Springer model theory an introduction (springer verlag)

David Marker Springer... Ribet Springer New York... Model Theory: An Introduction ÂN > Springer... go, IL 60607-7045 arker @math.uic.edu Editorial Board: Mathematics Department Mathem

David Marker

Springer

Editorial Board

S Axler F.W Gehring K.A Ribet

Springer

New York

Model Theory:

An Introduction

ÂN > Springer

go, IL 60607-7045

arker @math.uic.edu

Editorial Board:

Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San ncisco, CA 94132 Ann Arbor, MI 48109 tkeley, CA 94720-3840

axler@ sfsu.edu fgehring@math.lsa.umich.edu ribet@math.berkeley.edu

Library of Congress iataloging-in-Publication Data Marker, D (David), 195:

ISBN 0-387-98760-6 Printed on acid-free paper

re 2002 Springer Verlag New Yor!

All rights rved This k may “not tbe translated or copied in whole or in part without the w permission La the publis! sher (S pringer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010,

a except for brief excerpts in connection with reviews or scholarly analysis Use in connection any form of information storage and retrieval, electronic adaptation, computer software, or by ior or TainHlar methodology now known or hereafter developed is forbi idden The use in this publication oft ts ‘le names, trademarks, service marks, and similar terms, even if they are not identified as watts is ‘o be taken as an expression of opinion as to whether or not they are subject to proprietary ri;

Printed in the United States of America

54321 SPIN 10711679 www.springer-ny.com

Springer-Verlag New York Berlin Heidelberg

A member of BertelsmannSpringer Science+Business Media GmbH

1.4 Exe

2 Basic ¢ Techniques 2.1

2.2 Comple

2 : Back and Forth erci

te Theories Dow:

nd Rem

3 Algebraic Examples 3.1 Qt uan ntifier Elimination

interpretability ee mar

ark;

A

4.5 Exercise:

Indiscernibles

5.1 Partition Theorem Order Indiscernibles Le

54 A Dp ta Arithmeti 5.5 Exercises and Remark: w-Stable Th

6.1 CTneountably “Categorical Theorles 6.2 Morley Ra nk

63 Forki 1

4 £ Demme Madel B 6.5 Morley Sequence: 6.6 Exerci nd Remark: w-Stable Groups

71 TheD Vo Chain Cond 7.2 Generic Types

73 7! Lilie T4 in Aleebraically Closed Fieldls 7.5 Phúng ° a Haaa ee 7.6 ercises and Remark: Geometry of Stronely Minimal Sets 8.1 Pregeometr

89 Dp 1 ow oe pl a 8.3 Geometry and Algebra

4 Exercises and Remark: Set Theory

Real Algebra

References

Index

1 hót 1 1] 1 "

1 ha & 1 14] lafinable be £ Jon £, lac Traditionally there} been two principal themes in the subject: : " 1 toa] hac the feld of numbers, 1 and usi lel-il j lafinabl] ic 1 I f Teles 1 1 1 hat 1 1a] " Pp 1" le nf the & 1 Tonal 1 he fald of real

ma mbore Tel al 14] he 4] 4] 1 Bald te danidahble T]

cardinality «, then T | BỊ hia line |

1 Jed be Shealah [091 [pe] Leerelaned deep general classification results

1 alacet LI 114 laxel>»ed 1

tụt le of 4] 1 TTenebcseki2a F431 1 had ha M 1 £ function fields 1 " 1a} +} 1 im ad-

_ The first results of the subject the Compactnes Theorem and the

In S

1 1 Laat lee | :

1 ha dacidabil £ the +] n lee fald

° 44 ha hacl 1 font] hoa † La

1 £ Bla ad 1 1 Bhranfarc}

Ta AC 1a Te 4] BỊ j "

