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.
Printed on acid-free paper
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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
. . , an >
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 , .&
a
9 -
9
L
LL
(-1) $ 0 $ 1 > .
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 <&, a > 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 <_ i
5 n).
The axioms and rules of f i r s t order logic with equality a r e consonant with common sense, s o they need not be listed here.
One point
worth noting i s the finite character of logical consequence: if S then S 0
F for some finite S
0
t F,
C S.
S is logically consistent if no sentence of the f o r m F
&
--F i s a
logical consequence of S. Theorem 2.1. Proof.
S i s consistent iff S has a model.
If S has a model, then no contradiction is a logical conse-
quence of S, because every consequence of
s
is true in every model of
S. Suppose S i s consistent.
The construction of the model i s in two
G . E. Sacks steps.
F i r s t S i s Henkinized by adding new individual constants to the
language of S, and new sentences to S that force the new constants to be witnesses.. Let So
- S.
Sn+l consists of S together with all senn
tences of the form
where F(x) i s a formula in the language of S and c n' -F(x) dividual constant.
Let S be U {snln < w ) . ca
easily checked by induction on n.
i s a new in-
The consistency of S
For example, suppose S
gether with (1) yield a contradiction.
is
00
(=S) to-
o
Then So yields the negation of (I),
and so
so Since c -F(x)
t (fi)F(x) 4 -F(c-F(x))'
does not occur in S the derivation of -F(c ) from 0' -F(x)
S i s equivalent to the derivation of -F(y) 0 variable not occurring in S 0'
from So, where y i s some
Since S in no way limits y, the deri0
vation of --F(y) from S is equivalent to one of (y)-F(y) from S 0 0' But then
so t-
cEx)Fe)
4
(y)-F(y),
an impossibility since S i s consistent. 0 The second step i s an application of Zorn' s lemma made possible by the finite character of the logical consequence relation
t. Sco
is
* , a maximal consistent set of sentences in the language
extended to S
00
of Sm. Note that every sentence or its negation belongs to S
*
S
00
00
defines a model
of S a s follows.
.
With each individual
G . E. Sacks constant 5 of the language $
of S 00
03
C [J
=
, associate
{cllc = 2 E
the equivalence class
* 1.
S
00
a consists of {ulc c ). The relations, tions and distinguished elements of a a r e defined by: The universe A of
func-
Q)
6
Ri ( k l l , .
6
f.
J
. .,[en])iff ~
([c~],. . . ,kn])= [CJ
iff
*
~ (.. , 2 e n )~ E sQ), ~ . f
.(c
. , e n ) =CC S* , Q)
e = Cc,I.
Ck
An induction on the complexityd (i. e. number of steps needed to generate) F shows It follows
a
@
bF
* , where
iff F e S
F is any sentence of
00
$ . Q)
i s a model of S.
Theorem 2.1 i s the work of K. Gbdel, T. Skolem and A. Tar ski; the proof given i s due to L. Henkin. A theory T i s (according to the definition I prefer) a consistent
set of sentences.
Thus every theory has a model by 2.1.
means every logical consequence of T i s also one of T2. 1 means T1 C T~ and TZ T
F or T
C T1.
T2 T1 = T2
T i s complete if either
-F for every sentence F in the language of T. By
2.1 T is complete iff all models of T a r e elementarily equivalent. Corollary 2.2 (compactness).
Suppose S i s a set of sentences such
that each finite subset of S has a model.
Then S has a model.
Countability Proviso: From riow on assume every structure has at most
G . E. Sacks countably many relations and functions, and similarly that every language has a t most countably many relation symbols and function symbols. Suppose
Corollary 2 . 3 (upward Skolem-Lbwenheim). structure. K
f o r each
Proof.
Then & has a proper elementary extension of cardinality K
2 cardinality
6. The models
{C61
T,
of
G.
a t A'
Let T be the complete theory of
of a l l sentences of
Let
a i s a n infinite
6
7A
of T
t r u e in
6, where
7
that i s the set
i s the similarity type of
coincide with the elementary extensions of
C&
be a set of individual constants, none of which occur in
and let S be
S i s consistent because
G.
elementary extension of
6 of
Any model of S i s a proper
i s infinite.
cardinality a t l e a s t
K.
The details of
Theorem 2.1 imply S has a model of cardinality a t most cardinality of the language of S i s a t most
K,
since the -1
l.---
K.
A directed set < D , i > consists of a set D with a partial ordering f such that for any i, j t D,
d i r e c t system
{Gi, m. .)
there is a k
D such that i, j
t
5 k.
A
of structures and monomorphisms consists of
1J
a directed set , a family {m.. :Li + 1J
a.J 1 i 1.j e
D)
{Gil i t
D)
of structures, and a family
of monomorphisms such that m.. : Gi
i s the identity monomorphism and m
+
11
ik
= m
m . . whenever i f j jk 1~
ci 5 k.
