David Marker
Trang 2Editorial Board S Axler F.W Gehring K.A Ribet
Springer
Trang 4Model Theory: An Introduction
Trang 5go, IL 60607-7045 arker @math.uic.edu
Editorial Board:
S$ Axler F.W Gehring K.A Ribet
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
USA USA USA
axler@ sfsu.edu fgehring@math.lsa.umich.edu ribet@math.berkeley.edu Library of Congress iataloging-in-Publication Data Marker, D (David), 195: Model theo ISBN 0-387-98760-6 (1 per) 1 Model theo: 1 QA9.7 67 2 511.3—de21 2002024184
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;
Trang 9A 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
Trang 101 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]
: O24.) T 1 L L dq +} +
hea +} £4] ] £ laadabla TF £ Neral devel 1 Tet as lial 1 i 1 1) «al £ DN dafnable in +] 1 Bald tically well-behaved More recently, Wilkie [103] extended these ideas to 1 Jofnable in +} 1 +) Gd 1 Theat 1 1 14] «yy The M 7, hšet
hat igh 1 hie] 14] t, up to isomorphism, T has a unique model of : Jal of Vota ye LI Tena] mm
cardinality «, then T | BỊ hia line |
Trang 111 1 8ald 1 j 1 alacet LI 114 laxel>»ed 1 1 1: 1 " mỊ tụt le of 4] 1 TTenebcseki2a F431 1 had ha M 1 £ function fields 1 " 1a} +} 1 im ad- " Jerre he] 1N ha 1 1 e 44 1 14] £4] €1 11 h the | Lat j Tae of |] 14] ẤM F+he † Lot 1 126 bi "1 1 m 1 ea] 1 1 1 Tae @ 12nd 1 troduction to M* This i ï1 " ] 1 idea tl ill I led 1: _ 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
buil 1 li i In particular, we tudy
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]
models in Section 4.4 h 1 1
Trang 121 haf £4] 1 TY 1 j 1 LI 1 1 £ nà “ys we ae tabilit tabilit forking has a con- 1 CM on] 1 P
1 1+1 hej h 1 Sections 6.2 and
° 1 lnteal devel cay hị £ Mozl 1 he] 1 co lng 1 TL 1 Tre 1 1e] 1 1 1e] 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 7 1 |
Trang 131 Pp
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
ses 1 : i lee
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
diff lai h 1 @ 1 7 pA: ff
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] 1 wy 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 £
Trang 141 heats } hie hook; 1 iali ions 1.1, 1 2 and | 2.1 A 1 BỊ 1q °
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
by working them out i lly ] ] }
ne | w ° : a : 1
] h t+ahbilit } T 1 £4] †
I mark those exercises with a dagger.t Dp 1 tone] Tal Li 1 and attributions With a few exceptions, I tend t fe t 1 wy 1 : 1 1 I 1 hạn 1 Jnaeriha fart] 1 for further reading 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
Trang 15influence } I Id also lik hank John Baldwi Eb abeth Bou - Wu Harrington, Kitty - Holland, Udi Hrushovski, Masanori “Hai, Julia Knight,
MA MeAl a+ NA,
Nesin, Kobi Peterzil, A 1 Pill Wai Y Cl A
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
Trang 16Structures and Theories
1.1 Languages and Structures
In mathematical logic, fi der | d ib tl
atical s tructure: itivel i Ì |
lection of d hed functions, relations, and ele ments 1 1 1 lk al ha d
Trang 17Definition 1.1.1 A i I Ệ I g data: } ££, + 1, 1 J i} ef, f hRER, 11) a set of constant symbols C M1 h J d + ft ® } £ J tah] d
R is an np-ary relation A 1 ea] " Ta lee of] include: aL] fos :
= {+,-,-,0,1}, where +,— and - are binar f, : bói ‘| 14
] f ordered = £,U{<}, where < is a binary relation sy: mbo ];