1 le inf

€1 2 al 1 he id €1 1 a] 1el-tl i fier el We then prove quantifier 1s Ide of real and 1 fl 1 1

to study definable sets co A " Tel bald le af cl toa] 1a) +} Woe | 1 1 ical model theor

prime, saturated 1 | Jels In Section 4.3, we sh

1 1 lorahras : :

ee : j mm hor ed fields Tt lode af Cacti Ạ 142 i tad tab]

mm 1 1 £ N Tele The fret tee

£ Œ 5 l 1 in baal 1 sas ae Wa than iyi

of w-stable theories

_ Chapters 7 i led i ick but, I hope, seductive

di i bị It is often interesting

1, alot 1 h add 1 1el_+† c1 hapter 7 Lh aloal 1 hy loot 1 al Tự elosed Gelde We al 1 lad

Tm ehenrel 1 1 c 1] Lich 3 Tyad £ X21? +] Lal

cr 1 | le al 1 1

£ sloel 1 1 1 Lut w wg 24] 1 1 Jan] lool

1 Lh] trlnot W lad +] 4] THỊ 1 © of tha Mardell] 1

cyclopedic treatment, I | had k † f hl 1

1 it Some Interesting topl

1 lels of arithmet; ]

to the exercises Others, such modules, the p-adic field, or finite model

Bhranfacht_Tratead Cl q Liter Ty Morlew’ lt

1 £ 1 le Shalahva MẸ Moadele TH and the Paris—Harrington Theorem For t] 1 hematician 1 i Tà] +} 1 in applications, 1 I have tried to illus- elton): lười Tassi 1 toa] CT} 9 Level hod of

Igebraically closed 12L to dđeas fields T had 4+] and real closed fields One of the areas where fate t a] er, yy gebra In Chapter 4 losed Feld : i li BỊ ially closed fields Diferentialh na A : fy] : Cahn 3 £ 1 ta

mà er, tai ala £ Ï zi ions In Chapter 7, we look at classical fl ditional 1e]_+† : 4 mathematical tụ obj roup

w-stability We also use 4] : to of tr 1 lzebraically closed fields In : 2 F} £ : 1.) 4]

Prerequisites

1 hl the haeie defniti a] "

m 1 +7] biệt " th HE] 1

I ical logic should be abl 1 this book, I expect that most

fl TH } thị tartal hef The ideal reader will have : : lready tal iat fl Tat ape 1]

i with ] ical formal proofs, Gédel’s Complete-

TT 1 TY } tha haaica al has ShaenGeld’a Math rT lod] op Ebhbincl [94] Plum and Thomas’ Mort bed F1] [ot] 1 xe£

7 1 Jon | ¢, : wy 1

1 : ineluding Z oof hi ti M dinals, biet and "m cardinals Appendix A sum- 1 ta]

1 Jed in CH E1] Texelaned lately in 4] text

ẤM ee] Treati lee +] ay: : ¢, loahes Thea iden] } WY head 1 fl 1

1 fortable witl basics al mmutative rin nd falde PB 1 1 i m 14] leak £ | Geldeathat 3 de a eT] «dad

Li sors 1 NT neh [ESI 3 [99] fl £

1 " Tdasll 1 HT] 1

1 loahnas 1 1 hs rà] Jee]

1 heats } hie hook; 1 iali ions 1.1, 1 2 and | 2.1 A

1

of 2.4, 3.1, 3.2, 4.1-4.3, the beginning of 4.4, 5.1, 5.2, and 6.1 In a year-

1 he] fos Ls 1 ot toa £ the remaining text M hoi Id le include @ 5 6.2-6.4, 7.1, and 7.2

Exercises and Remarks

Each cl f, 1s with tion hallencine i 6 £ + ks The exercise : 1avel

h T la} tahsđod 3 1 | have left : 1 T thin " 4

Notation

ard T AC Rt that A b bset (Le, AC B but 4 z B)

fB.ang ÁACB If A is a set,

- HV m n=1

se) When I write & a € sá, I reall mean @ Te Aw’

- The z power set of A is

(a,

influence } I Id also lik hank John Baldwi Eb abeth Bou - Wu Harrington, Kitty - Holland, Udi Hrushovski, Masanori “Hai, Julia Knight,