G. E. Sacks
1
P
P
D.
Let A =U{ A X~ {i) i m. a = m b for some k lk jk < a , i > , and let A
03
Call < a , i> equivalent to if
Let [a, i] be the equivalence class of
be the set of all such equivalence classes.
directlimit of {&., m. .), 1 1J
.
whose universe i s A cb
G.,
D).
denoted by li_m (Li or
& , is
The
a structure
co
n
The relation Rb c b ([al, ill,
. . . ,[an, in])
.. .
holds
a , ,mi kan) holds for some k such that i < k for iffR (milk1 n t 1 t 5 n. The functions and distinguished elements of a r e defined
-
03
similarly. [a, i].
The monomorphism m.
:
la,
Note that m . m.. = m. Ja,
limit of {@
.,m.1J.)
1J
la,
ai &,
if i 5 j.
la,
Strictly speaking, the direct
and {m.
consists of (i
1
i s given by m. a =
+
03
la,
:
ai Ld . -+
Suppose { a i , m. .) i s a direct
Theorem 2 . 4 (A. Tarski, R. Vaught).
1J
system of structures and elementary monomorphisms. mia,
'
Proof.
I;i + La, i s elementary for all i. For all i show by induction on the complexity of formulas that
..
dmb
G i ~ F ( E ~ , ,.-n a ) iff for all F and all a l , . :
4w
. . , an E
I= (Ex)G(x,m.103a ) , where -
for' some j
E
D and b
E
A j'
A.. 1
a
E
( m,... ~ ,~ m . aa n~) 103
Ai. Then
Choose k so that i, j < k.
-
G is less complex than (Ex)G,so
I= (Ex)G(x,mika). -
~
For example, suppose
a, t G(%rnjkbJ Cr',
Then
But then
di
Then
mkzika).
ak
G(m. b, m a ) . Hence
2
(Ex)G(x,&),since m
ik
is
G . E. Sacks elementary Suppose y is an ordinal, and {Ch, la < y) i s a family of structures such that
a
aaC 6B
chain of length y.
when a <
fi <
y.
{ a a la < y) is said to be a
It can be viewed a s a direct system { & ff
where a < B and i
fffi
:
&,
C hBi s
,i }, ffP
the inclusion monomorphism.
The
ff
union of the chain {@. la < y) is defined to be the direct limit of a
{q, iag).
A chain i s said to be elementary if all of the associated in-
clusion monomorphisms a r e elementary. Corollary 2.5 (elementary chain principle).
The union of an elementar)
chain i s an elementary extension of every member of the chain. The extended completeness theorem serves to erect elementary
.
chains (as in the proof of 2. 3) whose limits a r e of interest thanks to the elementary chain principle.
3. B. g.
G. e b. g.
Model Completions of Theories
Today I will discuss A. Robinson' s notion of model com-
pletion of a theory, his f i r s t and possibly most successful attempt to generalize algebraic closedness.
My prime example will be the theory
of algebraically closed fields viewed as the model completion of the theory of fields, and I will use i t to illustrate how some of the direct arguments of elimination theory can be replaced by indirect arguments via the extended completeness theorem of first order logic.
G . E. Sacks Let T and T
1
be theories in the same language.
completion of T if for all (i)
if
&, @
a b T l , then
(ii) if
and
e:
&bT;
T, then there exists a
(iii) if $ C T ,
T i s a model 1
GCBi
and
63
such that
6)
T1:
Bi b~~ ( i = l , Z ) , t h e n (61. a >aeAi
<@2,a>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 <
Ql
Ik.
of T i s also a model of T 1 2'
such that CO= 0.
W)
< am,
so
(4, b Tl
is an elementary chain, s o
a0<
and
Define a chain
dZntlC T2.
Go ( Ow by 2.5.
Similarly,
by 1.
The theory of algebraically closed fields (ACF) i s the theory of fields (TF) augmented by n (yl). ' ' ( y n ) ( W [ x
+
YIX
n-1
+
.
''
+ Yn_lx + Yn =
01
for each n > 0 . Theorem 3.2 (A. Robinson). Proof. and
B2.
Suppose TO
see
h!
ACF is the model completion of TF.
is a field with algebraically closed extensions
a e A ;<@
elementary extension of
63 i
2 ' a > 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
. . ,cn >.
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 <_ j
<_ k.
1
D} of spaces of
1
is D
f..x = x. for all j ~1 j 1
where x. i s the i-th coordinate of x. 1
nxi. X rn
J1
Assign the product topology to
Define f
nxi.
