P. Mangani ( E d.)
Model Theory and Applications Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 20-28, 1975
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
[email protected] ISBN 978-3-642-11119-8 e-ISBN: 978-3-642-11121-1 DOI:10.1007/978-3-642-11121-1 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1975 With kind permission of C.I.M.E.
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T o the Memory of Abraham Robinson
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
2' Ciclo
-
Bressanone $a1 20 a1 28 giugno 1975
MODEL THEORY AND APPLICATIONS Coordinatore: Prof. P. MANGANI
G . E. SACKS
Model theory and applications
H. J. KEISLER
:
Constructions in model theory
9
55
M. SERVI
:
SH formulas and generalized exponential
)>
109
J. A. MAKOWSKY
:
Topological model theory
G . SABBAGH
:
Model Theory in algebra with emphasis on groups ( testo non pervenuto)
Centro Internazionale Matematico Estivo
"Model Theory and Applications (Second 1975 C. I. M. E. Session) Lecture Notes for Course (a) Theories of Algebraic TYD*
Gerald E. Sacks
Bres sanone (Bolzano), Italy
G. E. Sacks 1. Buon giorno.
Fundamentals
This is the first of eight lectures on the model theo-
retic notion of theory of algebraic type.
Some examples of the notion
a r e the theories of algebraically closed fields of characteristic p (p 2 0), real closed fields and differentially closed fields of characteristic 0.
The last example is the most important for two reasons.
F i r s t , it is the only one known whose complexity matches that of the general case.
Second, several results about differential fields, results
which hold for all theories of algebraic type, were first proved by model theoretic means. The key definition i s quite compact, but five lectures will be needed to unpack it.
A theory T is said to be of algebraic
if T is
complete, T i s the model completion of a universal theony, and T is quasi-totally transcendental.
In the brief time left before the onset of
formalities, let me indicate why the theory of algebraically closed fields of characteristic 0
(ACFO) is of algebraic type.
The complete-
ness of ACFO means that the same first order sentences a r e true in all algebraically closed fields of characteristic 0.
Thus a first order sen-
tence in the language of fields i s true of the complex numbers if and only if it i s true of the algebraic numbers. ACF
0
i s the model completion of TFO, the theory of fields of
characteristic 0.
To say TFO i s a universal theory is equivalent to
saying a subset of a field of characteristic 0 closed under t,
, etc.
is
G . E. Sacks a field of characteristic 0.
To see the meaning of model completion,
let 6? be any field of characteristic 0 and let F be any first order sen+ tence in the language of fields with parameters in
a.
(For example,
F might say that some finite set of polynomials in several variables withcoefficients in
a
has a common zero. ) To claim that ACFO is thq
model completion of T F
amowff$to claiming F i s true in all or in
0
none of the algebraically closed eGtensions of G . The property of quasi-total transcendality i s too complex to elucidate in a lecture on fundamentals.
For the moment think of it a s a den-
sity condition on simply generated extensions of structures weakly exemplified by the density of the rationals in the reals. totally transcendental, then each substructure
If T i s quasi-
& of a model of T has
a prime model extension, and ail prime model extensions of morphic over d
.
In the case of ACF
0'
a
this means each field
a r e iso. (i
of
characteristic 0 has a unique prime algebraically closed extension, namely the algebraic closure of
I: .
And now the fundamentals of model theory. i s a 5-tuple
A similarity type
such that 0 : I + N and (I/ : J-. N, where N
i s the set of positive integers. A structure
a
of type
(i) A nonempty set A called the universe of @
a
(ii) A family {Fti li
c
I) of relations.
Each R;
7
consists of:
. i s a subset of
A e(i)
k
(iii) A family {f. J
Ij
7
6
J) of functions. Each !f
J
maps
G . E. Sacks li; (iv) A subset {ck 1 k
Q
K) of A called the set of distinguished
elements of A. One often writes
The cardinality of 6 i s by definition the cardinality of A.
.
, .. ,
will be denoted by d, 8 ,
Structures
and their universes by A,B, C,.
..
Consider the structure -1 &= ,
+
where
and
a r e 2-place functions on A,
-
-1
and
a r e 1-place
functions on A, and 0 and 1 a r e distinguished elements of A.