Wilkie, Carol Wood 1 Boris Zil’t fc ligt

1 Alan Taul 1 Dal Peril lia ret fl 1 1 logic

Amador Martin P Dal Radin Kathryn Vozoris, Carol Wood | icularly Eric R

Ins ma Finally, I, lil Jel theorist of learned model ] fi Jerful books, C C Chang and H J Kei siep Mod el

Th } Camald Gal bmatod Model Ti} Mv debt to them for : ea] bộ Tha al this book

Structures and Theories

1.1 Languages and Structures

In mathematical logic, fi der | d ib tl

lection of d hed functions, relations, and ele ments 1 1 1 lk al ha d uished functions relation lel 1 1

1 1 Janad Gald af ran] 1 hài lf + fl ! n1 1 +] Jer] +

7 1 1 4] ¢, aie and multiplication, tl funct the binary order relation

14 7.4 be thị 13

1 hale f 1 and

1 p(z4t L p(y) =2) \— rr)

We ] “er el = si for all z and

Dã and “for all positive #, there i isa y such that e J 4.0.1) of th

1 yeas 1} dtetineriehad al 11 Th 1] £ Tri +} he] 1

1 f, Lal ¢ 14 1 Lale £ 14 VAL 1 : | « Ler L | ta aitl x=yt+yt+)), which L Ld 1n] hich

Definition 1.1.1 A i I Ệ I g data:

f hRER, 11) a set of constant symbols C

iii) the language of pure sets £ = 0; iv) the language of graphs is £ = {R} where R is a binary relation

ol

Definition 1.1.2 A tract M i) pt t M called th 4 domain, or underlying set of M; ii) a function f@: M"™ — M for each f € F; iii) a set RM“ C M"® for each RE R; iv) an ele cM € M for each c

i) n(fMar, -,4n,)) = PY (lar), :M(an,)) for all f € F and Q1,.-.,0, 6M,

ii < R™ if and only if (q(a1), ,7(@mn)) € RN for all

am, € os iti) me" ye = i for ec

4# J+} itl I M is a substructu ] 1 1 jj;

A, V, and 7, which we read as “and,” “or,” and “not”, the thuantifiers 3 Definition 1.1.4

i) c€ 7 for each constant symbol ¢ € c ii) each variable symbol uw <7 fori= iii) if ty, €7 and ƒe Ff, then es ne For example, s0 05,1), (tần, 2), ; ra, 1)) and sài +, +, 1))) term

standard notation vilvs 1), (ị + 92)(s + 1), and 1+ (1+(1+1)) wh nfusion truct (Z,+,-,0,1), we think of the term tet +1) for the functi for the el y)(¢+ 1) This can be done in any 4, while (v1 + v2)(vg +1) is a f-structure

Suppose ° thác

ariables Tạ VE, = xẻ We want to interpret ¢ as a function iM: wn MR orga sib of ¢ and @ = (a;,, ,a:,,) € M, inductively define s“ (@) as Me ii) If s is the variable 1,, then = = ai

t, e terms, then s“(@) = Pee asthe May) thet, retin dfn by a r9 a)

is, f= exp, g^ = +, and e⁄f = 1, Then

(ay) =a +1, t84(a1) = ele", and t34(a1,a2) = e+ + (ay + e%) now ready to define £-formulas

Definition 1.1.5 We fi la if } i) ty = te, where n and (fe are terms, or ii) R(t, \ eReRandt tn, are term

ep et 1 VA we 1 og mulas such th

i) if ¢ is in on then mới is in W, il) if ¢ an are 1 W, and iii) if 6 is in W, then J

Here are three examples of £,,-formulas

1 I tl hhird, wher j = bo ound

in both formulas We call a f 1 if it | 1 fy 1

or false in 4 On the other hand, if ¢ is a formula with f iabl

To simplify some bookkeeping we will tacitly restrict our attention to formulas where

in each subformula no variable 2; has both free and bound occurrences For example we will not consider formulas such as (v7, > OV de; v1 +071 = v2), because this formula could