Thus
i
oC,
E ~,
00
2 i},
is a closed, hence compact, subspace of
Then X
i
be
To form the inverse limit of {x., f ..}, let X
nxi &
x
{f .. : X 31 j
mi
: XoC, + Xi by f .x = x 001
it i s onto by virtue of the compactness of
i'
Clearly f
00 i
is continuous:
nxi. i
To see that {xoC,,foC,i)i s the inverse limit of {X f } in the i' ji sense of category theory, suppose Y is a compact Hausdorff space and {gi : Y
-
x.} 1
is a family of continuous onto maps such that fjigj = gi
G. E. Sacks
2 i.
f o r all j
Define g : Y
Clearly f .g = g 031
+
for all i.
i
X
03
by ( g ~=)g.y ~ for each y e Y.
The "universal property" of the inverse
limit i s merely the uniqueness of g: if h : Y f .h cQ1
+
Xob i s such that
= g. for all i, then h = g. Only two definitions separate us from the next lemma.
G : X (T)
+
Let
# be a contravariant functor. Thus if {&., f. .} is a direct 1
1J
system of substructures of models of T and monomorphisms, then {GQi, Gfij} i s an inverse system of compact Hausdorff spaces and onto maps.
G preserves limits if the following holds: suppose
the direct limit of {@.,f. .) consists of &. and {ficQ : G i 1 1J w
+
ha}, and
m and the inverse limit of {GQ., Gf. .} consists of l ~ G&. 1
{gwi
1J
1
: li_mGOi -. G & ~ )let ; g :G
(d,+l
p G 6. be the unique map such
that g .g = Gfim for all i; then g is a homeomorphism. cQ1
A category
pair j : & + @ , g :@
+
B
in
Lemma 4.1.
PC k:
admits filtrations with amalgamation if for each
&-.cin 2 , there exists a p a i r
(i) x ( T ) admits filtrations with amalgamation. +
(i) Suppose
structure
&.
L8 - 8 ,
such that f j = gk.
contravariant functor S : X(T) Proof.
f :
63
(ii) The
% preserves limits.
and C a r e models of T with a common sub-
For a model &' of T which extends both
@ and C ,
take any model of the following set V ,of sentences: T; all sentences of
dTB true in @ , where tences of
xTc true in e.
i s the language underlying B) ; and all senThe consistency o i V follows from the
G . E. Sacks elementary equivalence of
< B ,a >a c
A
and < C . a >
a r A'
which is im-
plied by the assumption T admits elimination of quantifiers. (ii) Let { a i . f. .) be a direct system of k ( T ) , and let lLmSOi, 1J
together with
igoo i
: li_mSai
-
~ 6 . 1 be , the inverse limit of {s(ti, Sf. .). I
Thus (Sfij)gmj = gi when i 5 j.
&oo ),
be the direct limit of
g .: S%-
Let
L? , together with {f. oo
{a*, f. .). 1J
:
ai
-
There exists a unique
1
lLmSCL. such that g .g = Sf. 001
loo
1J
loo
for all i.
If g i s one-one,
then g is a homeomorphism, because every one-one, continuous onto map between compact Hausdorff spaces i s a homeomorphism.
g i s one-one, suppose gp = gq. Sf. q for all i. loo
To check
Then g .gp = g .gq, and so Sf. p = 001
loo
col
But then p = q, since any difference between p and q
would be manifested by a formula belonging to F ( T U D ~ . )for some i. 1
Tomorrow I will develop the Morley derivative DS of S, and use it to define the concept of quasi-total transcendality.
5.
B. g.
G. e b. g.
Quasi-Total Transcendality
Today I will finish unpacking the notion of theory of alge-
braic type by giving a definition of quasi-totally transcendental theory, a concept due to L. Blum [I]. The simplest example of the concept, but notthemost misleading, i s the theory of real closed fields (RCF). The axioms of RCF a r e those of T F plus: (x) - ( x < x)
G. E. Sacks (x)(y)(z)[x< y (x)(y)[x < y (x)(y)[O<
8y<
A
0
(x)(Ey)[O < x + x = Y
+ zl
yl
n n-1 ( x n ) ( E y ) [ ~ xly
..
. y]
+o <x
+
21
XI
' x = y V Y<
(x)(y)(z)[x< y + x
(xl).
<
+x
+
+ .. . + xn
RCF i s the model completion of OF,
= 0]
(n odd).
the theory of ordered fields.
(OF
is T F plus the f i r s t five of the above axioms. ) Hence by 3. 3 RCF a d m i t s elimination of quantifiers, f i r s t proved by A. T a r s k i via his improvements of S t u r m ' s algorithm [3].
A l e s s direct proof via the ex-
tended completeness theorem in the setting of saturated models ( a s in [2], p. 92) requires only one algebraic fact: tension of an ordered field
every simple algebraic ex-
a (extension in the sense of ordered fields)
can be found inside every r e a l closed extension of
G.
