0 Background 1 Standard Model

7/28/2019 0 Background 1 Standard Model

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0Backsround

' cor rs ic leri t log ica l ly , 'va lc leza ic l .what is rnyprob lem? o ind Curhy.Whatc lo knowaboutCarhy? oth ing. ''Thatc loesn ' tounclogoocl , 'Marv i r r a ic l .'Br" r ti t i son ly ha l lo f theprob lern. rantec lhat know not l r ing boutCathy,whatckt knowaboutFinding'/ ''What? 'Marv inaskecl .' l t happenslrat knoweveryth ingboutFinc l ing, ' a lc lez a ic lr iur r -phant ly , estur ing i thh isgracefu lerracot taancls .For i t happenshatatnan exper t n theTheoryof Searches! 'Robcrt ShccklcylVlirtd.swrrlt

Although the reader s not expectedo know anythingaboutnonstandardmodelsof arrithmetic, e wil/ assurne little krrowledge f moclel heoryzrnd r i thmetic or rather, ecursionheory.)The purpose f th ischapter sto review hisbackgrouncl ater ia l .But f i rst ,a few general emarks.In the ext , ' s . t . ' i s n abbrev ia t ionor 'such ha t ' . ' i f f i sana t rb rev ia t ionfor ' i f and only i f ' , and I denoteshe encl or absence) f a proof. Theexercisesrre artof the ext,soshould e read,ancl referably t tempted!Alt "' agzrinst n exercisendicates hat the resr-rlt i l l be r.rseclater in thetext, a * indicates hat the exercise s sl ight lyharder than usual.Toprevent ossib le r i t ic ism n grounds f unfairness,hese wo categoriesfexercises sual lyhavehints,e i ther n the wordingof the quest ion r inbrackets afterwards. Some exercises equire some extra backgroundknowledge from algebra r set heory, or example).Theseare ntendedonly for readerswith thisbackground.Our metatheorycan be taken to be ZFC throughout,and our set-theoret ic otat ion s standard. he only set- theoret icesul ts eecledireelementatryesultsof cardiner l r i thmetic, uch as rct :rc for an inf in i tecardinal . Cardinal umbers re houghtof as n i t ia lorcl inalsn the usualw z r y .W e use he a lephn o t a t i o nN , , , N , , . . , N . , , . . . f o r ca r r d i n a l s ,r(very occasional ly)he alternat ive otat iono)u,0t, . . , (0,1,. . when wewant o emphasizehe ordinalcorrespondingo a givencardinal . husc,-r , ,can be thoughtof as he least r( ler- type f a wel l -orcler ingi carcl inal i tyN o .One f inal (non-mathenrat ical)ernark: n lv l in t lswop,VL\ l ( lezlocs vet l -tual lv i rrdCathv!


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0 . 1Buckgrottnd

B A C K G R O U N DM O D E L T H E O R YWe wil l assumehe readerhassomebasic nowleclgef nrocle lheory. nfact,JaneBriclge 's ook (Bridge 9l7) is clu i te i r f f ic ient, l though thertextsare a lsosuitable, uch as books specia l iz ingn nrclcle lheory (forexample,Changancl (eisler 1913,Chapter zrrtclartsof Chzrpters i incl3), orcerta in haptersrom booksaboutmathemzit icalogic n general forexample, arwise 977,Chapters 1 andA2). The purpose f th isscct iorris to f ix somenotat ion nd state he model- t l ' reoret icesul ts hat we shal lrequire ater.Modelsarestructuresor a f i rst-orderanguage, ncl oconsist f a notr-empty set (called he clomain) ogetherwith various elatiorts,unctions,andprivi leged lementscal lecl onstcutt .s).e shal ldenotemocle ls y thele t te rs , N" K , L , . . . . I f M is a mode l hen he c lo tna inf M wi l l a lsobedenotedM. The cardinality f M, card(M),denotes he cardinality f thedomain of.M.Finite sequences f elementsof a model (popularlycalled uples) ared e n o t e d , 6 , . . . . Thus aeM ' m e a n s ' as a se q u e n cef e l e m e n t sf t h edomainof NI of f inite ength' i . , , .: i!Tradi t ional ly,model heory s the studyof the id lb ipf iy between hemathematical tructureof a model and its first-order anguage.All f irst-order anguagesn this book contain he following ogicttl ymbols:Boo lean pe ra t i ons' a n d ' , ' o r ' , l l d ' n o t ' ) :A , y , - T ;equal i ty: ;quant i f iers: , V;

