierarchy f fornr-rla lztsses',,.r,,,A,,, . is calleclthe aritltrneti hierttrchy


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BO Collectiort\ . /

0 , 1, / \, / \' \ , /\^ ,,'. L \ t/ t \

t \t /\ , /A ,

, / \\ , /\ n /a l

f I r

n

t .

I .

L 1

[ , , A1 r,,Fig. 4. The ari thrnetichielarchy

"fhe classes ,, and il , , are closed under (finite) conjtlnctionsanddisjunctions. or example,whenever0'(x) and 02(i) are.I,, here are E,,formulas quivalent in the predicate alculus)o 01(x)A9r( i ) , and simi-lar ly or fT,, .To ee hissuppose 1(x)sa!,Yyt ' ' 'Q/, , cP(i .y) and03(x)sa | t v zz . . . Q 2 , , cp z@ , 2 )i t h e r , e z L o a n c lQ : l o r V a cco r d i n gowhether z s oclcl r even, hen by renaming ariables f necessaryo thatthere are no variables ommon o y ald 2 we see hat O{i)A0r( i ) isequivalentoart ,ztvr2,z Qy, , , , , (QG,y)A Et@,Z) ) ;

a simi larargurnent ppl ieso al l the othercases.If 0(t) is ,,, then_l0(;) is equivalent in the predicate alculus gain) oa I1, ,ormula.For examplef 0( i ) is3y,Vy2' ' Qy,, @,r) with q e A,, henl g i s e q u i va l e n to V y ' 1 a y t " ' Q ' ! , , - l r p ( i , t )w h e r eQ ' i s h e ' c l u a l ' o tQ ,i .e.Q' is V i f Q is 3 andQ' is otherwise. imi lar lyt 0( i ) is I , , hen 10(t)is I,, . This shows hat A,, is closed under negations,and by the lastparagraph ,, s also losecJnclerA ancl . (Flowever, or n > I neither t,nor II,, areclosecl ncler l. For example f .Ir(N) wasclosecltncler, everyr.e.setA g N wouldhavean r.e. complement, ndsowoulcl e recursive!)Usirrg he tuir ingunct ion x, , xr) : (x,- l -xr)(x,* x2l1)12+x2 see '62)and the cleriveclunctious x,, . . ,x,,)coding uples n PA we See hat zr I,,forrnulaay,vyr. . .Q/, ,q( i , t ) is equivale l t ( i1 PA) to t i formula n I, ,


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The arithmetichierarchY Bform btrt with only onecluuntffiern eachbloclc.For example he formulaabove s equivalento

az1V 2 ' ' ' Qz, , [3, < z ,3 iz { zz ' i Jy , , { , ,A i= , , (Z i( t , ) ) 'AE( t , l t ) ]where the formula in squarebrackets s clearly A1y. imilarly every II,,formr.rlas equivalent in PA) to a formula in fI,, form with only onequant i f iern eachblock.In fact the equivalencesn the last paragraph re provable n a ratherwezrkragmentof PA. All that is needed s

Vxr, x2al.y(xt,zl: yVyl !x1 xz(xt,xz): !Vx1, 2, ( (x , ,xz l | " ) x ,xz< 1l )

to be provable, nd t is a simpleexerciseo check hat these entencesreindeedprovable n 1A1ynly.The collectionaxiomsThe collection xiomsallowus to show hat the formulaclasses,,, f I , ,andA,, areclosedunderboundedquantification.Dnnrutrtox. Let e(x, y , 2) be a formula of .56. The collection rxiom o' Eis the sentence E, which s

Y2, t (V < EyE(x, y, 2) '3sVx < t31< sq(x, , Z)) '(Str ict ly,we should ndicate ,y rn the notat ionby'8, .r(P' ; however,noconfusionwill arise rom just using BE'.)

Notice hat the 'converse'of.BE,Y2, t( lsYx < E, < sE(x, , Z) 'V x < tSycp(x,, i ) )

is valicl n all $a-structures. lso notice hat B(-1E) mpliesY2, (ax< tV cp(x,, 2)*V sax< tV < sE(x, , z)) '

Thus the collectionaxiomssay hat a bounclecl uantifiercan be 'pr-rshedinside'an unbottndecluar-rtif ier.We shalldenote he induction-freeheoryPA- + {Brplrp n 9o-formula}


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PA- + {Bcplcps a J,, 9o-formula}of Col lwi l l be clenoted ol l , , . Here 'cpisa , , formula ' isunclerstoodn i tsstr ict ense,.e.meaning'e s n ), , f orm', andnot tneaning'(psequivalentto a ) , , formula ' .However, i f M FCol l , , nd0(x,y,2) is 2, ,(M), hen

lv lFV , , 2(0(x, , Z) cp(x, , z))for somecp(x, , 2) in 2,, n the strictsense, o sinceM FBE we haveM FB0also; hr.rshis d ist inct ionmakes it t le d i f ference. ut see alsoExercise7.1.) The theory BJ,, is 1A()+Col l , , ,.e. the theorywith axioms:PA-;inductionon A,, orrnulas; nd collection or ^I,, ormulas.Pnoposr r toN.1 . Le t ne N,0 (* ,y ) be a ) , , fo r rnu la , p (x ,y ) be a f I , ,formula,and r(y) be an So-term.Then the formulasVx