More generally
L. Blum [l] (or [2], p. 89) h a s given a necessary and sufficient condition
f o r a theory T2 to be the model completion of a universal theory T
G
namely, every simple extension of any model inside every sufficiently saturated model of T
2
The simple extensions of an orderedfield
of T
1
1'
can be found
which extends
a.
a r e of twokinds, algebraic
andtranscendental. A simple extension &b) of & i s completely specified by the polynomial equalities and inequalities over a polynomialover
6 satisfied by
b. (Note:
can havetworoots inthe same cut inthe ordering of
0.)
Any simple extension c ( b ) can be well approximated by a simple algebraic
*
*
extension a ( b ) inthe sense that b
realizes a preassigned finite portion
G. E. Sacks
of the specification of of
a ( b ) . In short, the simple algebraic extensions
constitute a dense subset of S @ . Morleyl s notion of ranked is a
generalization of algebraic, and a theory T i s said to be guasi-totally transcendentalif the ranked elements of SCL a r e dense in S& for every
I k c Pt(T). As in Lemma 4.1 assume T admits elimination of quantifiers. The Morley derivative DS of S : X(T)+ & is defined a s follows. each @ f :
+
E
&T),
p
E
DS@ iff p
c
S& and there exists a monomorphism
R(T) such that
@ in
each f : & -. $3
For
has a limit point in
in X ( T ) , DSf is the restriction of Sf to
that DSf maps DS@ into D S a , fix q c DS@.
~ d 3 . For
DSB.
For some g : fs
To see +
c,
1 -1 (sg)-q contains a limit point, hence (Sgf) Sfq contains a limit point, and s o DSfq
6
DSa.
Now fix p h :
-.
C
-
DSG to find a q
E
DS@ such that DSfq = p.
so that (sh)-lp contains a limit point.
such that 1 (Sh) p in
E
s ~ Jprojects '
~ 8contains '
onto both
s@
Choose
By 4.l(i) there i s a
and Se. The pre-image of
some limit point r.
The image of r in Sd3
i s the desired q. Thus DS : %(T) +)t- i s a contravariant functor. In addition DS preserves limits by essentially the same argument given for 4.l(ii) but making use of the fact that S preserves limits. For each ordinal a , the a-th Morley derivative of S, denoted by a D S, is defined by recursion: DS '
n { ~ " s & l a<
A)
= S; D~+'S = D(D~s);and DA S& =
when A i s a limit ordinal.
A Of course D Sf i s the
G. E. Sacks restriction of sf to D
Lemma 5.1.
A
s@, if
f :
-6
is in
- fk
F o r each a , D ~ :SX(T)
X(T).
is a contravariant functor
that preserves limits.
The proof of 5.1 is by induction on cu. tions of S and limits and
A(T) a r e
The details of the defini-
irrelevant; one needs only that S preserves
X(T) admits filtrations
with amalgamation in order to argue
a s in the proof of 4. I(ii) (cf. [2], p. 176). Suppose there is an cu (necessarily unique) such that p Da+l SQ,.
ff
6
D S&
-
Then p i s said to be a (Morley) ranked point of S& of
rank a , Lernma 5 . 2 (Rank Rule). f :&
-
8 belongs to
Suppose p is a ranked point of S@, and
X(T).
If Sfq = p, then q is ranked and
rank q <_ rank p. The rank rule i s an immediate consequence of the fact that Sf maps D"+'s@ Proposition 5. 3.
D ~ ' ~ S Q for a l l a. Suppose p c S &
integer n such that for a l l f :
- DSa.
-@
Then there exists a positive
in X ( T ) , the cardinality of
-1 (Sf) p i s a t most n. Proof. such that because
Suppose not.
QO =
Then there is a chain f
and (Sfn.
. . flfo) -1p
X(T) admits filtrations with
:an+ antl(n<
W)
has cardinality a t least n, amalgamation.
But then the pre-
G . E. Sacks image of p in
a00
i s infinite and so must contain a limit point.
The proof of 5. 3 applies to every limit preserving functor from
X(T) into 3 , and hence to
D ~ S for all a by 5.1.
If p c S d has
rank a , then the maximum value of the cardinality of f :
& -. @
ff -1 (D Sf) p, for all
in A ( T ) , i s called the degree of p.
Proposition 5.4 (Degree Rule).
If p
E
S a i s ranked and f :
a+@
belongs to ~ L ( T ) then , degp = ~ { d e g q l = ~ pf ~& rankq = rankp). The degree rule follows easily f r o m the fact that x ( T ) admits filtrations with amalgamation. Suppose T is quasi-totally transcendental. of T, denoted by d for all
and defined by Blum [I], i s the least a such that
T
G E U(T), the
ranked points of S @ of rank l e s s than a a r e
dense in S Q . Thus d
+
= w (DCFO i s d ~ ~ r O
The density number
RCF
= 1, a s indicated e a r l i e r in this lecture, and
the theory differentially closed fields of charac-
teristic 0), a s will be noted in the concluding lecture.