v a r i a b l e s " iV 1 1 1V r . . . V i . . . ibrackets: ) ;together with a nonlogicalsymbol for each function, relation, and con-stant. (Note that equal i ty s alwayspresent.) n pract icewe shal l usex , l , z , L t 1 u , w , . ( e t c . )as va r i a b l e sw h i chsh o u l dbe co n s i d e r e dssuitable ,s)andoccasionally ifferentstylebrackets uchas } and [ ] inplaceof ( ) whenwriting first-orclerormulasand sentences.The impl icat ion ymbols, > and


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Baclcgroundmoclel heory 3occurrences f x in t1t(x) ' .We also employ the notat ionsuAj '=,g,andf f i = 1 r l , f o r ( r p , A ( r p r A .. / \ ( q , , ) . . ) ) a n c l r p , V @ 4 V . . . V @ ' , , ) . . ) )respcct ively,ncl l f tenwri te *s in p laceof -T(r: . r) .l l &1 s a structureor the f i rst-orderangLnge l , E(x) is an J-. ' - lornrulaw i t h r e e - v a r i a b l e sm o n g: ( * u , x t , . . , x u ) , n d : ( n u , c t t , . . . o , , ) e M ,then we wri te tvlEcp(ci)o mean rpis true in M when eachvariable , isinterpreted y c, ' usingTarslci 'svel l - lcnownlef in i t ion f t ruth (NB, : isulwuy.rnterpretecly eclual i tyn &/.) Simi lar ly f O(t) is a setof formulasrn , M FO(,t)meansME cp(a)or eachEG) e O(t). If r is a closed erm n St( 'c losed': 'hels o free-var iables') ,&l enotests nterpretat ionalso al ledits realization) n M. Similarly, f s is a function, relation, or constantsymbol f 9, then,r ' l enotests rr terpretat ionn fu| .Usual ly, owever,wewi l l drop the superscr ipt and denote a or s ' t by just r or . r , unlessconfusionmayarise rom so doing.lf :J, Y' are first-order anguzlges,ncleverynon-logical yrnbolof I isalsoa symbolof !/ ' , then an l,-structure M nay be converteclnto an 9-structrrreM I g (callecl he reductof M to -9) by 'forgetting'about theext ra e la t ion ,unct ion , r constan t ymbo ls in ' . In the o therd i rec t ion ,an 9-structure N may be converted o an i4'-structureby adding newrelat ions,unct ions, nd constantsorrespondingo the extrasymbols nS'. For example, f re lat ionsR1 R,, , unct ions1 f l , , and con-stants 1,. , c lwereadded o N, the notat ion f the newstructure ouldbe (N, R ' , . . , R , , , . . , f , f r ,c , c , )andwo l r ldbe sa id o be tnexpun,sionf N. Of special rnportar-rces the stmcture(N, a),.r otrtainedby addinga constantor everya e N.A theory n a fir st-order angrvage is simply a set of sentences f 9.(Thesesentences re thoughtof as being the axiortzs f the theory.) Weassume hat the readerhzrs ncountered omenotion of a ormctlprrtof inthe predicate alculus, nd if Z is an 9-theory and o is etn -sentencewewr i teT lo to mean ' the resa proo fo f o f rom sen tencesn T ' . S im i la r ly ,fS is a seto f sen tencesf 9we wr i te7FS to n ' lean Fo f .o reach eS. T issaid to l 'teconsistentff for no -7-sentence do we have both Zl-o andTl* lo. The two turnst i le ymbols,F and F, are re latedby the fol lowingtheoremof Gocle l 's.TttEonEvt .1 (The completenessheorem).Let 7 be a theory in thefirst-order zrnguage ancl et o be an 9-sentence.Then Tto iff for al lmoc ie ls f .o r : MFT> MFo. D

