9 Satisfaction

7/28/2019 9 Satisfaction

1/26

9Satisfaction

T l i c t r t r t h i s r a r e y p u r e , a n c l n e v e r s i r n p l e .

Oscar WilclcTlte Irrytortunca of Beirtg Eartte.sl

This s he chapter hzi t o onc wanted o have o wri te : he mater ia l iere stechnical ncl difficult o clescribe learly, yet it is very necessary or laterrvork. have put i t off as orrgas possible , ut cannot c loso any onger.

Roughly, what s going on here s this : by the Church-Turing hesis heset S Nr defineci y

. S : { ( { t ( V ) , a ) 1 0 ( v, , , .V, , * ' )s I , a n d N f 0 ( ( u ) r , , ( r ) , , ) }

is r.e . (exercise: er i fy his) and so by Corol lary3.4 there s a ) , formulaSatr, (x , ) represent ing n N. Sat ; , (x , ) can be consiclered s epresent ing2,-truth, ol, as we shall say, of Zssutisfnc:tion.t is a '),-complete' formulii

in the sense hat any r.e. decision roblefflczlnbe redr.rced o decicling orwhat ye N is Sat ; , (n ,y) rue (for some ixecl e N depending nly on thedecis ion roblem being consiclered) . hat we hzrve o do here s to c lef inethe ormtrla Sat;,(x,y) precisely, tn.dshow t l tus all the expectecLropertiesin nrbitrary models oJ PA. The important property that we must clteckl iolclss Tarsl 1. The bulk of the work,hor,vever,s in clescr ibinghe A,(PA) formula Sat . r, , (x ,y) . ot ice hat , byTzrrski ' sheorem on the unclef inabi l i ty f t ruth (Exercise .7) here s no./o-formula Sat(;r, ) cleltning atisfaction or all 9rr-f.orntttlas thtrs: we

can ' t get any sat isfact ion!) his problern wil l be taken up in Chapter 15,where r,ve onsicler he conseql lences f aclcl ing new binary relat ior lSa t (x ,y ) o the anguage .

Histor ical ly, he motivat iol t or this worl< )n ) , , - ancl I , , -completet04


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Sequences 105

formulas was n showing hat he formula classes ,, ancl 1, , lo ncleecl orrna h i e r a r chy - tha t i s , fo r each t> - r the re are fo rmulas 0 (x )e2 , , no tequivalent o any r1, , ormula, ancl e@) err, , not equivalent o any r, ,formula. Howevel ' , he conseclLrencesor the mclclelheory oi PA go verymuch deeper, as we shal l ee n the rest of this bool len(x)Az :0) v (y < len(x)A (r) ,*r : z)

T h u s a n y n u m b e r c o d e s s e q u e n c e [ r ] , , , " ] , , . . . , 1 * ) , _ , w h e r e -l en (x ) : (x ) , , and (x ) ;* , : [x ] , fo r each < / . No t i ce ha t VxS ly ( l en ( . r ) : y )and Vx , yS lz fx l ,=z are bo th p rovab le n PA, so l en (x ) and x]"a re bo thprovably ecursive unct ions, ince hey both have A1, raph.

I t is clear rom this clef ini t ion nd he propert ies f (x) , is ted above hatfo r each eN and . r, , , t , . . . , x , , - , he re s a l e a s t cod ing he seque f l ce .E11 ,x r, . , X , , - ' n t h i s way, wi th en (z ) : i z . Thus we can make he o l lowingclefrni t ion.


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106- . - - - - - - -

I

lDeprNrnox . For: - - - - - * - _ - - - - - +

Satisfactiort

a l ln e N ( i n c l u d i n gz : 0 )

. , x , - r f r - t l e n ( z ) z \ A , . , ( [ t ] , : x , )Al *u ,, ,

Yw 1 z(len(w) nY W,. , , [w], xi) .' fhLrs

f*u, , , . . . ) x , , - r f=z clef ines farni lyof funct ions, ach of which sprovably ecursive n PA. In the case :0, the defini t ion bove proviclesus with the notat ion ] for the east ode or the sequence f length 0.

In a similar way we define he concctntenation perutionoby

i

i D e p r N r n o N . j xy z e z i s e a s t . t .

l en (z ) : l en (x ) l en (y )AVi< len (x )l z l , :

l , l ' )AV7 len(y) [z]r""r. l*i :y ] , .

The idea s , of course, hat z: xl I codes he sequence

[ r ] u , r ] r , . , [ x ] , . n r. . 1 - , ,y ] , , ,y ] ' , . . . , [ ] ] r " u r r r - r

of length en(z) len(x)+ len(y), and again t is easy o check hat x f-l is aprovably ecursive unction n PA.