It can be shown
that dT is always countable [4] despite the fact there exists a quasitotally transcendental T and an @ of SQ,
6
X(T) such that the
ranked points
do not have their ranks bounded by a countable ordinal. I hope that I have not made Morley rank rxlore mysterious than it
is.
If I have, perhaps the following two observations will help.
First,
the Morley analysis of S d does not require all of &(T) but only those
G . E . Sacks structures in X ( T ) of the s a m e cardinality a s
a. Second, the Morley
analysis of ,512 i s closely related to the Cantor-Bendixson analysis of S&
.
In particular there exists a
cardinality a s
# 3&
such that
@
has the same
@, and such that the Cantor-Bendixson assignment of
rank to elements of S@ i s the same a s that of Morley.
In the next two lectures I will use Morley rank to prove theorems about models of quasi-totally transcendental theories by induction on the rank of the 1-types realized in those models.
6.
B. g.
G. e b. g.
Lndiscernibles and Prime Model Extensions
Today I will show how Morley' s notion of rank can be used
to study s e t s of indiscernibles in models of quasi-totally transcendental theories.
Sets of indiscernibles very much resemble sets of algebraic-
ally independent field elements, and will serve a s "transcendence bases ' in the proof of uniqueness of prime model extensions lying in wait in tomorrow' s lecture. The elements of I a r e indiscernible in
. . ., l n )
2
F(&,
{jl,.
. . ' Jn) '
+ +
F(il.. . . , i n ) whenever { i l , ,
a r e n-element subsets of I and F(x 1'
in the language underlying @ , cernible over Y in
a
.
If Y
C A,
CA
and
. . ,in }
and
. . , xn )
i s a formula
if I
'
then I i s said to be indis-
if the elements of I a r e indiscernible in
y. Any s e t of algebraically independent elements in any algebraically
closed field K i s indiscernible over the p r i m e subfield of K. As in the l a s t l e c t u r e , assume T admits elimination of quantifiers,
&.
and consider
X(T),Y C A
F o r each 6 :,a, l e t p
of A.
Y U{a
E
Y
1
6
and a wellordered subset {a / 6 < a} 6 be the 1-type realized by a 6 over
< 6). (To be p r e c i s e , p6
s t r u c t u r e of
&
6
s @ ~ where ,
whose universe contains Y U {a
Y
BF,is the l e a s t 1 y<
61, but t h e r e i s no
h a r m in ignoring the distinction between the two. ) Suppose p Then {a 16 < a} i s said t o be a Morley sequence over Y in 6 i s a pre-image of p
Y
of the s a m e rank and degree a s p
0< - y < 6 < a. (Sfp6 = p , where f : @ Y
C 63 . )
Y
6
sub-
Y
0
i s ranked.
6?
if pF
when
Any sequence of alge-
braically independent field elements i s a Morley sequence over any algeb r a i c extension of the p r i m e subfield with p of rank 1 and degree 1 6 f o r a l l 6. Lemma 6.1.
Every infinite Morley sequence over Y in h? i s indis-
cernible over Y in $. Proof.
The sequence {a 16 < a) i s said to be o r d e r indiscernible over 6
~ i n I ; : i f
whenever
6 < 1
. .. <
. . . < yn and F ( x1"" ,xn ) i s a formula A compactness argument having < a,y > ye Y'
6 , yl < n
in the language underlying
little to do with rank shows any infinite s e t of o r d e r indiscernibles over Y , a11 of whose m e m b e r s realize the s a m e ranked 1-type (p ), i s a s e t 0
G. E. Sacks of indiscernibles over Y (cf. [2], p. 229). i s established by induction on n. a g l s satisfy p
0
SY.
6
The order indiscernibility
When n = 1, a l l i s well since all the
Fix n > 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 <
T, then
= h6
A ) when A is a limit ordinal.
(iv) If
h6
(iii)
is not a model of T, then by
7 . l ( i ) there is an isolated p a S a g not realized in k 6 ; l e t one such p be p
6
, and
be
let
h? (a ), where a 6 realizes p 6 6
6'
e be any model of T that extends 6?. By 7.l(i) the construction of the %! s can be mimicked inside e . Thus e contains a Let
chain { C 6 ) isomorphic to the chain must be a y such that is prime over
e.
&.
@ y --
because
C,l .
{ a 6 ] Since . But then
6Y
C
is a set, there
i s a model of T , and
is an arbitrary model of T extending
G. E. Sacks Suppose
a C (8 c X(T) and
n > 0.
Define S G , the space of nn
types of T U DQ, in the s a m e manner S& ( = S1&) was defined in
@
Lecture 4.
. . ,bn > E
i s an atomic extension of
if for each n and
n , b > is i s o B , the n-type of S @ realized by < b n l'.'. n
lated.
7.4. and
Proposition.