The mostbasic esultn moclelheory s thecompactnessheoretn.Tr reotrev .2 Compactness) .f f is an 4thcory, hen has a mocle levery f in i te subsetSc Z hz is mocle l .

lf . M, N are rnoclels or the silme first-orcler lztnguzlge , then M ts zt;ubmodc lo f N (o r a subs t ruc tu re f .N ) ,M cN, i f f t he c loma i r r f M is a

iffn


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4 Buc'l igroundsubsctof the c lomainof N conta in i t tq hc cotrs ta t i tst l 'N i tnc l losecl t lc lcrthe funct ionsof N, ancleach non- log ica l ynrbo l t ' r / ,1s in terpretec l n lv laccorc l ing o t hc res t r i c t i o r r l ' i t s i l t t c r l ' l ' c t l r t i o t tr t M . ( t J r r l css t i t t cc lo t l rer rv iscfulc :N r ,v i l l te iu . lM is r s t - t i r tn t lc lc lf N' r l t t l tc r l l i t l l thc c lo t l t l t i t tof &1 s irtclr-rclccln thc clotnit inol 'N'.) M is irn dcflr l l l4l .yi l lb1ryJlglol N'Mcarr l ( lVl)>A,hen here s a properelementt try xtensiorr > M of Mwi t l ica rc l (N) :" . n

A few worcls bout he rnachinery scd o proveTheoret ls0.3 ancl .4are n orcler iere. t is uscl ' r . r lo havcsorne ttr tof cr i tcr ion or clccicl ingwlrena n-rocle l is t rnelementary ubrnocle lf anothermocle lN. Thefol lowingurns .r t- t to b e useiu l .Trtponev 0.5 (The Tarski-Vaught est for e lementary xtensions). etIVI -N be lnoclels or the sill.ne trngr-rage'. Then the lollowing tl l 'ee q u i va l e n t :( a ) M < N ;(b) for eirch il,.rfortnttla(t, y) ancl or ettclt 1e I,

N F a y q @ , y ) ) t h e r e e x i s t se M s . t .N F q ' ( l t . d ) . nTheorcm .3 o l lows y pu t t ing . jus tr tuughn toM to sa t is fy i r rc l ( ,&1) : , tanc l onc l i t ionb) of T l l co rcm .5 '' l 'heprool 'of he Llpwarcli )wenhcir t i -Skolert tl te t l ret t tests l l t l tc c lcaof thec,omplctaiugt 'urnf a nrocle l , that s, hc set l f i t l l se r tetrcesrLtcin the structure i l : ( \ i l , ( t ) , , , r t . f N sat isf lcsl tc ct l t l rp lcte l iagr i rn l f 'V l 't l rcp l iere s a clrrqpical r lbeclcl ingf NI----N iverr y sencl i t tgacl tueNIto tf tceler lcptg i N Lcal iz i rrghe cttnst i t t - t torrespol. lc l i l rgo rr . I f l / is thcrec luc t f N to theor ig ina langLrage ,hen t is easy o scc hat 'scnds iV1


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B tt: tgro t n cl rcc s o thco r)) 5i t t t to u t t c le t t rcnt i t t -yubstrLrcturc[ ' / / . - l 'hcl t roof 'o1 'T l rcorcr l .4 now gocsby s l r t t r .v i r tgl ta t t l tc rc is t r t tN sat is fy i r rghc contp lcr tc l iagr i tnr f ' M ant lhav ing thc i t 1 - rp r< tp r i atea rc l i r r a l i t y ,. r s i nghc co rnpuc t r r csshcorcnr . TocnsLu 'cha t N t s? p r ( )pe t ' c l c rncn ta ryx tcns ion , vc Lrs t nukcN su t i s l y hcs c t o l ' s e r t t c n c e sc * u l u e f u l l i o r a n o t h c rn c w c o n s t a r r ty r l t r o l ' .Wc c l . such l t i r t :(a) % contains the zero anclsucc'r,. i .rol 'unctiorts,