The
concept of field can be formulated so that every field has the same similarity type a s 6 ,but
a
need not be a field since the relations, func-
tions anddistinguished elements of
6 need not satisfy the axioms for
fields. A monomorphism m : & -. dj i s a one-one map m : A -. B such
(i)
a
Ri (al,.
CL
(ii) mfj (al, (iii) mca
k
63
iff Ri (mal,.
...,an ) = fJB. (mal,...,man
I and n = W)). (je
J
and n=rL(i)).
k
6, and
i 8 a r e both of type
a substructure of
63 ( d C & )
map iA : A C B i s a monomorphism. morphism.
. . , m an
= c B (k E K).
(It i s assumed that
6 is
. . , an )
7.
)
if A C B and the inclusion
An isomorphism i s an onto mono-
An isomorphism is indicated by m : &.
- 63
+
or by
a~
8 .
G . E. Sacks Each similarity type
7
gives rise to a first order l a a ~ u a g e&
whose sentences a r e interpretable in structures of type tive symbols of
7
7
The primi-
7.
are:
(i)
f i r s t order variables x, y, z,
(ii)
logical connectives
-
(not),
... ;
4 (and), E (there exists), and
= (equals); {iii) a B(i)-place relation symbol Ri (i t I);
(iv) a @ (j)-place function symbol f . (j t J); J
(v)
an individual constant- c
-k
The t e r m of
7
(k t K).
a r e ge#e,rated by two rules: all variables and
individual conetants a r e terms; if f . i s an n-place function symbol and J tl,
. . . ,tn
a r e t e r m s , then f (t j 1".
. , tn )
i s a term.
The atomic formulas are: equations such a s t = t2, where t 1 1 and t2 a r e terms; and R (t i 1'"' symboland t l , .
. . ,tn
,tn ),
where Ri i s an n-place relation
a r e terms.
The formulas a r e generated from the atomic formulas as follows: if F and G a r e formulas, then -F,
F
d: G and (Ex)F a r e formulas,
where x i s any variable. '
(or), -. (implies), * (if and only if), and (x) (for all x) a r e
abbreviations: F for (F
-
G)
V G for -(-F
4 (G -. F), and
The predicate, x is a
-G),
F -. G for (-F)
'
G, F -G
(x)F for -(EX)-F.
free variable
of the formula F , i s defined
by recursion on the number of steps needed to generate F: if F i s atomic and x occurs in F, then x i s a free variable of F; if x is a
G. E. Sacks f r e e variable of F, then x is G
&
free variable of 5 F s of F
&
G and of
F; if x is a free variable of F and y i s a variable distinct from
x, then x 12 a free variable of (Ey)F. The only way to kill a free variable x of F i s to prefix F with (Ex). A useful convention is: all the free variables of G(x, y, z) lie among
X,
y, z.
A sentence i s a formula with no free variables.
Each sentence of
has a definite truth value in each structure
%
7
G' of type 7 . A s an aid in defining truth, consider the language obtained by adding a new individual conbtant language
dt' 7'
The formulas of
a7A
a
7A
for each a c A to the
\
a r e merely the formulas of
7
with some of the free variables replaced by individual constants naming elements of A .
Each constant t e r m (no variables) t of
d ?A
names
some element at of A a s follows: (i) 02 = a and oc
-k
(ii) ufj(tl,
= cL k'
. . . ,tn) = f J. (atl, . .. ,a tn ). Ii
Let H be a sentence of
?A'
The relation
@ bH
:H i s true in
k ) is defined by recursion on the number of steps needed to generate H
&7
from the atomic formulas of G?
~ :
tl = t2 iff otl = a t
2'
6 Q b ~ ~ ( t ~s t.n .) iff R. (atl , . . . , atn ).
..
&b~. iff i t i s not the case that a b F.
O ~ F G &
@ b5 F
iff
c b ( B ) F ( x ) iff
C L ~ F and
@-
F(a) - for some a e A.
G . E. Sacks If the sentence H i s not true in
a , then it i s
satisfies (or realizes) F(x1,.
& k ~ ( 2 ~ , ., -n a. .