@I
Suppose
@,
i s atomic over
atomic over
e , and
atomic over
$3 .
Proof.
@
(i) Suppose c
then
C
Lf (?
i s atomic over
i s finitely generated over
c
taining F ( b , x ) , where b S a isolated by G(x).
C Z)@ I&.(i)
i s atomic over (A
a.
(ii) If
a , then
,
is
C
is
C r e a l i z e s the unique member of S@ conE
B.
Let b realize the unique member of
Then c realizes the element of SLk isolated by
(EY)[G(Y) F ( y , x ) l . (ii) It i s safe to a s s u m e
= @ ( c ) and
83 = 6 (b).
Let H(y, x)
isolate the 2-type of S C1?, realized by < b , c > , and l e t q be the 1-type 2 realized by c over 7. 5. Lemma.
every @
E
X(T). Let @ &.
Proof.
fi
Since
extension of
Then H ( b , x ) isolates q in
s@.
Suppose the isolated points of S & a r e dense in S@ for
i s atomic over
(
@.
Ik,
=wig116<_ y})
be a p r i m e model extension of
i s prime over
h!,
&.
@
Then
i t suffices t o find an atomic model
7. 4(i) and an induction on 6 show that the model
constructed in the proof of 7. 3 i s atomic over
k
6
.
Y
a1
G . E. Sacks
7.6. Lemma. of
Suppose
d , and 61
Proof.
e 2fl 3%.If
i s finitely generated over
It is safe to assume
Recall that
realizes an isolated point of S k 6'
* I 6 < ",
implies a
6
and
*
*
7 . 7.
E
Thus
realizes an isolated point of Y
a
* =8, C * =
6
c = u { C ~ * I ~ 7~. 4~ ) .
*
sC6
for all 6 < y.
go also
i s prime over
@ C 63 C C E R(T). @
i s normal over
But then
C
shows
is
Lemma (L. Harrington).
dense in SO for every
&a
in
C
if for
C
belong to
6.
Suppose the isolated points of S & a r e
X(T). If C
i s normal over
Q.
I
s&, all o r none of the realizations of p in
of b?, and
63.
C,*.
Suppose all p
i s prime over
aY ( =u(66 16 5 y)) a6, = (26 (a6), where
C
Define:
C 6 + , = C 6 (a6).
the argument given in 7. 3 that prime over
a, then C
i s the model
constructed in the proof of 7. 3.
u{C,
i s a prime model extension
&
in
i s an atomic model extension
C , then
The truth of 7.7 i s fairly evident when
is atomic over
&
fl.
i s countable; thus most
of the work of proving it consists of a reduction to the countable case via a downward Skolem-Lowenheim construction.
(The basic downward
construction shows every structure has a countable elementary substructure. ) F o r details see [ 2 ] , p. 206. No further lemmas a r e needed for the proof of Shelahf s uniqueness theorem, but one definition remains. closed subset of
s@. F
Suppose
b!
c
X(T) and
F is a
is said to be ranked if every p c F is ranked.
G . E. Sacks The ordinal rank of a ranked F is the least y
2 rank p
for all p
E
F.
If the ordinal rank of F = a , then the compactness of S a implies a t least one, but not infinitely many, p a F a r e of rank a. Let n be the highest degree attained among those p of rank a , and let d be the number of p' s in F of rank a and degree n,
The rank of a ranked F
i s (a,n, d). Let F(x) be a formula in the language of TV DL. F(x) i a ranked over A -
if {p 1 F(x) a
is a ranked subset of S@.
1
F(x) over A is the rank of {p ~ ( xa ) PI.
The
rank of
(The ranks a r e wellordered
by "first differences". ) 7. 8.
Theorem (S. Shelah).
6! (respectively extension of
&
* 6 )
E
Suppose T is quasi-totally transcendental,
%(T),
(respectively
dj (respectively &j
6 * ),
and f : A -. A
9
) is an atomic model
9
is an onto map such
that
Suppose further that every s e t of indiscernibles over
*
[i ) in
-8(respectively 8 * )
formula over A.
i s countable.
(respectively
Let F(x) be a ranked
Define
c = ALJ
{b lb
e
B
&
8
Then f can be extended to a n onto map g : C
Proof.
&
~(b))
+
C
9
such that
By induction on the rank (a, n, d ) of F(x) over A .
Choose
G . E. Sacks p
E
SQ, such that F(x) 6 p, rank p = a and deg p = n.
F i r s t suppose
d > 1. Let G(x) c p isolate p in D ~ S I L . Define
D=A u
Then f
{bib E B & 63
F(b)& -G@_))
* * D = ~ U { b / bBt 8 a* CF@)&-G@)). * can be extended to an onto map h : D D such that +
<8,d>dc
j < @ , f d > d sD, since the rank of F(x) & -G(x)
(cu, n, d-1).