0(x ) 0 fo ra l l xeN,S(x) ;r -l-I forall r eN

i lnc l , or each l< i


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(t ,tuc:kgro t l( b ) i f . / ' ( x , , x , x1 ) anc lg ' ( I ) 8 , , ( y ) t rc a l l i t r ' ( ' t hc r rso i s thccotnpo,sit iol o f ./ ancl8r, . . , ,(r,,

h(y) J G ' ( - r ' ) ' , (Y ' ' .g r ( /w i t h t he co n ve n t i o nha t f a ny o f g , ( y ) , , g J t ) i s r - r n c l e f i n e c l ,r i fJG , Q ) , , g , ( t ) ) su n d e f i n e c l 'h c r r o s r ( v ) ;" ' i . i ? is closecl-rnclerr int i t iue ec'ur, iort ,.e. 1 '(x)ancl ( i , "y, z) zrrcr t( i thenso is /r(x,Y) define lbY

h ( i . 0 ) : f ( i )h(iy+

{ii";J; li' !,'),r, ) s .r,,delirrecr;

Pr{o l 'ost t - toN.7. The fo l lowi t tg( a ) + y ;( b ) ' v ;f 0 i f x = 0(c )x - I - 1 - r o t l t e rw ise ;t "f 0 i lY > x( d ) - y : \ . .[ . r , Y ot l tcrwise;

(e ) rnax(x , ) :

(c I ) ( / is c losecl nc lermin inut l izc t t i r t r t ,.e . , i f g (f .y) is in ? thcn so ish $ 1 : ( , r , y ) ( S ( i , y ) : 0 ) c l e f i n e d Y

h ( t 1 : t he e a s t su ch ha tg ( x , ) ) , i , l ) S ( r ' Y )a r ca l lc lefrrreclndg(x,Y) 0

li( i) is rnclefitreclf there sno sr-rch'

Der-rNrrrc>N.he classof primititte racursiue.f'tutc:tiorts,/;!/1.s thc lcastc lass f funct ions o l ta in ing , S and Ui ' fo r each


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( ' ) n r i n ( x , ) ; ( l( g ) " q ( t , y ) : {l 0r l(h ) t ( .v ,) ' i, ,L "

Prropostrror.r.8.i . e . , f g , f u , t , . .

[] uc'tgro t l rc c' t ,; o theo v

i f . l : - yi t x * y , ;

i f x < ; ri f x > . y .Both !/l/l ancl 6 are closeclunclerclefinition,f i are in 9'l/1,resp. /) then so s

, , r , : {/ , ) ( x )f g ( x ) 0| ( i ) i f g ( - r ) : I

if^ - , ( i ) l 'g ( , r ) : r If t ,( i ) t gG)> kunclel inccl,l 'g(x isuncleficcl.

nby cases,

nnininnlizution,noposrrror.r .9. T'JI. and '6 trrecloseclunclerbounclecli.e. if g(i, y) e 0.61tresp. 6) then so is

/ ' ( . f : ) : ( , r r y z ) (g ( . f ) , ) 0 )clcfrncclry

f(x,z1:t l re east < z such hatg( i , 0),g( i , I ) ,g(i, y) are all clefineclncl (;r,y) : 0,i f such exists;z . i f g ( i , t ) ) , ( ; . 1 ) , . . . g ( i , z )areal l def ineclrnd otr-zero;ru def i ecl , therwisc.