We will also require he restriction f a sequence to length y, x I y,

defined by

F+ ;q w: x ly* w s east uch hat

l en (w) y AYi< l en (w) [ r ] , : [ x ] ' ) ,

so that , f x cocles t] , , , t ] r, . . . r [x]r""r*l-1nd ylen(x) , then (since we def ined t ] , :0 for i >len(x))

x f y i s h e e a s t o d e o r h e s e q u e n c e x ] , , ,t ] , ,. . . ,

[ l ] r " n r ' r - r ', . ' ' , 0 of

l eng th .V, ' padc l ec lu t 'w i th ze ros , s neces sa ry. nce aga in t i s ea sy o

check hat , r I y is provably ecLrrs iven-PA.The next clef ini i ions ihat ot l r lytzl \which is the least code for the

sequence

[ r ] , i ,t ] r : . . r l * 1 , - t , Y,x ] - * r ,

i f z


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Secluence.s 101

if z> len(x) (where his as t seclLrencc as ength z) . Thus xlylzl is theresul t of replacing he z th entry o[ the secluence oclecl y r, [x ] . , withy-padding the sequence ut with zeros [ necessary. ctre onlal ly:

;DerrNrr ;oN.i .y x[ylz] y s east uch hat

len(w) merx(lenr) , t + 1)A

Vi< len(w)

Once again lylzl is a provztbly ecursive unct ionin PA.The la s t de f in i t i on f t h i s sec t ion s tha t o f t he ' pa r t -o f ' r e l a t i on , en l .

x ep ! is to hold f x codes sequence hat forms a 'contiguor.rsblock' of thes e q u e n c e o c l e d y y. T h u s : [ 1 , 3 , 2 , i ] = r [ 9 , B , , 3 , 2 , 7 , 1 I ] . H o w e v e r ,s ince given equence ay have more han one cocle , here s an annoyingtechnical onsiclerat ionhat mr-rs t e made. and he ful l def ini t ion s

DEprNr-rroN.Y py

len(y) > len(x) f i < len(y) len(x) Vi< len(x) [ t ] i : [y] ' - ; lA

VzI ,Z, , (PA) and n, , (PA) are closed nder 'par t -of 'quant i f icat ion

and

Proof If 0(x, y, z) is-I,, ndn> 7 thenY (x c.,. ,!-->0(x,y, z)) s equivalentin PA to the ormula

l lu : l en(y)AVi ,< l l i* j< l - - ->x{len(x) jAVk


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108

Exercisesor Section .1

Satisfaction

9.1 Def i le a provably ecurs ive unct ion F(r,y) s t rc l t ha t the fo l lowi t ig r re

orovable n PA:

( a ) YLr,, y [ l e n ( a ) yAVi < len (a ) [ a l ' < x ) - -

3u < F(- r, ) ( len(u) len(a) Vl< len(a) [u l , : [ ' ] , ) ) l

( b ) V x , z f z : F ( x , y ) + ( l e n ( z ) y n Vl< len (z ) [ z l , x )

A Vry< z(len(u') len(z) 3i < len z) lt),+ [ '1, )) '

g.2 Definea provably ecursive unction G(,r)such hat he ollowing s prcvable

in PA:

Vx(G(x) oxAVY qpx(Y< G(x)) )

g.3 .(a) Define {.r},,: the yth digit of the base u representation f lr, where

* : Q , , u ) : ( u + u ) ( t t + u + 1 ) 1 2 * u . S h o w h a t h i s h a s h e p r o p e r t i e s f ( . r ) , e -

scribed n p. 105. Hint: use x),, o define .r}' n PA')( b ) D e f i n e [ ; r ] r ,e n ( x ) , l e n ( x ) , x e p l , e t c . a s i n S e c t i o n g . l b u t u s i n g { . r i . , i n p l a c e

of'( .r)n,

ancl trow hat the followinguseful property, Yx, y(xepl--->x


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Syntux

Table r\ssignrne t of nzrturzrlnrrmbers o syrnbols 'i 51,1

-7.',1-synrtrol,r na tu ra l t t t r t t bc r, r ( s )

I 0 9

0I

-r

0IL

3tl?

-5o7B9

1 0l lT2

( 1 3 , )

:

-l

fV(

)

h a s a u n i q u e G o d e l - n u m b e r o l e N . F o r e x a m p l e , + l i s l 2 l , ' 0 : l r i s

[0 ,5 ,1] ; v i t i s

[ (13 ,1 ) ] ;3 ( r i s

[ 9 ,11 ] ,and l x , y ar e Gode l -number s hen

so i s the i r conca t enz t t i on ny . (Here [ r, , , . , xu - t f , n , e t c . a r e therecursive unct ions defined n the previous sect ion.) n part icular fGN(x)AGN(y) , we can c - r rm he Goc le l -number : t ( l n rn t1 . rnyn r ) r.We shal l implify he notat ion n this ast example onsiderably y wri t ingt ( , -Fy) , for r(1 l* f l r-pr y n r)r, with similarabbreviat ions or other coff l -monly usecl construct iot ts , o t(r+/) t c lef ines (provably recursive)funct ionwith two argumentS , y. (Thus, he symbols I are used n twoclifferent ways: firstly as a function sencling a string s11" , s,- | of9o-symbols o their unique Godel-nutnber . s , , ' s , - r r N, ancl econcl ly

zrs clevice orintroclucing ew provably ecursive unctions nto P,A, sLrch

as ( . r+ y) t . )The notzrtion n the zrst aragraph oulcl eziclo confusion n the cttse I

expressions uch as (x-v,)r, so ve mnke he couvent iotr hat sar ts-ser i fare ixecl ymbols f 9o(i.e. not arguments o fr-rnctiotrsuch as