We often write (v1 ] licit the fi iabl We efi ] fc +,) to hold of (a a,) <M”

i) 1 ia (0 AB), tien ME 4G) EM = Wand A 0),

v) If dis Vv v6), nd lạ mm 1F MLE uo) or M | (a) vì) lf ở is duyý(, ei M | ¢$(@) if there is b € M such that

ME vGb vil) If di is ven nh +} NA Ị FAN af AA Ị a,b £ Wh EME M satishi a @) is t M Remarks 1.1.7 « Tl | f useful abbreviati I ill

e In addition to we will use w, 2x, y as variable symbol

If ¢ is the variable v;, then é )= 0 Suppose | that ý = /(, , ba), whee fie is an n-ary function symbol, th, e terms, and iMG) = tạ @) for ,n Because M CN,

ME¢@ = S 00: ¬ = 04), ee) € me

ME ¢@) = MF Yo(@) and M - (a)

NF to(@) and M - i(@) E28) then it also holds for @ and ¢ A Because

equivalent NHA ite M= Nữ 1 4L |E ¢ if and only if VE ¢ for all £-sentences ¢

We Ie et h(A) the full theory of M, I f 1

MA The

lt al hat Th( AA proof uses the important technique of “induction on formulas.”

an isomorphism Then,

uppose that 9 Proof We i fi las M | ¢(a1, ,@n) if and only if N |= MG (a ee) for all formulas ¢

Wr

Oe ee an M, we let j(@) d j j(an)) Then 70@) = = “G0

We prove this by i

i) Ift =«, then 7 “0 ole "= =WG@)

ae cu he ee = j(aj) = wale i)) 4028) = GMOM@, t0°@))

œ E20) ii) If ¿(@) is R(Œ, ,Éz), then

MEd(@) = (4 @), ,t%

= (GM @),- 5G @)) € RY

M Ee dfe) AMA |

1v) óis ý A6, then

Mi ¢@) = ME¢@ and \{ E0)

= NE oG@) and WE OGG

v) If 6@) is Jw (9,00), then

š ¿(@) = MEdG,>) for somebe M

® W|E0(@),e) for some e€ Nbecause j is onto

Let £ be a language An +1 A41 Talat J 1 £-th AA | fm1é3Ƒ A4 + We mm"

Example 1.2.1 Infinite Sets

by

derydre dtp A 4 SẺ đị

£<j<n The senten hat 1 ] nd an

= {<}, where < is a binary relation symbol The class of linear _ is axiomatized by the £-sentences

Va a(n <

VavyVe (@ HAT +2 <2), VvVW(œ<wVœ=wVw< s)

ay ( Ụ € VĐ )))

he +} ey: " 1

VỤ, ld al 14 1 n Fl 1 £ top or bottom elements

Example 1.2.3 Equivalence Relations Let £ = {EF}, wl I lence relations is given by the sentences

L {R}

+ nef] 7 1 by th aR(z, en

Tt will often | fal to deal with addi 1 of

Let £ = {ts <, OF, where + isa binary function symbol, < is a binary

are

the axioms for additive groups, the axioms for nen orders and VwVWVz(œ < #+z<t+2) Example 1.2.7 Left R-modules

Itipli identi Let £= {4+,0}U{r:re R}

1, O is a constant, and r is a unary

d where Li

+ r(+ a a 5 +r(y) for each re R, : (r+s)(œ) =r(#)+s(œ) for each r,s € R, vette me #)) =rs(œ) forr,se R, Example 1.2.8 Rings and Fields

by

Xã & —U=z©z=w+2), Var

ver © (w-2) = (œ9) <3) Ver-l=l-2=2a,

vavye 2+ (y-+2) = (w-3) + (y2),

VaVVz a a+ wee = 2)+(y- “

lud n the langua;

or will be useful bông, W _—_ y=

Va y=) field axioms the sentences

p—times has characteristic p For Pe > Oa _ Prime, let ACR, = ACF tp} and ACF = ACF | LH Tú; : nn I mm" +1 ve]