D g n N I t - r o N .some t> 0)

nThc rcctrt'.s iuctrnctioris re thc trttt t l rt t tct iot ls':NA---' (f t lri r r%'.A sctA q-N/ ' s rcc' t t r ,v iucff i tscI i t racter ist ic'Lrt tct i t i t t

( l i f i c - Ar ' ( t ) = 1 , i t . y e ASirrril irrly sct zl isprintitioc ectrt ' .s' i t tcl l y1e .'/ ' . / l'tha t hcc lass '6 t l inc ic lcsx i tc t l v i th a l t hc o thcrc lc f i r t i t i t l t l s' r r . rt ) l "pa r t i i r lco rn l . t r t i rb lcL r r rc t io t t ' (b i tscc lt t t thc i r l tL r i t io r t

l s fccLrrs tve.I t is a fact

o f the no t i r


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.3 1luc:lcgrountLc lescr ibec lt t hc bcg inn ingo f th i s sec t ion ) ha t have been sL rggcs tec ltva r ious i rnes . l i i s cac ls l s t l t hc fo l l ow ing hcs is .C r ru t


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B uckgr o tncl rccu s c t thect yWe fin ish f f th issect iorr i th a [ewworcls bout elut iue :omputubi l i ty.l f A , B a resubse tsf N we r ,v i l l f tenwant o cons idcr he ther r no t B isrec:ursiueelot iueo / (usLral ly,ve Lry:(cLtr .sioen.z{) neani l tg,oughly, i fz l wcrc recurs ivchenso r ,vou lc l be ' . ' l ' h is s ach ievcc ly the fo l low i r rgc le f in i t ion .

Der.r ivr-rrc. lN.f / cN/ ' fur some c>1, then i ' t is the east lass f part ia lfunct ions n N contairr ing ,, , he zero.sLrccessorncl ro ject iorrunct ions,and c lose nc le r o rnpos i t ion ,r im i t i ve ecurs ion , nc lm i r r i rna l i za t io r r .set B q N/ is recttrsiuen A iff Xn'64.Similarly we can define 9!AA and hence he notion of a set B beingprintitiue ecursiuen A.In all thesenotions.4 s referred o as an oracle,terminologyhator ig inatesrom the deaof an deal izedomputer unninga program hat stopseveryso often to consultsomethingike the Delphicoracle o determinewhetheror not a certain uple of integers is in ,4 orno t .


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tThe standardmodel

We nevcr know how I r ighwe areTi l l wc i r recu l l c t l o r i sc 'Anc l ther r f we arc t rue to p lanOr- r r ta tu lcs ouch he sk ies-

E,n r i l y i c l i i uson

Throughout his book 916wlllclenote he ir.st-ctrderurtguuge f arithmeticwhose onlogical ymbols onsist f the o l lowing: he constant ymbols, lQand ; the h r ina rye la t ion ymbo l , l< ; nc lhe wo b ina ry funct ionymbo ls ,I +1 and . . F rom one po in t o f v iew th is book is abou t a par t i cu la r9o-structure, , anda pzrrt icu larxiom ystem y which heorems bor.r tcanbe cledr.rced.The structureN (called he ,stundurd odel) is the #/,r-structure hosedonra ins he se to f non-nega t iven tegers ,0 , ,2 ,3 , . . . } , anc lwhere hesymbolsn 96 are given heir obvious nterpretat ion. he axiorn ysteminvolveds the formalsystemof.Peuno rithmetic PA) whichconsists f asmal l umber f basic xioms ogether i th heaxiomscheme f mnthemu-tical nduction.Much more will be said about PA later. This chapter sconcerned ith introducinghe reaclero someof the cleashatwi l l comeup again nd again n the course f t l t isbook.To discovermore abor-rt ancl he strengths ncl irnitations f P.4 r,vewi l l concentrateur at tent ion n those ,r-structureshztt ook very muchlike N from the point o{ view of first-orclerogic,br.rt renot isomorphicoN. (Suchstructuresare saicl o be npJ$llndPLCJ.)I t wasSkolemwho { i rs tshoweclhat rrclnstanclardocle ls f a l l t rue i rst-order. /o-sentencesxist Skolenr, 934).Here tncl hroughout he booktheunclual i f ieclorcl t rue 'a lwaysmezlns' t ruen N'.) Nowac' layskolern 'sresul t s provedusing he compactnessheoretn s o l lor,vs.Let f t ( \ ) clenotehe complete i- theory of the stanclarclt rocle l ,.e.Th(N) s hecol lect ion f al l t rue o-sentences. or each re N we et r l bet h e l o s e c le r r n ( ( ( l + l ) + 1 ) + ) + . . . + l ) ( , l s )o f , 7 o ; i s u s t h econstantymbol . We now exparrclur anguageIoby aclcl ingo i t a newconstant yrnbolc, obta i tr i r rghe ner,vanguttge ,, ancl consiclerhefol lowing ,.- theory i th axioms

o ( fo reach e Th(N))10


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l 'he,y uncLud rnocLct '> r t ( for each re N).