(-v: ui)r),whereas all stbsu'ipts o v (such as i in the case above) nnd uLl otheruar iahles *,y, Lt) u)w,i , , k , . . . ) are argtunents ol 'unct ior tsntrot lucacl ythe - . .1notat ion. o with this convent ion, (x :v, ) t bccomcs r onvenicntnotittion or the functic' ' ,n

x , i r - +( t n . t n r - r n ' (13 ,

l ) tn r ; t


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1 t 0an(l not the function

J , V i D r ( l n - r n r - I n V - n ) 1 .

Also : in s ide he l " ' l s i gns , we n t e rp re t , l , * , ' , ( , ) whe re os s ib l ea.t fixed -9lo-symbols so I(x+ y)l denotes the function x, J et ( n y o r 4 r n y n r ) l a p p l i e d o x , y , a n c l o t t h e u n c t i o n F + r ( t n z n r ) la p p l i e d o z : x + Y .

Note a l so ha t , i f o : . t o J r ' s , , - 1 iS a S t r ing f f ixed 9a - symbo l so '

denotes oth the Goclel-number f o and he constant unct ion with no

arguments) hose value s the number ore N. Thus the r ' rse fr" ' r to

clenote unct ions an be regarded s an extension f the Gddel-numberingfunct ion f we ident i fy 0-ary unct ions nd constant erms n the obviousway.

Withthese preliminaries ut of the way we can now show how to define

an 9o-formula erm(x) denoting x is the Godel-number f an 9o-tertn'.

DEr'trutrtoN:ermseq(s) enotes he 9o-formula

[ s ] ' :tOtV

[ s ] , : r1 tVVi< len(s) fi < s([s], rvir)V

aj , k


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Syntax

Thus, f ' s c le termines erm(x) ' s rue, hen s codes maximal ol lect ic ln fsubsequerrces f x , each f which s a term. Note hat x i tselfneed not be aterrn.) The ic lea s that i t should be possible o prove n PA that VxSs (sclctclrninesernt( ; ) ) , a l td hat he property ' term(x) 'czrn e reacl f f any. lde t e rmin ing e rm(x ) , by examin ing whe the r o r not x : [ s ] , fo r somel< len ( . r ) .No t i ce oo tha t ' s c l e t e rmines e rm(x ) ' i s L (PA) by Lemma 9.1since he quantifier Vy is bor-rnded y y cox. We aim to prove hat term(x)i s equ iva l en t o Vs( s de t e rmines e rm(x )+3 lc t en ( s ) x : [ s ] , ) , wh ich sn(PA). To clo his we need a technical ernma.

LEuvn () .2 (Unique reaclabi l i ty f terms) PA proves the fol lowingsentences:

( a ) Vx ,y ( t e rm(x ) t e rm(x y)* l en (y ) 0)

(b) Vx ,y ( t e rm(x )Ate rm(y r ) - l en (y ) :0 )

( c ) Vx , , r , s ( t e rm(x )Ate rm(y )Ate rm( r )Ate rm(s )

ii r(xxy)lc oI(r 'r, 's)lwhere F, t , ' ree i t he r or . , t hen :

ei ther (x*y)rco r

orr(x ' rY)l

o sO f X : r Ay : S A ' r : , r ' ) .

Proof (a) We use nduction on z in

Vx ,y ( l en (x ) zAlen (y ) < zAte rm(x )Ate rm(x y ) - l en (y ) g ; .

Suppose hat his s rue or some iven (i t is obviously rue or z:0) and

sLrpposee rm(x )Ate rm(xny) wi th l en (x ) and len (y ) bo th {z*1 . I f

l en (x ) : 1 hen : r0 r, l 1 , or rv r rfo r some7 , nc l o en (x y ) : 1 ( s ince heonly Goclel-numbers f terms with f i rs t symbol 0, 1 , or vi are those ofl eng th 1 ) , hence en (y ) 0 . O the rwi se en (x ) 1 so r : r ( r ' r s ) rand x n y :,(p , r 'q) . where , s , p . q are al l erms and ', 'F ' are or ' . i f len(r)< len(p) ,

t h e n p : r ) w f o r s o m e h /w i t h e n ( r ) , e n ( w ) { 2 , h e n c e e n ( w ) : 0 b y t h eindLrc t ionypo thes i s . im i l a r ly,f l en (p )< len ( r ) t hen r :pn w fo r some. ry, z tnc l en (w) :0 . Th t r s en (p ) : l en ( r ' ) and hence p - - r ' Th i s in t t t r ni m p l i e s : r < : , F / a n d : s o 4 f o r s o m e . B y t h e n d u c t i o n y p o t h e s i s g a i n

l e n ( a ) : 0 , h e n c e : p a n c l : x o y, t h t r s e n ( y ) : 0 , a s e q u i r e d .