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 r R, and c We off ] Me Pe Ms fe T Re RR and ce c) § We will the notation A,B, M to refer to A.B M.A For example, supr ] 1 ps We might use the language =f 1k el ] bi £ + hal Tỏi ymbol A + C=~(A & › c6 1) 4 ct TuIpp : 1 binary relation -Ÿ and a distinguished el oF 1 ức, ;])
1 +1
= and &§ =1 Also, N= =(N,+, 0) i is an Le ~structure where “Y =+ and = 1 Ofe course 1 Ineal lind] 1 aa 1, le Gf 1 4] 1 as an ordered field Id ta he f ord ly and
Trang 18i) 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; of NV or that NV is an extension of M For example: )) a " ne isa a substructure of “ +, 0) ñ) ln: bedding of đo o 0) into si n9 f AAG AFI + J; 1 f£ 41 T : M[ —> vier Idi I linality of NV is at least, the caedinality| of M W k 1 thị ariable symbols 2, v: lit 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
11) If s is the term f(t t i 1
Trang 19is, 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 ev =OVv > 0 @ Ave ve + ve $ Với (0 1=0V ¬ =) 1 1 " 1 mm lam 1 the third | 1 +] 14 Pp 1 4 VW TAL 14 } £ 1
to be true in a structure, | Ì 1 | Jifficul While in any be 8 £ 1 hird formula 1 will either | fal 1 ea] Từ +} tructure (Z,+,—,-,<, 0,1), 1 1£ 1 14 4 £0 hut falen af Ww 1 "n teelsi £ le A af ite not ined
¬ 6 1 1 1 71 For e ample ÙỊ is
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
Trang 20We often write (v1 ] licit the fi iabl We efi ] fc +,) to hold of (a a,) <M”
Definition 1.1.6 Let ¢ be a formula with free variables from (Vi Vig and let @ = (ai, in) © M™ We inductively define M $(a) as follows
D If? is ty me Me o(a) if t{(a) = (a) ii) If dis R(&, ne hen M M o(@) if (¢M(a), , 4 (@)) = RM i) HE in then AF) EM ee (@)
i) 1 ia (0 AB), tien ME 4G) EM = Wand A 0),
v) If dis Vv v6), nd lạ mm 1F MLE uo) or M | (a) vì) lf ở is duyý(, ei M | ¢$(@) 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 use: 1 hh f, 1 Tứ * £ ( ( 4 TỊ ] ] TW W, ld] Jered KI d (agAnwp) and ¬ 1 watt 7] (su 8) 1 her ind ì £ 1 because it eliminates the V and V cases wb; and Vw for túi A A„ and by Vv Vn, respectively
e In addition to we will use w, 2x, y as variable symbol
Trang 21If ¢ 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, đảm Ths MG = FMM O),.- t2°@) = #Y0°4), ,t2“8)) = YE"), @) = WO If ¢ is ty = tg, then M i= ¢@) +14 @) =4@) = @ = @ eNEO@) If dis R(t, tn) 1, then ME¢@ = S 00: ¬ = 04), ee) € me ©œ (Ý(), , tà (3)) cR = NI Thay +} 1 1 q 1 £ 1+ 2) Then, Mk 76(@) M | = m | 1 ition i f d d that to Aw Then,
ME ¢@) = MF Yo(@) and M - (a) NF to(@) and M - i(@) E28) then it also holds for @ and ¢ A Because
Trang 22equivalent NHA ite M= Nữ 1 4L |E ¢ if and only if VE ¢ for all £-sentences ¢
Trang 23M Ee dfe) AMA |
T Ƒ
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 N = 6G@) 1.2 Theories Let £ be a language An +1 A41 Talat J 1 £-th AA | fm1é3Ƒ A4 + We mm" L + oF : L ,
di ] lels of 7 We say that a theory is satisfiable if it has a model Kì 1 Ct Lee if +}
+ m h‡ + SAA 1
L Je
One way I Th(M the full theory | of an L-
tructure Ad In this case, tl | if MA
le the cl £ 1 1 MA ne typi
I] +1 £ 1 £
VAL 1E +‡+Ek 1
clementary class Wy ea] Tà] 1 will return to frequently Example 1.2.1 Infinite Sets
Trang 24= {<}, 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) Far (: (z= \\) 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 Va E(z,2), Vavy(E(2,y) + EQ, 2)), VzVuVz we wa eo) — E(z,2)) If w d the sentenc: 1 The theory of equiva- ¬ xa] 1
zSw(¬E(,u) AVz(EŒ, 2z) V E(,2))) and the infinitely many sentences