i f f N F n { n zi f f N E n + n t . :ki f f N F r u ' n t : k

iff lvl Frt 1. tn ,i f f M F n l t n :i f f M I F n ' t n :

I I

T l r is l rco ry s cons is tcn t ,o r cach in i tc ragrnen t f i t i s con t i t inc t ln7 'r T l i (N) u tr '> rt ln k\

f lo r on ie e N, unc l lea r ly he l , -s t ruc tu reN, /c )w i th c loma inN,0 , 1 ,+ , anc lsat isfres1.Thr.rs y the comperctnessheoreUrur,rZ1scttnsistetr tincl asamoclelM,. .The irst h ing o not iceabor-r t , is thatM, .Fc> . t

for r , r l l z N, and hence t conta insa l t ' in f in i te ' in teger . In par t icr - r larhismeans that the rec luct , M, o f M, to our or ig inzr l angua1e9,+ is t ro tisornorph ico the stanclarc lnodelN, for any isomorphisrnz:N--+ lV1 or- r lc lhave to sencleach n e N to r t tu tthe element rea l iz ing he c losecl erm r i intul) and M EVx,Y(x>Y-- -+-1 .ul ) ,s i r rcehissentences rue n N. I t fo l lows hat heelement eal iz ing : rt M,.cannot be in the image of h, and so M is nonstanclarcl .y theLowenheirn-Skolelnheoremswe czllt akeM,. anclhenceM) to be of anyinf i rr i te arcl inal i ty e i l M (which, ' ry ur corrvent i t ln , e czI l . llow regarclas i ln inclr . rsictr tg M) has t t t r t l thernrpt l r ta l l tproperty re lat ing o the


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t2 The :;tttnclurclmodeI

Fig. A nonstattdardodel f arithmetico rder< . No t ice ha t < is a l inear rderon M wi th a leas t lement andno greatest lement. Thesepropelt iesare expressed y a l l rst-order9o-sentencehat s rue n N, andhence rue n M.) Now for each e N wehave

M F V x ( x < " > ( x O y x : 1 V x 2 ' ' ' V x : k - t ) )since his sentences true in N. I t fo l lows rom this that N is an i t t i t ia lxgruenl-ofM, andM is an end3.f,lt_ns-iQ.n*of(in symbols:N c"tVI); i.e.,the nclusion c M has he property hat

for all ze N and or alla e M,M F u < n ) 4 N ,

so hatno newelements re added elowanygiven le N. We candrawap i c t u r e o f t h i s s i t u a t i o n ( s e e F i g . l ) . ( I n F i g ' 1 t h e o r c 1 e r N. ) One o f the ma ingoa ls f th isbooks o nvestigatehis icturenmuch reaterletail.orexample:ret l rere nyother nlerest ingni t ia lsegmentsc-"fu| ly ing etween anclM?(Theanslverurnsott t o be an emphaticyes' ' )How cansuchnonstanciarclodels e of interesto anyonewhosemainconcerns N itsel f?The key is in the existence f inf in i te 'e lementsn anonstanclarcl ancl the first-orcler roperties hey rnay possess'For