(b) is proved n the same way as (a) .

u l


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I 12 Suti.sf'uc'tiorr

(c ) Firs t ix tcrnrs , J ancl r, ' : * or . . We prov(lby incluct ion n z that

V . t , , r.u( ternr(r. ' ) / \ lcn(r. ' )zAu I t ,o vs , ' I(1 ' t" . i ) r

- ' ( l en ( r l ) l c r t ( ru ) 1 ; t

l cn (u ) l cn ( r ) r2V

l e n ( r , u ) > l e n ( , i )2 ) )

fo r a l l z , hence

Vu (term(u)A u c, , ( r ' r"s) l -+ Gp V r, lGp tV {, t ( r r"s)

( O ) i s o b v i o u s l yr u e o r z : 0 , 1 , s o a s s u m i n g t i s r u e o r z > lL t , , w w i t h e r m ( r ) A l e n ( u ) : z * A l r n u f r w : r ( r ' r " 5 ) 1 .T h e n ufo r some e rms , y and , r. : o r ' , anc l en (x ) , en (y ) l e n ( . r ) 2o r ( 3 ) l e n ( r , r . t/ n r ) 1 n ) 2 l e n ( s ) * 2 a n c l

l e n ( a n r ( r n ' n r : r . r )> l e n ( r ) + 2 .F l o w e v e r, e n ( r. l n t ( r ) anno t equa l en ( r )*2 , f o r i f i t c l i c l ) t : s f o r s o t n e/ , h e n c e x : s b y (a ) above , wh ich i s i rnpc t s s ib l e . i n r i l a r l y en ( t )1ow)*

len( , r ) 2 . Thus ( l ) and (2) above mply len(r, r ) len( r ) * 2 anc l en(w) >len(s)*2 respec t ive ly. We will show that in case 3) we l l lLls t 21ys r ' : ' t . ' ,

x : i ' a n c l y : s w i t h l e n ( a ) : l e n ( w ) : 0 , B t t t , n c a s e 3 ) , we t n t t s t a v e

l e n ( r , r , ry n r ) rn * ) : l e n ( s ) + 2 a n c l l en(u r ( rnr f l r ' r. r ) l en ( r )+ 2

since therwise en(a n ,n r ) > len(r(r , ' s ) t ) . t fol lows hen hat r , : t , 'anclt 1 n : r , s : y n t , f o r s o m e 1 , , , h e n c e : x a n d : y b y ( a ) a b o v e , e n c eu : t ( r ' t , ' . r ) 1 , hence l en (a ) : l en (w) :0 , a s r ec lu i r ed , omple t ing heproof

LeN{vrn .3. PAlVxa: ; ( s cietermines erm(x)) .

Proof . Fix x. We show by incluct iol l n y that

Vy< len(x)3s(s le terrnineserm(.Y y)) ,

whichsr-rff icesince s cletermines errn(x) 'only clepertclst l the seqLlencccoclecl y x, ancl not on thc part icLrlar ocle x (so . l ' c le ter tni t resterrn( . r ien(x))rs clctermirreserr t r ( r ) , ven hotrgh arrcl r I len( , r : ) r l tybe c l i f f e r en t ! ) .

!


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Syntux I 13

I t i s c l ea r rom he de f i r r i t i onsha t . i : [ ] c l e t e rminese rm(x 0) . Fo r heinc l r " r c t i ont ep suppose c l c t e r rn ines e rm(x y) , y< len ( , r ) , x f ( -y+1) :(* | y ) n [ r, 1 . f [ a ] i s r0 r, r l r, o r ' u , t fo r so rne 7 , then s

n[ r. r l l e a r ly

c fe t en t r i ne se rm(x (y - r 1) ) as no c rm oc t ,x | (y t l ) can havc eng th>2

anc l a s t sy rnbo l , I o r v, . I f [ a ] r ; 1 , t l t en uppose

x I y : t t r ( ln

[ t ] , n * 'n [ s ] ,

where , r. : * or ancl , j< len( , r ) .fhen ,r ' r n t([ . r ] , r, [s] , ) tc le terrnines

t e rn r (x (y+ 1) ) , s i nce c l ea r ly e rmsec l ( s ' ) o l c l s , nc l he max i rna l i t yproperty of .s ' fol lows from unique reaclabi l i ty, nc l the maximali typroperty f s . Final ly f Ir. r ] : ) r ar td . r y is not of the above orm, or if a i sany th ing l se , hen s i t s e l f l e t e rminese rm(x | (y + l ) ) , a s can eas i l y echec l ccd . n

Levrrvrn.4. PA proves he fol lowing:

Vx, i , ( . r e termines erm(x)Ar cletermineserm(x)

>Vi< lcn(s)37 len(t) s] ' [ r ] i .