Trang 26Let £ = {ts <, OF, where + isa binary function symbol, < is a binary
relation symbol 1 TH i fi 1
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 I „0, 1}, where +, —, and - are binary 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),
Trang 27p—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
function and 0ï nsta We think of functi œtr+#-L] i fi itl + s() 7 ve mac dy s(y) = 2), Va a+0 Va Vy a+ - (s(x) = sứ + 9), Wœ z-:Ũ= VaVụ œ s(y) = = (- U) +2; h É ] 3), where Ind(#@) 1s the tence [ Pt w (sa) wy)) (sv) (rw) ;10))) (x, w)] f 1 It asserts that if aeM, {meM:ME d(m,a)}, 0 € X, and s(m) X whenever meX, chen x- M ™ Logical Consequence
Trang 28Prop 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
Tp I Ƒ ¬ 4 a 1
Proof
a) Suppose that Ad = (M,4 0) i dered Abelian group Let acM \ {0} We "7 th
" Neat ee ‘one eo eno Bown
a, Deere 0 " a=O+a<a+a and again hà Œ1 rl 9 I 7p ¬1 db 1early, + 1 1 1 In general to cl that T | + ae 1 a] ¢ construct a mo 13 Definable Sets and Interpretability Definable Sets Definition 1.8.1 Let M= = (M, 3) be a an f-structure We say that X C la đ(Đ1,
€ M™ such that X = {ae : MLE2(,b)} We cay ‘that #0, 5) Safines X We say that X is A- “nate or definable over there is a formula eG Wy WI » and b < A! such that ve b) de
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}
Trang 29Let 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)} d FLX] 1\ 4 f 1 Ta] 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 Tế 1 bì c : lef : 1 74] 2 — „8 =#”+ 1l =z Suppose = lf f and Tự h that c f fe +1 Then t+ (f(t), 9) Cc: rn Le 293441 ButE ] 1 ] c le (ORT) 4] 1 |9] Cie¢] 1 £ 1 le, £ suek that f and f+1 hath ¢& 4] ™ 1 1
J 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
Zp
_ 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
Trang 30The dafnah] : lex Từ le there 4 f 1 hthatNET Fangd se at 1 ay Jedd hal T ¢ 1 111) Thus the T 1 hal j fol]
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
p(Xy Xn Yin) ach that
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
Trang 31iv) for flip en {ứa, ee Me Cayen M wXe Dn » then Mx Xe Day 2) he and intersection; ow iy X Dns and m Mr+1 5 M®” is the projection map Ly Do (#1 Ln) then mX)yeD nở TX" Dain 4 andbe M™, then {02 M™ (0 bbe X}ED, roof We first sl he definabl fy the cl proy i)-viii) B | lest h these cl every X € D,, is " le
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
) If X M is defined b X is defined b
%(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
1 (tứ =
} € 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
Trang 32Syd (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 LK (6,0) o ME (cb), 0@) & ME (cb), 2) › > >a) T 1 T yor ] Tự ¥ ] e give a sample application 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
Thy › R Tp JR: coh] :
Ct hi The situation
Trang 331 1 44 1 he def " fae Th le let Ko] Sald and Ø1 K f tía,b,e ad—be + 0} Let f: X? 4 X by
Par brent), (@2,b2, ¢2,d2)) = = 1 Ld bo + dido) Clearly X Tự Jofin al} ¡ % 1
f + TK 1 1 cy 4
We that A 1£ bh
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 "
Trang 340 B={geG: 98 = ws =00)={ (4 1) 240} Clearly, A 1B 16 B acts on A by conjugation z 0 ly zg 0\ _ /1 # 0 1 0 1 0 1/ \0 1/ˆ The action (a,b) H fine tl i: A\ {1} Bhy Ce eon ehe ee that is, i 1 zø\_ (/z 0 0 1/ (oO 1s? Define an operation * on A b: ep fi@aGiQ)) + fb xT axb mg TT,
1 T 1 + that (Fo a1);
isomorphic to (A, -, +, ore Interpreting Orders in Graphs