1 8ala Example 1.2.9 Ordered Fields fae fields,

the axioms for linear orders, e+

Example 1.2.10 Differential Fields

I _ U fẩ} | fi i bol The class of differ-

en tal fie lds i is axiomatized by the of fields,

Vay dle + 9) =ð() + ð(), Vay ð(œ - 0) = ø -ð(u) + y- 6(2) Example 1.2.11 Peano Arithmet:

et = (4, 48 1n where + and - are binary functions, s is a unary

Prop ition 1.2.13 ) Let ={4, ,0} d let T be the theory of or- dered Abelian groups Then, ( Ị ; Taasa-al

of T b) Let T be the th £ 1 1 k} h Then

Proof

a) Suppose that Ad = (M,4 0) i dered Abelian group Let acM \ {0} We "7 th

the language of rings

e Let M = (R,4,-,-,0,1) be a ring Let pk) R[X] Then,

= {x € R: p(x) = 0} is definable Suppose thatp(X) = Soa Xt,

£=0 Let đ(0,too, ,+0„) be the formula

10p © 2c <0 -E -Ƒ t0 © U-E 0g =0

—— n—times the fut 1 Ly haf 1

as “wyv™ + wyv+ wo = 0”) Then, @(0, ø ay) defines Y Indeed

Y is A- definable for any AD 2 (ao, 1 +s On}

# Let M = (R, field of real Let $(2,y) be the formula

Let 1) be the rng of integers Let X = {(rmsn) €

zi <n} Then is deta tndead đ y Lagrange’

let Sứ y) be the formula

0 Lz2a 421 424 2) 1T+22 1 + 4), then X= = {(m, n) « c72: M FE 2(m, n)}

4? +; over F Then F i | definable i in M Indeed 11 đoEnedl F is tk f f F(X]

o Let M = (C(X),4 u (C(A), +, be ¢] ld of 1 lf hat © te defined in COX) by the ¢ ] dedy ye =vAe?+l=av

hat Cie defnahl Co lal cy X

he the Geld nfoad 1 ™

@ Let M = (Qe, +, - 1 „ 1h ring of padic integer: bl l for E: 1.4.13) and o(x ) is the formula Ay y¥ ae We clai ] efi

_ Fits, suppose that y2 = = pat +1 bet v y denote the padic valuation

= 30(a) ; if v(a) < ateger and u(y?) = u(p Tin =

an even integer Thus, if Ad &

On the other hand

On the other hand, 4?) = 2u(y),

$(a), then v(a) >Osoae h

uppose that ø Let FO) = — (pa? + 1) Let F be the inci of F mod p Because v(a) > 0, 0) > 0 and F(X) = X? and ee mi FO) = 0 and FO) #0 by Hensel’ Femme ther 0.1

e Let M = (Q,4 1) be the field of rational l L

be the formula

aSb1c x22 + 2 = a2 + x2 — tục? and let 2(z) be the formula

only if N & Js Tie, 2, s), so tÌ : 14 1 : f halti 4] £ le I94) Thị leads to an interesting conclusion Proposition 1 3 2 able (i.c #h thon that anh f th ; tural + b lecid tay will always halt answering “yes” if N a and “no” if N E —¬/)

let be th nten: 1s 7(1+ +1,1+ +1,3) e—times øœ—times

lem to [24])

for any recursively eee set A C N® there is a polynomial

A={zeN?:NE 3i dưm p(Œ,9) = 0} Tha fAllaw} 1 +1] 1

Lemma 1 3 3 bet Ly be the language of ordered d (R,+,-,°,

i field of Suppose that X C R® is A- definable Then, the t logical cl f X is A-definabl oof Let d(v, Un, @) define X Let (ry Un, W) be the formula