Proof . Let . r, determine erm(x) . We show by indLrct ion l1 that

Y < yaj < len(t) [s], [,f]; (' f

for a l l y


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174 SutisJ'action

Strppose :f ) holds . Then by Lemma 9.2 there is some s determiningterm("r) . By (1,) there is I < len(s) with [s ] i x ancl henceterrnseq(( , r i )n[x]) , so term(x) holcls . Conversely, f term(x) holclswewant to show here exis ts onle cletermining ernr( . r )with the property] i < l en ( s ) ( [ s ] , : x ) , o r t hen a l l such have h i s p rope r ty y Lemma 9 .3 .Our a s sumpt ion , e rm(x ) , g ives s some wi th l en ( t )> 1A[1] r. , , r, r: j rAtermseq(r) . ut then by incluct ion n y up to len(i t ) n

Ss t e rmsec l ( s )AVi l en ( , s ) ( l , r ] , cnx )AVi y ( [ r ] , , , x - -> 7< len ( s ) [ r ] , : [ r ] i )

there s s sat isfying

termseq(s) V, < len(s) [ s ] , , x) A 37< en(s) [r] , : r ) .

We c la im de t e rmines e rm(x ) , . e . , ha t . l s ' r nax ima l ' . hi s s p rovec l yincluct ion n y up to len(s) n

VuV cy(( termseq(s [ , , ] )A u c , , [ t ] , ) ( o )-- f i < i(a: [r] ,))

Thus s is maximal , ince s] i :x for some/ . We leave he ul l detai ls o thereader o check; he reason hy he nduct ion tep or (o) works s because

rf u co [s],with u*frsl;, hen a so [s]r: [([s]r 'r[s]1.] for some , l(< i and r: +o r - , henc" v tec [ s ] , r u - , , [ s [ , by Lemma 9 . ' l ( b ) .

The 'moreover ' par t ol lowsdirect ly rorn he clef ini t ion f terrn(x) . n

Next , we shal luse he same pproach o define ther syntact ic ot ions r tP,4, such as he Godel-nr"rmber f aformula, a Z,,formulu, or a fI,,fornzula.formula.

DEprxrrroN: ormseq(s) s the formula

Vl< len(s)

3u , u


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/(b) Vx, r, s l ( fo rm(r )n orm(r )Afor rn(s ) ) - -

Iifx =, , (r ' r 's)rwhere t ': AotV )1\

I then c , , V x cp sv. r t Q+,s ) t .f'

'gyntrtx I 1-5

s rletermirtas'[orm(x) s the formulir

Vi < len , i) [s],q nx) A fo rmseq s)

/ \Y u(u c , , xAforrnsec l (s [ r ] ) - - - '3 i 0A fk < s( [s ] , rvvl[s] ; - ,

) ) ]

Finally, forms,,, (x) and form r, . ,(x) erre s formseqr-,,, (sn

[xl)formseqp,,, ,(rl [x]) respectively.

PRoposrrrow .9. For each ne N, form;, , (x)ancl ormr, , , (x) reand moreover PA proves he fol lowing entences:

Vx( orm5,,(")V fonn,-r,,(x)- form (-r))

Vx(form;,,(")- formly,,, ,(x) Aformrr,, ,(x))

Vx , k(forln:,,(x) - fo rnrl-,,(3vrxl ) )

Vx, /c(fo m n,,(r) fornl',,( rVvrxl ) ) .

Exercises or Section 9. 2

9 .4 Using he provably ecurs ive unct ions (x ,1 t ) nc lC( .v) n Exerc ises .1 .9 .2 ,shorv hat

PAtYt f t e rn r ( r ) - ' 3 , r [ t e l rnsec l ( . r )A[ . s1 , . , , , , , - r : /As


14/26

SYntttx I l1

f o l r r r i n ' ) i n t ' t l , s u c h t l i a t P z l l ) r o v c s t h a t , f o r a l l, 7 . / c , a l l c r m s r , . t , i t t l c ll l e t - n r s

o r f o r r n u l u s , I r , 0 .

i r t ' t ' ( r v, t i 1 . - 1 : 1

- l f r c a 1 1 1 1 t ,i ) n - l f r e e ( r 1 r ,i )

f r e e ( t ( t t ' r ' t ) ) 1 ,) . t f r e e ( t t , i ) y l ' r e e ( u , i ) ( w h e r e i ' i s * , ' , ( , A o r V )

f re e ( t l r r t , )


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118 Satisfactiott

ot rp,-1cpor some atomic ormula cp rt !l7nor f , i s ' c le r ivec l 'rom , , , 1 , , ( j , k


16/26

Semunlic.s I 19

A p r o o f f , , f . , . . . , f / i s u p r o o f o / t h e f l 6 - f o r n r u l a 0 i f f f l i s j u s t t h e s e q u e n t w i t hone element .