. . ,xn )
said to be false.
in
a
if
1-
It is now quite simple to say what a field i s .
The similarity type
of afield is exemplified by the structure @ : IC.
a , .& .
The nonlogical primitive symbols of the language associated with the similarity type of fields are:
+, .
, -,
-1
, 0 and 1. The theory of fields
(TF) i s the following set of sentences: (x)(y)(z)[(x+y)+z = x+(y+z)I. (x)[x+O =
XI.
(x)[x+(-x) = 01.
,
(XI (Y)[x+Y= Y-I. (X)(Y)(Z)[(X. Y). z = x. (Y. (x)[x. 1 = (x)[x f 0 (x)(Y)[x'Y
41.
XI. +
X. X
- 1 = 11.
= Y'XI.
(x)(y)(z)[x.(y+z) = (x. y)+(x.z)l. Ofl.
d is a field iff i t has the similarity type specified above and every sentence of TF i s true in
k /means:
@.
.
is first order (or elementarily) equivalent to
C
F iff
6
F for every sentence F.
( cf @ )
(It is assumed that
G . E. Sacks & and @ belong to the same similarity type 7 , and that F is a sentence of
.) 7
3
In the next lecture it will be seen that any two algebraic-
ally closed fields of the same characteristic a r e f i r s t order equivalent. More generally it will be observed that any two models of a theory of algebraic type a r e first order equivalent. An elementary monomorphism m : &
-
5 fl
i s a map of A into B
such that
. . , -n a
Lab for every formula F(x 1'
'
) iff
. . , xn )
...
dj ~ F ( E ~ , ,-man)
and every sequence al,
. . . , an c
A.
An elementary monomorphism m i s necessarily a monomorphism, since ( 2 k a l = a 2 iff
d l k ~ ~ = ~ ~ .
Note that a map m of A into B i s an elementary monomorphism of
@ into 6 iff A ~ < B , m aA > .~ ~
(The similarity type of ac A is X T A . ) Proposition 1.
Suppose f :
-. 63 and g : 63 -.
c.
(i) If f and g a r e elementary, then gf i s elementary. (ii) If g and gf a r e elementary, then f i s elementary.
& is extension of i
A
an elementary substructure of
6.) if &
63
is a substructure of
: A C B i s an elementary monomorphism
(or
.63
53
i s an elementary
and the inclusion map
(CE- ( B). In Lecture 3 i t
G . E. Sacks will be shown that every monomorphism between models of a theory of algebraic type i s elementary.
2.
Crazie, e buon giorno.
Existence of Models
Today I will describe two approaches to the construction of
B. g.
models, the f i r s t via the extended completeness theorem of f i r s t order logic, and the second via direct limits.
&
model of a s e t S of sentences if
(A structure
@
i s said to be a
G for every G r S. )
A formula F is a logical consequence of S (S
F) if F i s
among the formulas generated f r o m S a s follows: F r S; F i s anaxiom of first order logic; F is the result of applying some rule of inference of first order logic to F1,.
.. ,Fn
when S
t Fi
(1 aeAi
aeA. Theorem 3.1 (A. Robinson).
If T1 and T2 a r e model completions of
T, then T = T 1 2' Proof.
By 2.1 and the symmetry of the situation, i t is enough t o show
an arbitrary model
{ Gn In < w ) (62n In
a ~A'
of cardinality
K,
use 2. 3 to obtain where
K
$3 1 i'
an
> card &. The
G . E. Sacks identity map on
0,can be
extended to an isomorphism between
e1
and
e,2, since the latter a r e algebraically closed fields of the same transcendence rank over
&.
Now apply 1.1.
A theory T admits elimination
of quantifiers if for
each formula
F (in the language of T ) , there is a formula G without quantifiers such
that T
F-
G.
A formula is universal if i t is of the form (x ). 1
(x )H, where n >_ 0 and H has no quantifiers. n
..
A theory V is said to
be universal if there exists a theory W such that V = W and every member of W is a universal sentence.
T F i s a typical universal
theory.
heo or em
3. 3 (A. Robinson).
If T i s the model completion of a uni-
versal theory, then T admits elimination of quantifiers. Proof. sal.