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Thearithmetichierarchy 830el. In part icular X,, s PA- * induct ion for a l l ) , , formulas, ncl1I I , , sPA- + induct ion or a l l I1, , ormulas. hen,since very ormula s equiva-lent ( in the predicate alculus) o a -X,,ormula or somen.eN, PA iseclrr ivalento the heory1Jr+ IZr+ 1J.+' . . , rvhich s a lsoeclu ivalentoI nt+ I I I r+ I I I r+ . . . . We provePRopostrroN.2. 1), ,FCol l , ,or a l l n.71. HencePA -Col l .Proof. Let MFIZ,,,satisfy let cp(x,y,2)be a ), , formulzr, nd sl lppose

,te M

M F V x< t a ycp ( x , t , A ) .In the case ,>2 we shal la lsoassumencluct ivelyhat IZ,,- tFCol l , , - ' .Wemustshow hatMFfsVx t.Now f n :7 t ! (w , a ) s I , s ince p is I I , , - r A r ) , nc l i f ) l thenVx< wa j


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84 CollectionLevrue7 4. For al ln>0 I f I , ,* Col l , ,* , I tu*t .

P roo ! ' . Le t ae lV IE In , ,+Co l l , , *1 , nd le t 0 (x ,a )e2 , , r , .where0(x . ?) saytp(x,y , a) and pe fI,,. Supposealso hat

M F-0(0, )AV x(O(x, ) -->0(x+ 1,d))We nrtrst how that MF0(b, r?) or all b e M.So et b e M be arbi trarv.Clearlvwe have

M FV < b + l3y lp (x , y , a )vY | -hp(x ,Z,a ) ) ,and the formula n squarebrackets ere in i l , ,+t , so by the col lect ionaxioms n M there exists e M such hat

MFYx < b + lAt < c(tp(x,, a)VY2-1tp(x, , d)) .This shows hat

M FV < b(aytP(x,, a)*a, < cV(x, , a)) .But 3y ap(x,y,a) is equivalent in M) to a fI , , ormtt laQ(x,c,a) (bVProposi t ion .1),andhence rom (* ' )we have

M Fcp(O,, a)AV x < b E(x,c, a) -->p(x 1,c, a)) ,so by l/,,-induction n the formula

x> bV (x< bA cp(x , ,a ) )we haveM F E ( b , c , d ) ,h e n ceM F S t ! ) ( b , t , a ) , as e q u i r e d ' n

Leurrvra.5. For all n>0,I Z,, l I II,, and I n,,l 12,,-

Proof.To prove \,tln,,, let MFIZ,,,E@,y) a I1,, ormula,aeM andsuppose i Ee, A)AVx(E@ a) +E(x+ | , A)).Wemust how hat, or al lb ' l , t M Fq(b a). So supposehis is not the case, 'e' that h eMF-\cp(ba), andconsiderhe ormulap(x, ,a),which s

( . )

(*> b)V (.v b AAyQ * x : bA-l cP(Y,1)))'


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The arithrnetic ierurchy 85Since p s IT,, , IE is a. I , , formula,and as-I , , s closedunder A' y anclexistent ia luzrnt i f icat ion,/ is equivalento a ) , , formulzi .Moreover,SinceM Flcp(b, ) we have VI ?(0, b, a), andsince

MFYx(-1cp(xr l , a) - ->1EQ,A))we have

M EY (t l . r (x, , a) -- p(x+ 7, b, A)).ThusMFVxt l t (x ,b ,o ) nd n par t i cu la rv lFQ(b ,b ,d ) , enceME-1 'p (0 ,a ) ,a contradict ion.The converse irection which s not needed or the proof of Theorem7.3anyway)s proved n a simi lar ashion y consider ing

x> bV (x < b AVy(y x: b -r-1cP(Y,)) )for q e 2,,.

Exercisesor Section7.17.1. (a) For some uitableGdclel umberi lg ' l on formulas f 9d as n Chapter3, show hat or eachn e N

{tq,lE is a (strict) X,, ormula}is recursive, nd hence he theories12,,,Btu, III,, Coll ,,,etc' are recursivelyaxiomatized.(b) Assumehat hereexist ets q N thatare .e.but not recursive'Thiswil l beproved n chapter 9.) Using corol laries3.4 and 3.5 show that there is no -x'iormula p(x),such hat, for al l ya-formulas9(x), N Frl(tE(x)r) ff E(x) is -x1(N).(.Hint:Let e: {neN IN F0(rt)),whereg(.r) s x1 ndconsider-1tp(r-)0n)y-10('r)r)asn var ies verN.)7.2 Define o.,nuiaclassesL n: as ollows: ,i : l -I, i :A.; for al lneN,x;]-*1isthe least classof /a-formr-rlzts onteliningIi ancl closed unclercor-rjtrnctions'disjunctions,ouncleJ uantif ication,ncl unboundecl)xistential uanti f ication;ff j-, is he east lass ontaining.Iiancl loseclonjunctisns, isjunctions,ouncleclquantif icationnd(unbouncled)niversal uanti{ icai lorr.Show t l . r a t fo ra l le Nanc l fo rz r l lo i l nu l i s0 ( f )e X, frespec t i ve l y , l - l , i ) t he re i saformulaO(t)e),, (respectivelyI,,) such hat