Ve Je > 0t Sya, n (90,39) A À “(mí — ti)” < ©) t=1

tM be an L-structu Suppose that D,, 7s coll

iv) for flip en {ứa, ee Me Cayen

M wXe Dn » then Mx Xe Day

i) M® is definable

il) The graph gat is 5 definable by ŒI, ,#a„) = 9 iii) The relation Rei is defined by R iv) The set {2 « M” = ai) is defined by ve = Up

%(s +1,9)

a TẾ xẻ ue is defined by ¿(5,ø) and Y € Mƒ* is defned by (5, b),

en M\ Xi is defined by =#(v,a), X NY is defined by 2(9,3) Ad(@,b) and

x Y is defned by $(ø,a) V ú(ø, b)

vi) Tf X C Aƒ2+1 is deñned by A(v1 z1 ø), then x(X) is deñned

by ng ge a) vill) TEX C_M"™*" is defined by ở(, ,#„uu,8) and 5 € M™, then _ {ae M i Uns b,e) Thus, if X € Den then X is definable

c then X We first show

by induction that if t(v Un) is a term, then {(@,y) ¢ M"+1 :t“(@) = M) € Đa+ + 1 Sim My) me henry { ):€ Des By iv) and phú) {e#⁄ e Dị Thus, Ì fv), {@e): M* < Dros

} € Days, but this follows easily from i) and iv) (Sup pose that t= f (ty tm) By inducti I

h of tM! M” M Let G € Dimyi be the graph of vat Then, the graph of t™ is {ew : dey dem [Aw EGARWE c)}

Syd (MAE) = w Ai) =zAw= Let ¢ be R(ty, ,tm) Then {% € Me ME ¢4@)} = De

m

#e M” : 3a đem \ tM @ =a rte RM i=l

J D Because PD i 1Ø _-đeBnahl

tion will often be useful

Proposition 1 3 5 ket M be an L- structure xX = MP ï is A- definable, (that ts, if o is an automorphism of M and o(a) =a for alla © A, then o(X) =X)

Proof | = =) ha +} £ la defnine X whereac A Let o bean

1 £ A4 s;‡tE > › đletb han

f of TI 1.1.10, we showed that if 7: M => N is an isomorphism, tl M | @) if and only if \’ | 9(a)) Thus

Corollary 1.3.6 Th fl 1 bk dofn abl tho Bold nf complex numbers

Proof IfR i ble 4] ld he deafinal

1 C he aloahpatcally ind | A with re E and sế E

Ls £O | Ate the identi 1

is much different for R A ti hism of #] 1 field fx th lon] 1 Dp 1 Jofin ab] 1 1 + Liem B 1 ] RP 4 1

1 fald he MW 1 cD lafinabl]

280 £ TP and only 3% : wa Theo án are 2 rf possible definitions), but we

tiên 18 1 oy l2#nable T

TP and anlar if 1 61 a dafnable } Jafnahl S Asn £ 1E for some 7m and unehi ì X 6 cam (el

“definable” Jafnahle nei 1 Tes triek isomorphic to VV mỊ le al 1 CT /(KỒ Am "

T = {(Œi, s2), (Mì,0)) 6 MỸ? x MỸ : gì = yi} Then, Á/ & (M?, R), so VV is definably interpreted in M Int 1 +: Field in the Affine Groun Pp

Tat FY] oe a 1 the group of matrices of the form

(6 1)

where a,b € Fra # 0 Thi is i hi ] f affi + 4 h wh he Fand 0

\ 1s} hat te defnssbl 1 oT o=(h H)aa=(5 9) where 7 40,1 Let

i 1 zø\_ (/z 0

0 1/ (oO 1s? Define an operation * on A b:

ep fi@aGiQ)) + fb xT axb mg TT,

isomorphic to (A, -, +, ore

Interpreting Orders in Graphs

1 ErenT 1 1

in a graph We 1 le of +] : 15 7 T TC Ybeal Joe We weal) beaild

follows For each a < A, Gy will } ti af

+ [Di wl RP bị We will d 1 1

Ệ 1 Hợi le] of Tia (2, ¢ 1 ler A

Ne } 1 1 £ he that dean} the first two diagrams 1 lathe f 1 He 1

and (v w)s 1 ] olving vertices u,v, and w Note tl Ị { E ll A

it (2) is dedydudw ole, tes 2,W),

3 is 3e3u2u2lu bla, y,u 9sœ lu ;2)-

If a, wit , then

Ga | (a) A @1(2t) A G2(25) ^A 0a(z3) and

Ga K Os(up) A Oalus”) A Os (08)

Proof Let x bea a vertex in Gy If fa le th of

om AR 1 7 £ 1 £ |

a ver £ 1 | 1 £ la that holde Te

1 hen 8, 1 £ lea that

where a,b € A,a <0, and? < 3.1Ft 1 ] fi ] lene: then 45(2) If there 1 holds of 1 | htt a lence 1 1, then 3(c) is form that holds of x

i) metric and irreflexive, ii) for Pall 1 # “exaotly one 6; iii) if fo(e) and đo nh then “Rr Ws iv) 3 (2,

Lemma 1.3.8 7ƒ (A, <) is a “ order, then (Xe,,<œ„) ® (A,<) Moreover, Gx, = G for any G = Quotients

1 lefnabl Cand a def, | eut W

thee | “definable” bl induced structure is isomorphic to wr Let K be a field Let

t X/E with th

ak i=0

quotient we ~ is Pro, jecti X„) be

1 en waa het FAG)

A? F(Z) for any \ and 3 Let V = ee € ne f(@) = O} Because f is

1 lanl with all : 1 m Although

1 1 weal] + ey CI 1 fax slir thị material for the time being Ww j lk lat £ wd bod structures

Let S be a set TI i f 1 ucbure Á/ with soris S

N that i itioned lisjoi Le Sy Bor each rary

My fM= n Mea ai) If N ¡ Men in M iv) If o is an automorphism y Mé&4, then a is an ina a

) if t @ an automorphism of Me such that ¢ =6|M

1.4 Exercises and Remarks Exercise 1 4 1z a) Suppo e that ¢ ¿ fi ] d ;

én Then there is § C P({1, ,n}) such

Fee VL Aer Ae

that

Q10 QuaUm 4Ú, Exercise 1.4.2 a) Let £ = {-,e} be the language of groups Show that the ere is a sentence ¢ such that M Eở 9 if and only | if Ms Z/2Z x 2/22

Exercise 1.4.3 Let £ be any countable language Show that for any

¬¬ Tel c + : Li £

đnaly K

AAlet T 1 % is an axtomatt ation

of T if M Mt i Ti if and onl if MA Ị T' for am", -S~Lructure M Sup

for all £-sentences ¢

Exercise 1.4.5 SĨ banld : ] he followi : ] ] I } a) Par vel orders

b) L

c) Boolean algebras

d) Integral domains

e) Tr

{n c€ NT: there is AI E ith |M| =n}, where Nt is the set of positive natural numbers

1 1 the f £ set of positive even 1 numbers

3 Suppose that f : M" > M is definable and one to-one Show that f-1 is definable

mì and ae A such that

| d{yeM:M $(y,@)} is finite We let acl(A) = {2:2

is algebraic over A}

5 I 1(A Show that there are # h that if

automorphisms of M fixing a b) Show that acl(acl(A)) = acl(A)

d) Show that if A C B, then acl(A) C acl(B)

1 Red L T Ai is an r+} lej get an £-structure Ad We ILM duct of M dM ; £

or £L

definiti f (R.4 0 Sh that if Ada ij 1 Jef

en every subset of M” definable in AA, is definabl

in ike structure M

Exercise 1.4.16 Suppose that V is interpretable in M1 Say VV (X/E, ), where X and E are definable in M4 We say that the inter- TI£ † £/¥ m Jf hlej NA ja hle j A