Let
Suppose T i s the model completion of V, where V is univerbe any substructure of any model of
showing T U DC i s complete.
T with the intent of
(DC is the diagram of
e , the s e t of all
atomic sentences o r negations of atomic sentences true in < C?, c > c e c.) Every universal sentence provable in T must be true in
c;hence
V. By clause (iii) of the definition of model completion, a l l models of T U DC?
a r e elementarily equivalent.
Now let F(x) be a formula in the language of T, and l e t S be the following set of sentences:
T;
F(c),where
2 does not occur in T;
and -K(c), where K(x) i s any quantifierless formula such that
G . E. Sacks T
t K(x)
-
F(x). To s e e S i s inconsistent, assume S has a model
Let
be the least substructure of
T b DC:
t F(g), since
and
(i
&
with c a s a member,
a.
Then
T U D C i s complete, G i s a model of TCI DC,
F(5). But then T
K(2) -L F(c), where K(2) i s finitely much
of D t ; clearly K(2) i s quantifierless and not occur in T, i t follows that T then the definition of S requires
K(2). Since
does
K(x) -. F(x), a contradiction because @
t'J= -K(c).
The inconsistency of S implies T
b F(x)
+
J(x), where
J(x) i s
the disjunction of finitely many quantifierless K(x)' s , each with the property that T
b K(x) -. F(x).
Corollary 3 . 4 (A. Tarski, A. Robinson).
The theory of algebraically
closed fields admits elimination of quantifiers. Two related consequences of 3 . 4 concern algebraic sets and solvability of finite systems of polynomial equations. An n-dimensional, complex algebraic
ficonsists
of all complex solutions of some finite
system of polynomial equations in n variables.
By 3 . 4 the projection
of an n-dimensional, complex algebraic s e t on m-dimensional complex space is a finite intersection of finite unions of m-dimensional, complex algebraic s e t s and their complements. Let S be a finite system of polynomial equations and inequations in several variables with coefficients cl , . .
. ,cn.
The assertion S has
a solution i s expressible by some f i r s t order sentence F ( z l , . By 3 . 4 there i s a quantifierless formula H(x
1' '
. . , x n)
. . ,cn).
such that
G. E. Sacks ACF
(1)
f
F(xl,.
.. . x n
* H(xl,.
. . .xn).
If (E is an algebraically closed field containing cl,
a
a solution in
iff
Cd
H(cl,.
. . ,-n c ).
. . . , cn '
then S has
H constitutes an algebraic
criterion for the solvability of S, because computing the truth value of H(cl,.
. . ,-nc )
i s equivalent to evaluating finitely many polynomials (with
coefficients in the prime subfield) a t .
The existence of H
was established in this lecture by little more than the extended completeness theorem of f i r s t order logic and the uniqueness of an algebraically closed field of given characteristic and dimension.
To compute
H f r o m S in an efficient fashion, one must appeal to Kroneckerl s elimination theory.
A crude procedure for finding a n H that satisfies
(1) is to enumerate all f i r s t o r d e r proofs based on the axioms of ACF. The procedure is effective, because the usual axioms for ACF f o r m a recursive set.
C. e b. g.
4.
B. g.
Isomorphism Types of Simple Extensions
Today I wish t o talk about the leading role played by the
notion of simply generated extension of a substructure of a model in the study of theories of algebraic type. substructure
6
of a model of ACF
field of characteristic 0 , since
Consider the example of ACF
0
0'
A
is nothing, more nor l e s s than a
a must include the distinguished ele-
ments 0 and 1, and must be closed with respect to addition, multiplication, additive inverse and multiplicative inverse.
The simple
G . E. Sacks extensions of
6 , save
6
correspond to roots of polynomials irreducible over
for the unique simple transcendental extension of
properties of A C F
0
a.
The
established in the previous lectures were imrnedi-
ate consequences of the nature of the simple extensions of fields. that the axioms of A C F
0
Recall
allude only to polynomials in one variable, a
restriction that i s no accident according to the following theorem. Theorem 4.1 (L. Blum [l]). universal theory.