Forrnalize his notion nsicle P,21, btaining provably ecursive ormula pro-of(p, x) for 'p is a proof n PA of the 14-forrnula with Gtjclel-number ' . and a ),

for rn t r la rovable("r )or' .v

i s provable n Pr l ' . ln te rp le t ing 0-r l ) as(10ytp) ,

verify hat provatrte(.v)s a provability1;reclicateor PA, as delinecln Exercise .8.

9 . 3 S E M A N T I C S

Having hown we hope) hat PA can aclecluatelyancl le yntax,we turn tosemantics . he goal i s to give, insiclePA, some restr ic tecl ers ion of' f i i rski ' sdef ini t ion f t ruth. The f i rs t s tep s o show how o eval t rute erms.

Deprurrror. r. a lseq( , . t t ) is the forrnuler

termseq(s) len(r) len(s)

Vi< len(s)

([s], r0rA [4r 0)V( [ s ] , : r l r A [ r ] , : 1 ) V37len(y) , so we woulcl xpect al(x, ) to be a wel l -c lef ineclunct ion.

Pnopos r r roN .10 , va l (x ,y ) s a p rovab ly ecu r s ive unc t ion n PA, tha t s

the ormula val( , r, ) : z is I , (PA) anclP r l lYx , ya l z va l (x , ) : z .

Moreover, PA proves:

V Y ( v a l ( r 0 rY ) : 0 Av a l ( I l t , Y ) I)Vy, ( v a l ( r v, r ,) : I y ] , )Vx, , z (va l ( r (x y) t . z ) va l (x , )+ va l (y, ) )

Vx, , z (va l ( r (xy ) ' z ) : va l (x , ) va l (y z) ) .


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120 Sutis.faction

Prctof Clear ly al(x, ) : z is ) ' (PA), s ince erm(;r) s A,(P,zl) , nc lei lso

PA, |Vx, y z(- l te tm x)+ (val(x, ) : z -+ : 0)) .

SLrppose e are givex, y with terrn( . r ) olcl ing. hen ermseq( , r ) or sor le iwith err(s) I and [ . r ]1 . , ,1 , r-x . To show hat val( .v, ) exis tswe t .nl ts t howtha t he re ex i s t s wi th va l sec l (y,, / ) , f o r then c l ea r ly a l (x , ) : [ / ] r. , ' r , r - r.

We prove 3 r r a l s eq (, s , t ) by induc t ion n w L tp o l en ( , r ) n

3 r ( l en ( r ) w Ava l seq (y, I w, r ) ) .

For w:0 th i s s r i v i a l s ince e Inay ake : [ ] ) . Suppose hz t t en ( r ) :wv a l s e q ( y, s w, r ) h o l c l s f o r s om e r . v c l e n ( , s ) . T he n [ s ] , , , i s e i t h er r 0 r , r l t , t v ,or I([s] , ' r, [s] , , ) ,r, * or ' ) for some


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we have

Sernuntics

[111. , ,1 , r- ,l t ' r" , ' ( r ' )r

s i l tce[s]r" , , ( , , r- ,[ . t ' ] r" , , ( . , , ) -r .(e) is obvious or rv:0. Suppose hat t is t rue for some w < len(r) and

that there exis ts < len(t ' ) with [ , r ] ," : . r ' ] i .Since ermse (s) holds , r ] , :r 0 r , r 1 t , r v , , l , o r r ( [ s ] , , r , [ . t ] , , , ) t ( , * : + o r . ) f o r s o m e /c < s , a n c ls o m e , m 1 w.S imi l a r ly,[ s ' ] i : r 01 , r l l , t ' , / 0 , 1or r ( [ s ' 1 , , ' r , ' [ s ] , , , ) lf o r some k '


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I22 Satisfuction

We are at as t n a posi t ion o defineSat6, , (x ,y) , he ' t ruth defini t ion '

Ao ormulas .

Deptutr toN. atseq^, , ( t , l )s the clrmula

formseqo,,(t)

V / < l e n ( r ) a i , , , < r [ [ 4 , ( i ,Z , w ) A i < l e n ( s ) Aw < 1 A

{a t t , u '< s ( t e rm( rz ) t e rm( ) A[s ] ,t ( a r t ' ) 1

(w : I al(u, z) : val(u', z)))

f 3 u , a ' < s ( t e r m ( " ) A t e r m ( a ' ) A t ] '' ( t ' { r r ' rA

(w I


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Semuntics I23

dis juncts nsicle he formula n curly bracl


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124 Sati'sJ'uctiott

(2) I f [ s ] ; : r ( [ . r ] ,A[s]e r for some ' /c i lc t

s a t s ec16 , , ( , i '( i + 1 ) , )A[ r ] , : ( / , l , t ' t ' \

s a t s ec1 l , , ( . r( / ,+ l ) , r ' )A[ r ' ] , , ( y . y , w ' )

whe re w, ,w '< 1 , < len ( f ) anc l '< l en ( t ' ) , anc l l l he se umber s l r e -ou t r c l

using f ie nclr-rct ioniypothesis ( i- l ) .' fhet1t" : to [ '^ [( i ,y, r 'v")] at isf ies

sa t s Jqo , , ( s( i+ l ) , r " ) whe re w" I i f w : w ' : 1 and w" 0o the rwi se ' The

cases t1.n ls] ' t ( [s] iV[s ]^ 1 ancl s] ,t - l[ s ] i rare rezr tecl imilar ly