P t - + CollFV-r(0(-t) '--( i) ).Show hat he function/: r0(x)l '-tr6(j) l maybe terkeno be t'ecttrsive'J .3 ' ' (a) Given hi r t 8 I1 , , s 1A,, \B0 l0e i l , , \ , show hat Bf I" is equivalentoBZ, , , ro r a l l t t> - l \ .

tr


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86 Collection(b) If LI- denoteshe heory onsistingf PA- together ,vith l l nstancesd of theleast umber r inc ip leor 9e l , show hatL. I , , l l [ I , ,and Ln, , l I '2 , , for a l ln>0.(c) Mocli fy the proof of Proposition4.2 to show that 1lI,,FLf,, and (usingProposi t ions.1and7.2) hat 12, ,1 n, , .(d) Declucehat the foltowing mplications etween he theories18,,, 2,,, BEn,etc .hold or a l l n>0: 'i''-'

B2n+reBn, ,.u1I,,e I II,,e L,,,e Ln,,.

7.4*' Verify he following ersion f the Tarski-Vaughtest or -X,,-elementaryextensions:M1r.,N i f f Mlo^N and, or al l r?eM and or al l 9a-formulL\sP(I ,a), fa i tp( i ,a) s , , andNF ip(*, a), henNrp(6, a) forsome e&1.

7 . 2 C O F I N A LE X T E N S I O N SCollection s a powerful ool in t he model heoryof PA. We will i l lustratethis here by study ingcofinal extensions.If M sN are modelsof "PA- we say that M is cofinal n N, N is a cofinalex tens ion f M ( in symbo lsM s. rN) i f f YaeNlb e M (NFb>o) -ThusM-,,N) iff thereare elements f M arbitrarilyhigh up N.ExnnaplE. f MFPA- isnonstandard,let81,,1.,e he anguage btained y addingconstantsor eachaeM and a furthernewconstant . Let beM be nonstandard.Thenby compactnesshe theory

{E@)la MEE@),E an gyformula} {c a la e M} u {c< bihasa modelK which(by consideringhe mapsending acha e M to the elementreal izinghe constant orrespondingo a) we may assumeo be an elementaryextension f lvl . Let Ns K be the nitialsegment ith domain r/e Kl 1(F


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Cofinal extension.s 81Proof We show hat M < N by nclLrctionn the complexity f formulasasmeasrrrecly their posi t ion n the ari thmetic ierarchy.SupposeM


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BB CoLlectiottso sLrppose7ev l ,q@) isYyazcp(u ,y ,z )i th p f l , , - , (w" n layassLrmey. z areboth single ar iables,tsing he pair ing trnct ion xr , x t l : l whichi i a s t h e r e c l r . r i r e c lr o p e r t i e ss t a t cc ll l p . B l i n b o t h l V l i t n c l N s i n ce< t . N ' )Lct b e tVIbe arbi trary.We shal l how l t t t t

M FYy< b lzcp(d , , z ) ) NFVy< bSzcp( t1 ,. z )whichsuff ices ince vlc, ,N. Br.r tt MFVy


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Cofinulextensiott,s

the exe ' rmp len p .86 , l e t K>&1 w i thKFc


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90Exercisesor Section .2

Collection

7.5 Le t M.1 ( . l / be mode ls f PA- such thz r t v l1K=,N=M. Show ha tN ! - -o , ,N , nc lhence he useo f the IVTRDPheorem n Coro l la ry .1 I can bee l im ina ted .7.6 Show hatCoro l lary .10 s actua l ly qu ioa lerz lo Resul t .8 n the o l lowingway.Suppose esul t .8were a lse. hen:(a) therewouldbea Al l or mula ( , r ) hat snotequivalentn PA to anyV' fornru la( i .e .oneof the ormVyi2(x, l )w i th l ,G, ) quant i f ie r - f ree) ;(b) hence he theory

T: PA + {cp(e)cp s V1 and PA + 0(a)rcp(e)l 10(e)(in the anguage o expanded y aclding ew constant yrnbols ) s consistent;(c) for anyM FT, by consideringhe theory

S PA+ 0( i ) + {x@,c) x quant i l le r - f ree,e v lFt / t (d ' ) }in s t6u NIu{e} , there exis tsN=NI wi th NF PA+0(e) , and l tence or no Ksat is fy ing . , , Kc. N is M < K.7.7 (a) Let MFPA- and et R be thecorrespondingl iscrete lyrc lerec ling.Byconsideringhe non-negativelements f R[X] for some uitable rder,show hatM hasa properend-extensionEPA- .(b) Let M11=lv l1c. lv lzq.. bemodels f PA-. Show hatM:UiMi is a lsoa moclelo f PA- .( " ) A model MFPA- is sa id o be as l ike i f f card(&I) : r ' and for a l l aeMcard{b MlMFb ay

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