Let V be the model completion of some
Then there exists a theory W such that V = W and
every member of W i s of the form (y ). 1
. . (yn)(Ek)F(yl,.. . .yn,x),
where F(yl, . , . ,y , x ) is quantifierless. n Blum' s proof of 4.1 i s a typical application of saturated models (cf. [2], p. 89). From now on assume T i s a theory that admits elimination of quantifiers,
Let
models of T. tension of of
C
R,(T)
Suppose
be the category of all substructures of all
6 C8
c
&.(T). @ i s said to be a simple ex-
if there i s a b c B such that
@
i s the least substructure
J! whose universe contains A V{b), in symbols @ (b)
and C ( c ) a r e isomorphic over f : &(b)
=-
=a. a @ )
if there exists an isomorphism
&(c) such that f 1 A = lA and fb = c.
Such an f i s unique
since each member of G(b) is named by a t e r m t@), where t(x) i s a t e r m in the language of T
u D a.
Since T admits elimination of quantifiers, the theory complete, that i s all models of T containing
T U DG is
satisfy the same
G. E. Saclts sentences of the language of T with constants added to name the elements of
@ . The isomorphism types of simple extensions of @- c o r -
respond to the 1-types of
T U D a . Let F1( T U Da) be the s e t of all
formulas in the language of
T U D a of the f o r m F(x). Call F(x) con-
sistent if T U D G k (Ex)F(x), and call a s e t
SCF ~ ( T U D & )consistent
if the conjunction of any finite number of members of S i s consistent.
A l - t y p e p is a maximal consistent subset of F ( T U D ~ ) .b i s said to 1
realize p if for every F(x) E p, F@) i s true in every model of T which extends
a@).
Let S & be the s e t of all 1-types of
T U D ~ If. p~
S a , thenthe
compactness theorem (2.2) implies there exists an @(b) such that b realizes p.
Conversely, for each
a@)the s e t of a l l
F(b) holds in any model of T extending
F(x) such that
a@)i s a 1-type of
S@.
Furthermore, b and c a r e isomorphic over & iff they give r i s e to the same 1-type. Suppose i :
-. @ i s an extension of (2 to a model
i s said to be a prime model extension of f :
-e
d.
of
that i = gf.
& to a model
if for every extension
of T, there exists a g :
F o r example, the algebraic closure of a field
acteristic 0 i s a prime model extension of
of T.
&
-.
k
a
such
of char-
in the context of algebra-
ically closed fields of characteristic 0. My seventh lecture will be devoted to showing that for theories of algebraic type, every substructure of every model has a unique prime model extension.
Both existence and
uniqueness will be derived f r o m the Morley analysis of S @ construed
G . E. Sacks a s a compact Hausdorff space whose clopen sets form a base for its topology. A typical basic open subset of Slit, denoted by U where F(x) 6
{ p l ~ ( xE)
Let f : @:
F(E,,.
-
..,an.x) E
F~(TU D&). Define Sf : S@
@ belong to #(T). (Sf)q iff F(fa -1'
continuous and onto.
F(x)' is
Let
'
,fan, x )
E
+
S@ by:
q, where q
s@. Sf
E
is
"
be the category of compact Hausdorff S : #(T)
spaces and continuous onto maps.
functor which assigns to each structure
-.
#.
is the contravariant
$ of &(T) the compact Haus-
dorff space S@, and to each monomorphism f :
+
X(T) the
@ of
continuous onto map Sf : S@ -. S& . Direct systems in #(T) tems of
#.
a r e transformed by S into inverse sys-
An inverse system in
&
consists of a directed s e t ,
a, and a family
-
Xi 1 i z j }
a family {x. 1 i
E
of maps of
such that: f.. i s the identity map on X.; and f = f f 11 ki ji kj
whenever i 1. By induction there i s an isomorph-
ism
Let p be the type realized by a6 over Y(a , n 1 type realized by a
over Y(a
'{I
'n
,. . . , a
).
...,a6
1, and q the
n-1
It suffices to prove
"n-1
(Sj)q = P. By the rank rule (5.2), rank p > rankp 0rankp
0'
rankp6 n
.
Hence r a n k p =
since {a ) i s a Morley sequence. A similar approach via the 6
degree rule ( 5 . 4 ) shows that deg p = deg po, and that p i s the unique pre-image of p po.