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The stantlurcl tctclcl 13exzrnrple,uppose (x) is an /o-tormulztwith only one free-variable anclsupposeulE0(u) or someue M where&1FTh(N)and rr s nonstandarcl .T lren MFax)@) anclso NF3x0(x) (for i f NFVx-lO(x) hen so woulclfu lFYx-10(x) i t t ce v l l - ' t ' h (N) ) . u t even r l ru lc s t ruc : i t r reN, thenMFa ln s ince is non-s tandarc l ,o

tu l ax@(x) x > n) ;herrceNF3x(9(x)Ar> r i . I t fo l lows hat therc are nl in i te ly turty ceNsat isfyirrg Fg(&). Conversely, uppose l rereare inf in i te lymany / xA0(y) ) , we hz ive

tu tFVxSy(y>x 0 (y) )Thus for any b e M, and n particular or any nonstanclctrde M, there sa>b in M sat isfyingMF?(a). .Wq Io,L. p_rov,ecl . thati4y,nons[ i rndprcl*,v1Th(N) a1a gtt1ot1glg"?MF-0(;Iifffrerg renfuritglvany, eN, ia t iSfyingNF0(&). This observat ions the basrs t rnar)y legant 'nor.r-]tot'iouia;proofs-of heorems boutN.A second xample hatwill concernus hrough his book s the ideaof anonstandardnumber ae N gglfu a (possibly nfinite) subsetS of N.Supposeo :2 , p t :3 , Pz:5 , . . a re he standarc l )r imes nd et 5beanysubset f N. We construct nonstandardmodelNI asbeforeby expancling9o to 9,., but this time consicleringhe theory 7 with axioms

T h ( N )U c> n l ne N}U V x_ lt r * ' * : c ) l k+S }u { 3 x ( p r x : c ) l ke S .

Notice hat every inite ragmentof T is containecln one of the theoriesT1givenby axiomsT h ( N )u { c> n l n < l }U Vr t (U- x : c ) lk+ ScL tc


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t 4 Thc stttnchrrLmrtclelI t iseasy o see hat heJi-structLrreN, r) withc: nterpretecly the ntegerr sat isf ies . Thuseach in i tesLrbsetf Thasa moclel ncl o Tis consister l t .LetM,.Ff, ancl et v lbe the recluct>tM, to the or ig inal angtragef n,trnclsu p p o see fu l e a l i ze s t h e co n s t i t t r t c i t t & 1 , . . T h e nC x l e s - t h e S e t , S , i ' e . , w ecrrn ecoveroLrr et t as ol lo,,vs:

t : {ne NIMF3x(p , , x : c t ) ) .Thus since !c N wasarbi trary) hereare nonstandard odelsMFTh(N)w i the lementsi>N cod ing he se to f p r imes, r the se to f compos i tes ,rt h e se to f p o w e r s f 2 , . . . and so o n . We co n c l u c l eh i sch a p t e r i t h asimpleappl icat ion f these cleaso the numberof countablemocle ls fTh (N .T r r r - o r < e v t. l .Th (N)Proo.f.Clearlymustshow hatt f MFrh(N)the set

There 1re exact ly2uunon- isomorphic ountab le nodelsof

.S , , {ne NIM Fax(p , , ' : u ) } .SoS, , N and(sinceM is countable) t mostcountably lanysubsets f Narecoded y some eM. We have hown, owever ,ha tanyse tSgN iscodecl y somea in somecountablemocle lMFTh(N). I f therewereonlyrc


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ThestunclardmocLelcons ider inghe for t t tu las

l 5T , 2 B Y

l l

3x(r * x * ' ' ' +.r : Y)ancl

V x l ( - r * - Y * ' " + x : Y ) ,show hat thereare 2N,,non- isornorp l i icountab le tructuresat is fy ing l l t ruesentencesn the anguage hose n ly non- log ica lynrbo ls * .1 .3 Class i fyup to isornorph isrn)l l c t ' runtab leodels f Th(N,0, l , S) in thelanguageonsis t ingf constants, l , anc l unary unct ion ynlbo l , .5 ' , h ich sinte preted n N as he.sttccc.s.rorperaticl tt:

S ( r - ) : x * I 'In par t i cu lar , e r i fy ha t , up to isomorph ism, here are on ly N1y ountab le node lso f th is theory .

0 Background 1 Standard Model

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