And h can be extended to the desired onto map g : C -. C
over A is
*,
because the rank of F(x) & G(x) over D i s a t most (a,n, 1). Now suppose d = 1. F(x)
E
Thus p i s the unique q r S(k such that
q , rank q = a and degq = n.
A I1transcendence b a s e u
{b616 < y ) for the realizations of p in C-A is defined by recursion: 0 (2) Choose b
(1) p 0 = p. then y
0
= 6.
6
(3) Choose p
E
C-A to realize p6; if no such b6 exists,
6t1
E
Sd(b
Y
I y 5 6)
to be a pre-image of p
of the same rank and degree a s p6; if no such p y
0
= 6t1.
6t1
6
exists, then
(4) p h = U { P 6 1 6 < x).
If y0 i s infinite, then {b6 16 < y0 ) i s a s e t of indiscernibles over
& by
6.1, hence countable.
Reorder the b h l s s o that {b6 16 < y ) 0
I
becomes {b m < a ) for some . a m 0 izes a 1-type over
a@m ( m < a0)
<_ w .
Note that each b s C real-
of r a n k < a, o r of rank a and degree
*= * * c SQ, . Repeat the above in C to obtain * m < cr * ), and observe that a * because B(B*) i s atomic jbm I 0 0 ="o * over a(&?) and , A <@ ,f a > Let p
I
..
a s A'
g = U {g m < w ) i s defined by recursion on m. m
Let g
0
= f , and
G. E. Sacks
2 0.
fix rn
Assume domain g
over domaing
m
m
C C,
(rangeg ), and < B , c >
m
* c domain .
CE
domaing
The definition of g
m
first puts some b a
* c
6
* C
atomic
-m
-
has t h r e e c a s e s .
m
The
in the domain of g: The assumptions on gm yield
whose type over range g
domain g m' m
k
c*, @(a*)i s
range g m C
Let g b =c mtl k
*.
m
i s the same a s that of b
k
over
Then 7.4(ii) implies the assumptions on
lift to g rntl' The second case puts some b
* k
in the range of g, and is similar
to the f i r s t . The purpose of the third c a s e is to add t o the domain of g (range
*
of g ) every b r C (C ) such that b realizes some q over domaing (rangeg
m
) of
rank < o , o r of rank o and degree < n.
be an enumeration of all such q ' s .
Each q
Y
Let { q y1 y <
rn
T )
i s isolated in S(domain
g ) by some F (x). Happily the rank of F (x) over domaing is m Y Y m l e s s than that of F(x) over A. Let q all q
E
* Y
be
(sgm)-lqy. Then
{q
Y
*/ y< r )
i s an enumeration of
S (range gm) of rank < a , o r of rank a and degree < n, such
that q i s satisfied by some b gmtl
{gLtl1
=U
and fix y 2 0. and '8'b'b
<
7 )
Assume @ (
-
E
E
C
*.
Define F
Y
* (x)
i s defined by recursion.
*)
= <' dornain g~ mtl
*
similarly. Let
mtl
=
gm'
i s atomic over domain g Y (range g Y ) mtl mtl
' b'b r domain g~
.
The rank of F (x) Y
mtl
over d o m i n g Y i s l e s s than that of F(x) over A, mtl
'mtl
can be
extended to gytl in such a way that the new domain i s the old plus m t1
G . E. Sacks {b I #3
F
@)I, the
Y-
elementary equivalence of "l gmtl.
By 7.7
{b 128
new range i s the old plus
4
$3 ( B
$3
*
4
F @ )), and the Y
8*
and
-
over g Y now holds over mtl ytl ) i s atomic over domain g "l since m t l (range gm t l ) '
domaingytl (rangeg Yt1 ) is normal over domain g Y (range g Y m t1 mtl m t1 mtl). Y =u{gmtl 1
A
~ e gt mtl domain g so mtl' Theorem 7.9. @,c
over
R(T).
Clearly domain gX i s normal over m t1 A i s atomic over domaing mtl'
63
< A).
Suppose T i s quasi-totally transcendental and
Then any two prime model extensions of
@
a r e isomorphic
a.
Proof.
Suppose
7.2 and 7.5 each p
and [ a r e prime model extensions of (h-. E
S & realized in
8
By
i s an isolated point of S C ,
and i s in fact isolated by a formula ranked over A , since T i s quasitotally transcendental.
Consequently 7. 7 and 7.8 make i t possible to
build up an isomorphism between normal extensions of
b!.
in f i
53
and
and
by proceeding through
d'.
Tomorrow I will discuss the notion of differential closure for differentially closed fields of characteristic 0 .
8. B. g.
G. e b. g.
Differentially Closed Fields
Today' s lecture will be final but not complete.
In the brief
time left before the cessation of formalities, I will t r y to cover a s much territory a s possible.