.. .
in SY(a , ,a6 ) of the same rank and degree a s 61 n-1 But the same holds for q , so q i s the unique pre-image of p in 0
SY (a 1'
, . . . ,a
0
) of the same rank and degree a s p
'n-1
Consequently
0'
(Sj)q = p, since Sj, being a homeomorphism, must preserve rank and
C]
degree.
Lemma 6.1 i s useful for generating indiscernibles over Y in & . One simply begins with a suitable p quence in
a. A suitable
p
0
0
c
SY and generates a Morley s e -
i s ranked, has many realizations in @ ,
and has a pre-image of the same rank and degree a s p extensions of Y in
0
over various
& (cf. [ 2 1, p. 232).
Lemma 6 . 2 (S. Shelah).
Suppose
&(b)C C 6 h!(~),the
1-type realized
G. E. Sacks by b over
e.
Then there exists a finite
&@)(J)
in
Proof.
Ih. in
a in
i s ranked, and I i s a s e t of indiscernibles over
JC I
such that I- J i s indiscernible over
c' .
@,(I) is the direkt limit of the finitely generated extensions of
a(1). Hence by 5.1 there exists a finite
and degree of the 1-type p that of the 1-type p
I
J
JC I
realized by b over
realized by b over
such that the rank
@(J)i s the same a s
@(I). )
To verify the indiscernibility of I - J over Q@)(J), l e t f be a one-one map of
5
onto 12, where
5
and I C I - J . 2
f extends to an
isomorphism *
f :
which is the identity on
&.
&(J), since I i s indiscernible over
p. be the type of b over 1
a(J)(5)= a(J)(%) Let
@ ( J ) ( I ~ (i ) = 1,2). It suffices to show
Sfp2 = P1' The rank and degree rules imply p and p have the same rank 1 2 and degree a s p
J'
Sfp2 has the same rank and degree a s p
is a homeomorphism.
But then Sfp2 = p
1
2'
since Sf
by the degree rule.
Today's lecture concludes with a proof of Lemma 6 . 4 , which r e quires some facts about prime model extensions summed up in Lemma
6. 3 and proved tomorrow, facts that have little to do with rank. Lemma 6. 3.
Suppose T i s quasi-totally transcendental and
Then there exists a g r i m e model extension of
d.
E
R(T)
Every such extension
G. E. Sacks
6
03 i s prime over every finitely generated extension of
in @
, and
every element of it realizes a l-type of S@ which i s both isolated and ranked. Lemma 6 . 4 (S. Shelah).
Suppose T i s quasi-totally transcendental,
(LEX(T),and @ i s a prime model extension of
&
indiscernibles over
F o r each formula F(x) in the language of T U D ~ B , Lemma 6.2 im-
plies either (a) @
f o r a l l but finitely many i
dj F ( i )
- F ( i ) for a l l but finitely many i
@
F(x) that satisfy (a). p in
i s countable.
Suppose I i s an uncountable s e t of indiscernibles over d in
Proof.
@.
6
in
I
E
dS
Let
I o r (b)
be the s e t of a l l
S@ and i s said to be the "averageI1type of I
finitely many i that
in
E
I.
i s ranked.
sB,
call i t
Hence Choose
D
D'
I
~ C E C Bp,,
e C 63,
e
E
in
has the s a m e rank
8.
S e and every i
0
63
LY
I
.
By 6 . 2
by a l l but
It follows f r o m 5.2
Then for every
&,
&?
so that
such that
and the same degree m.
K CI
such that K i s indiscernible
Sequences {p ) and {i ) a r e defined by recursion: n n E
K realizes p
realized by a l l but finitely i =
i s realized in
63 .
finitely generated over
By 6 . 2 there i s an uncountable over
in
i s ranked by 6. 3.
h a s the same rank and degree a s p
C
Ch-
be any finitely generated extension of
the projection of
Pm
I.
E
E
tB . Let
po
6.Then every s e t of
and deg p = m. n
E
.
i 0' n
E
K and realizes p ; n 'ntl
K over e ( i O , .. . ,in ).