(3 ) I f

[ s ] , : t 3v^ ( (u^ a ) A [s ] , ) r or [s],rvv*(_lu^< u) V [.t] ,t ,

we use a seconcl nclgct ion rgument ogether with the assumption ( i- l )to produce he recluired . Let U=val( t t . ,y) . The idea is to prove by

incluct ion n p uP o U that

3 t [ s a t s eqo , , ( s( i + l ) , r )A len ( t )>0

AVq< p3l< len(r)3rv 1([r] , : ( i , v[qlk) , h , , ) ) l

This seconcl lc luct ion s very easy. For p:0 tatke al ly f sr ' tch hat

satseqo, , (s ( i + l) , r ) with [r] , : ( / ' y f} lk) , w) for some w'l ' us ing the

f typotfr l i i tO(;- 1) . The induct iontep s proved sirnply y concatenat ing

,o*. previously btainecl r,vi th r ' sa t isfyir lg atseclo, , (s( i + 1) , . r ' ) and

I t ' f , : i j , y lp+ i t t] )ho lc l so r some u i t ab ly hosen : 0 o r l .

This concludes he proof of Lemmer ' l l ' t l

LruuR 9.12. PA Proves he sentence

V J , , s ' , ' , , i ' , , l ' ,w, w ' , y f s a t s e q o , , ( s ') A s a t s e q a , , ( s "' ) A[ . s ] , : s ' ] , ,A[ r ] , ( i .y . ] t )n

I t ' 1 , , : i ' y , w ' ) - w: w ' ) -

w ' s a t i s f y

( ' ' )

Proof By incluct ion n max(/ , ' ) 'I t l : l ' : 0 , [ r ] , : ( 1 ,, v v ) , l t ' \ , : ( i ' , y , r v ' ) h e n s ] ;a n c l s ' ] i

m u s t b e

a t o m i c , i . e . o f h e f o r r n ' ( r r : i r ' j t o r ' ( t t z -r ' ) r f o r t e r m s L t , L t ' . l v l o r e o v e r i f

[ r ] , : f r l ] , , , hen s ] , :t ( a t r ' ) ' [ s ' ] ; ' o r s ] , :

r ( a.


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Sa ta , , ( r ( r : . 1 ' ) l

Sato , , ( r ( r s ) r

S a t o , , ( r ( a A ) t

S a t o , , ( r ( u D) l

Sato , , ( r- l r r l

Sa t . r, , ( rv, ( (v, < r) A rr ) l

S'enruntic:.y

, ) , )vzt l ( r, ) : val( , r, )

, y ) * va ( r ,y ) < va l ( s y )

, y ) - rSa r to , , ( u ,)ASa t6 , ,@, )

, "y) Sato, , (u , ) ySato, ,@, )

. ),)-]

Sato,,(rr, )

, ) , ) * 3x ( va l ( r ' , ) Sa t , r, , (, y [x l i ] )

125ancl ornr ' - r las . hus

\w : I v a l ( u , y ) : v a l ( r . r ' ,) ( g n , ' I

i r r hcl i r s tca se , nr l

r. y : va l (u , y ) < va l ( r. r 'y ) eyv ' - - |

i n the seconc l .The argLrment or the inclLrct iontep spl i ts nto sevcral ases , nclwe

s l r a l l l o wo of t hem hcre , eav ing he o the r s o t he eade r. f I r ] , : ( i ,y, *1 ,[ t ' ] , , : ( i ' y , r, r ' ) , r r c l[ . i ] , : s ' ] , , hen [ . r ] ,s bLr i l t r pn f in i t e lymi lnyposs ib l eways co r r e sponc l ingo A, y , -1 , I o r V. Suppose [ s ] , : t ( r,Ar r ) , whe rer , [ . i ] , t anc l , : [ , 1 ] , , o l some 1 , .< i . Then a l sc l[ . s , ] , , : , ( r, n12 ) r he rer ' , [ , r ' f , i ,z : [ , r ' ] ,1or some ' , , ' .1 i ' ,by the uniclr. reeactabi l i ty I formr-r las .

M o r e o v e r h e r e r e 1 1 . . 11 a n c l i l ' , < l ' , a n c l r 1 , ) 2 , i , w, , s 1 s u c h h a t[ t ] , , : i t , f w, ) , [ t ] , r : ( i z , l , w, ) , f t ' ) r t : Q i , l , w, 1 ) , [ t , ] , i : Q \ , y , w l ) .h e n ythe inc luc t ion ypo thes i s 1 :w | e tnd wz :w '2 , anc i w:1 i f f wr : l e tnc lr r t : I i f f r . u j : a r r c l \ : I i t f w ' : l , S o w : w, .