1 ErenT 1 1
T 1 Le] 1 1
in a graph We 1 le of +] : 15 7 T TC Ybeal Joe We weal) beaild
Trang 351 in] TỊ 1 Ỹ (z‡ rey) i.) teh) fora © A and (a, 04”), (vi.¥3"), (0Ì), G5.) we Jn Gf Asa 1 1 1 b is the graph + [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 1 le 4] 1 1 1 +] 1 i 1 le 1 1 N 1 ot Ï 2941 7 , 4b ra a,b — a,b của 7 1 when- ever a <bin A,
Trang 36Proof 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 1 £ Ta TI +] 4 a,b fy 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 et 1 II
i) metric and irreflexive, ii) for Pall 1 # “exaotly one 6; iii) if fo(e) and đo nh then “Rr Ws iv) 3 (2, v) if 3u cu Be, Fugdvedwe (a, z ve bate và)
vi) if Đa) and oly), tt dudviw p(y, 2, u,v, w); vil) if d(a, u vill) if p(x Tf( ei A ta Ti Jar then (2, | a interpreted in a model of 7' @ HE +} +/Ỡ | T " Ta‡ X w) and Suidoiu(, 2, wa, 903), then oda mm t0”, 401), then % = tí, yeu! and w = w'; : 1 1 1 axloms IV} tG > 6o(x)} Because G 7' h if ] T every model of Tf’ 1 1 Jale anf T Wal 1 £ Lemma 1.3.8 7ƒ (A, <) is a “ order, then (Xe,,<œ„) ® (A,<) Moreover, Gx, = G for any G = Quotients
1 j ‡ For example, suppose
Trang 37thee | “definable” bl induced structure is isomorphic to wr
Let K be a field Let
t X/E with th
ak i=0
nhĩ b d defi hị d th
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 ứŒ 0,1) D } h 1 J dase ca
in the field Q, We saw ab that Zp i definabl bset of Let U={re 1 Zp: Aye E>liesti Z: 1} be the units of Zp Then, (Z,+) is p TT OW, Jatine +} Tat on QX/U by z/U >w/U 5€; Many-sorted Structures and Me 1
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 | 1 R, tl S such that RY c N*tx x%.N* 1 bol ƒ, there are ch that /ựX Nx x Nv N®
Tet Á⁄4 NT, 1 £ (o 28 m
Trang 38My fM= n Mea ai) If N ¡ Men in M iv) If o is an automorphism y Mé&4, then a is an ina a M AA 4h ) 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 XES igX that by) GỊ 1 £ 1 t+rhat£ 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 1 +} Af | £ AZ/ + Ị d 7 A 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
that 7" 1 £P Qt that Ti | Te
for all £-sentences ¢
Trang 39{n c€ NT: there is AI E ith |M| =n}, where Nt is the set of positive natural numbers Tet f E1 + + J J] † 1 | 1 1 i 1 1 the f £ set of positive even 1 numbers f£ +] 1 * £ RI+ 1} } * } finit Das" n> Ầ bh } 1 dh 1 ;
i) ip" : pis prime and n > oy nh) tp pis prim neh tg 8 Fand onlxy ‡£ tỉ 1 7 1 Me 1 1 Ệ M hal Fand only if X IRemark 1 her 1 Ệ trum is a finite spectrum Thi bl i it ival ] son of whether tl 1 bị 1 Exercise 1.4.8 Let £= at 0} Show that Z@ Z ế Z Exercis 149 tf:M" > M™ is
Trang 40| 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
hi £M ll A then (2) ~ 24 for
automorphisms of M fixing a b) Show that acl(acl(A)) = acl(A)
QL +1 TC l/A +1 l/A f, fini A
d) Show that if A C B, then acl(A) C acl(B) A197 8 ry aleal of K Show that the field F is i bl field K [Hint: F is a finite-dimensional K-vector — 1 1.4.13 † SI : £ yes ta dafnahl]
the fie ld of 2-adic numbers a, Exercise 1 1.4 a) Prove Lemma 1 3 5 that th ia Ti £ 1 h +t le] of lel of Ty every Ì o£ 7, + del 1 † dela of Ts of 1inal 1 1 £ dale At 1 £ Tư 1 ẹ 1 h th 1 Ệ Lemma 1.3.8 holds 1 Red L T Ai is an r+} lej get an £-structure Ad We ILM duct of M dM ; £ Mz a) Show that if X C M” is definable in M, then it is definabl hb CO} 1 +1 b ) Supp that M { w] fi | bol # R 1 £Á p4ũ Jafinahl M Tn +} or £L WM lofinst 1 Jat 1 Sela (R 4 1 1