Note that rank
is
G. E. Sacks According t o 6. 3 e ( i In < w ) has a p r i m e model extension n
@*c $3.
Since
@ i s prime over C by 6. 3 , there i s an elementary
monomorphism f of 4
Then K
,
@
into
63
is indiscernible over
*
such that f lC = 1 C'
C
realized by a l l but finitely many i
E
Let K
*
be f[K].
* * t o be the type in $ . Define p n * ). Clearly K over C ( i O , .. . ,i n -1
* * 4 pO = pO . Assume p - pn to s e e pn t l - 'ntl' The rank rule implies * < since p * i s a pre-image s f p * . Let p ~ *be the rankp ntl ntl n * "average type of K in @ . It is safe to a s s u m e I has been chosen in
among a l l uncountable s e t s of indiscernibles over minimize f i r s t the rank and then the degree of p But p
*
* ntl
implies
K c(io,.
=
* p
. . , in ) '
-- P n t l '
ntl
hence r a n k p
since
* p
ntl
* ntl
I
.
63
so a s to
*
Thus r a n k p K > _ a .
> a . Now the degree rule
-
and p arepre-imagesof p ofthe ntl n
= deg pn = m. same rank a s p and since deg p n' nt1
* < w}. The uncountability of K =u{Pn In < w) = V{Pnln * * require some i E K to realize p . By 6 . 4 p i s an i s o W W
Let p
W
and 6 . 2
lated point sf S e (in In < w).
Hence t h e r e i s a formula F(x) E p
W
such
that for every G(x) E p , W
Fix n so that F ( x ) E p to pw
.
But i
n
n
.
Since i
*#i
n
, the
formula x f i -n
satisfies F ( x ) , hence must satisfy x
#i , n
belongs
an absolute
impossibility. Tomorrow I will make a weak use of rank to prove the existence of prime model extensions of substructures of models of quasi-totally
G. E. Sacks transcendental theories, and a strong use t o prove t h e i r uniqueness. G . e b. g.
7. Existence and Uniqueness of P r i m e Extensions B. g.
The f i r s t half of toklayl s lecture will be devoted to Morleyl s
proof of the existence of p r i m e model extensions of substructures of models of T, where T satisfies a density condition much weaker than The second half will concentrate on
quasi-total transcendentality.
Shelah' s proof of the uniqueness of the prime model extension, a proof whose original version [5] a s s u m e s total transcendentality but with trifling changes extends t o the quasi-totally transcendental case.
7.1.
proposition.
Suppose
h! (T).
@,E
then the 1-types of S @ realized in 1-types of SCi: realized in
Proof.
(i) Suppose
Ip 1 ~
Then the completeness of
6
@.
C Ik
(i) If
a r e dense in
a r e dense in s&, then
( xb) , 6
a I= T,
and
s@.
(ii) If the
@bT.
i s a nonempty open subset of
T U D@ implies
a b F(5,b)
~$3.
for some
a r A. (ii) Extend
h?
to
, a model of T. An induction on the logical
complexity of sentences shows
C
(Ex)G(x). Then
Ip 1 ~
( xc )
b! ( e.
F o r example suppose
i s a nonempty neighborhood of S&,
and s o contains some p realized by some a
6
A.
But then
d b G(5).
G. E. Sacks 7.2.
Proposition.
If T is quasi-totally transcendental, then the iso-
lated points of S@,
Proof.
a r e dense in S& for every
Let N be a nonempty neighborhood of S a .
p a N of the least possible rank.
=
DOS@
7. 3.
6 a %(T).
fl M, where
Theorem.
There i s a neighborhood M such that
= rank p.
= Nn M.
But then
The isolated points of S(k a r e dense in S a for
6 a X(T) if
every
(Y
Choose a ranked
and only if every
k! a 31 (T)
has a prime model
extension. Proof.
Assume the density condition holds in order to prove the exist-
ence of prime extensions, duction to countable
(The converse follows f r o m 7.l(i) and a r e -
d ' s ; cf.
[2], Lemma 21.2 and Exercise 32.12. )
A chain { c 6 ) and a sequence {p } a r e defined by recursion. 6 (i) If
= &. (ii)
&, L6
d , =U{ h 6 16