Suppose ow tha t [ s ] , : I l v, , ( (v, , {a )A[ . r ] , r fo r some ,< i .Then byr"rniclueeac ' labi l i ty,[ . r ' ] , , 1 : lv^((v, ,< r r)A [r, ] , i t where [s] , , [s , ] ,1 ,or somei ' , i ' . Let u : val(u, ) . I t fol lows rorn he clef ini t ion f satseqo, , (s ,) that

ancl

n, I) lz < Ual ,< ( ( i , , f z l ] , l ) : [ r ] , , )

w ' IQaz < ual ' t4 ' ( i " , l z lk l , ) : [ r, ] , i ) .

Btrt hen t fol lows hat r t , : I i ff w' : I , by the ncluct ion ypothesis ncl hedefini t ion f satseqo, , ( r, r ) ,lnclhence w: w'

All the other curses re provecl n a s i r l i l i r rway. n

We have now clone l l he harclwork. Al l that s required s o col lect hefacts ogether nto a theorem.

Tueot


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126 Satisfuction

Sato,,(rVv,(-lv,< r) V u)1 y)


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Senrurttics 127- fr Eotrrvr 9.14. For each n7 l , Satr ; , (x , ) i s 2, , (PA), Satr, , , ( ; r,) i sn,(PA), ancl Pz l proves he tol lowing entences:

V, r [ fonn ; , , , ( t ) - Vy(Sa t r, , ( " r,) *Sa t , . , , ( x , y ) ) \

Vy(Satr,,,(x,) n51t.r-,,- ,(r, y))]

Vx[ fo rm1 , , , , ( t ) *Vy(Sa t . ; , ( x ,)Sz l t1 r, , , ( " ,) )A

Vy(Satr,,,(xy) - S2rt11,,,,( t , y))]

Yx,y, k(Saty. , ( r3v*,rry) n 3z Satr, (x y[zl lc ] ) )

and

Vx, l, k(Satr,,,(Vv rx1, ) +- Vz Satr7,,(x,lz kl))

Proof . By an easy ncluct ion n ne N. n

As a corol lary, we have he fol lowing mportant property of Sat t , , (x , )and Sat1,,,(x, ): P.4 proves

V r , , , , x k - 1 , y ( S a t r , , , ( ' r p ( u u , . , V r - ' ) 1 ,

[ " , , , . . . x 1 , - ' ]" y ) < - > t p ( x 11 , .. , x r - ' ) )

for a l i 02, , ancl al l tpefI , , wi th only the free-var iables hown.We close his chapter i th one very simple appl icat ion, he heorem hat

the 2,,1[I,, lersses f formulits orms a hierarchy.

Tr, reonErvr.15. (Due o Kleene or NI N). Let MF PA bearbi t rary. henfor each n > I the formula lSat2.,,(x,[x]) is I , , but not -I, ,(M).Hence all theinclusions n Fig. 4 on p. B0 are proper.

Proof If 0(x) is f,, ancl

M FY (0(x) I Sat ; , , (x , , r ] ) )

an d

t hen

lvlE0 t0 v,, ) ---l

Sat5,,(0(v, , , I g(v, ,) ' ] - ' I 0( 0 v,, )


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t2il Suti,sfuctiort

Exerci,se.sfor Section 9.3

9 .7 Le t scN. show tha t t he re s a fo rmula ( x ) t h a t i s A , , * , ( N )such h a t5 : { r z e NlNf 0(n ) l f anc l n ly f S i s r e c u r s i v en { ( / c ,i ) l N F S a t ; , , ( k , r ) } .9 .{ ' i Le t T be any cons is tent x tcns ion f Pt l . Show that sa to , , (x , ) is notec lLr iva lentn 7 to any A, , ormula 0(x ,y) .9 .9 Prove n PA tha t V,r,y(va l (x , )


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Sentuntics t29ancl

t , - Th(N) {oeZ, lN Fo, o Lv1l / .1-sentence} .

Suppose : - - .( a ) Show h a t I , , - T h ( N ) F Z , , ' r T h ( N ) .(b) Show hat here s i A, ,* , (N) ormula ( - r ) uch ha t or a l l rze N

N F O(ru )c=-n - o r fo r some o l , , - Th(N) .

( c ) L e t 0 ( x t , . . . , x t , y ) b e a I , , * l f o r r n u l a s u c h h a t N F Vi S t y 0 ( i , y ) . S h o w h a t h efunc t ion / : N( ->N de f ined by I i s I , , r, - r ep resen ted n I I , , -Th(N) .(d) Deduce that for al l consistent heories Zextending I, , - Th(N) such hzrt here so Au* 1(N) forrnula p(- r ) with

N F rp(m) nt - rrr fo r some r e Z

fo r each nzN, the re s a i l , , * l sen tence such ha t 7*o an d' t* -1o a re bo th

cons i s t en t . s e ( a ) to show ha t NFo .( l - l i n t :Fo l low the a rgumen t n Sec t ion3 .2 . You nray inc lExerc i seg .T he lp fu l . )

Documents

9 Satisfaction