G. E. Sacks The language of differential fields i s that of fields together with a 1-place function symbol D.
The theory of differential fields of charac-
t e r i s t i c 0 (DF ) i s the theory of fields of c h a r a c t e r i s t i c 0 with two addi-
0
tional axioms:
DFO i s a universal theory originated by Ritt [ 6 ] , but DCFO, the theory of differentially closed fields of characteristic 0, i s a l a t e r invention of Robinson [7].
He proved somewhat indirectly that DF
0
has a model
completion, necessarily unique by 3.1, and then defined DCF the model completion of DF axioms for DCF
0'
0
to be
Subsequently Blum [l] found the simple
given below.
0
Let I; be a differential field of characteristic 0. A differential polynomial over
.
G
in the variables xi (1 5 i
in the variables Dj x.
( j '_ n i )
5 m)
i s a polynomial over
( D ~ Xis the j-th derivative of x. )
Let f(x) be a differential polynomial in one variable.
Assume f(x) i s
in the following form:
ID".)
d
t g l ( x ) ( ~ n x ) d -tl
...
t gd(x).
where g. (x) (1 5 i '_ d ) i s a rational function over
The o r d e r of f(x), o r d f ( x ) , is n, and the degree of f(x)
D ~ X( j < n). is d.
Let
~ be a differential field that extends a. Suppose
b i s a g e n e r i c solution of f(x) over h(x) over
in the variables
k
of lower o r d e r than x.
a
b r B-A.
if fro) = 0 but h(b) # 0 for a l l
c c B-A
i s differential algebraic
G. E. Sacks over
a if
otherwise c
c i s a solution of some differential polynomial over
is differential transcendental over
Bluml s axioms for DCF
:
a.
[l] a r e :
0
h(x) # 0] (ord f(x) > ord g(x)),
(Ex)[f(x) = 0
(Ex)[f(x) = 0] (ordf(x) = 0). The next proposition, extracted f r o m Seidenberg [8] and sketched in [2], p. 298, contains all the algebraic facts needed for the applications of model theory to DF Proposition 8.1.
0
Let
made by Robinson [7] and Blum [I].
&
be a differential field of characteristic 0 .
(i) If b is differential algebraic over
@,
then there exists an f(x)
&,
such that b i s a generic solution of f(x) over generic solutions of f(x) over differential polynomial over sion of
a r e isomorphi> over @
&
.
(ii) Each
has a generic solution in some exten-
I;'. (iii) & has a simple, differential transcendental extension,
and all such extensions a r e isomorphic over Theorem 8 . 2 (A. Robinson).
Proof.
and such that all
DCF
0
G.
i s the model completion of DF
F o r details of the proof of 8 . 2 s e e Blum [l] o r [2].
approach is a s follows. language such that
Suppose T and T
TC T
4
*
0'
Blum' s
a r e theories in the same
, T i s universal and every model of T can
be extended to some model of T
*.
Then T
*
is the model completion of
T if and only if every simple extension of every model well approximated in every model
*
of
* T
@
of T can be
which extends
l8.
To say
G. E. Sacks B ( c ) i s well approximated in
@
i s t o say for each quantifierless
formula F ( x ) in the language of T is a b
* r B*
@
such that
*
u DB such that
bj (c)
F ( c ) , there
~(b*).
8.1 (ii) implies each differential field
of c h a r a c t e r i s t i c 0 has
a differentially closed extension, and 8 . l ( i ) and (iii) imply every simple extension of tension of
i s well approximated in every differentially closed ex-
n
&.
A theory T i s w-stable if SG!
&. E ; ~ C ( T ) , and
i s countable for e v e r y countable
i s totally transcendental if every 1-type of Slh. i s (Mor-
ley) ranked for every
&.r
X(T).
Proposition 8. 3 (M. Morley).
T i s w -stable if and only if T i s totally
transcendental.
Proof.
Suppose p
assume
&
E
$ r x ( T ) i s not ranked.
and a l l other s t r u c t u r e s occurring in the following construc-
tion a r e countable.
p must have distinct unranked pre-images
p1 in Sk?. for some 1 1
h!
s i m i l a r fashion in some obtains an
5.1 makes it possible to
d?. 00
1
extending
bo.
0
and
1 po and p1 must t q s p l i t in l a 1
d2 2al. By continuing on t o the limit, one
in which p has uncountably many p r e - i m a g e s , each of
which corresponds to a path through a binary t r e e .
( F o r the converse,
c f . [2]. Theorem 8. 3 (L. Blum). DCFO i s totally transcendental.
Ci
G. E. Sacks An immediate consequence of 8.1 and 8. 3.
Proof.
F o r each differential field
Ik
of characteristic 0, l e t
unique prime differentially closed extension of 7.
9.
&
provided by 8. 3 and
,?f
Ld .
is called the differential closure of
be the
3 a 3 a.
Suppose
Call f(x) irreducible over d if a l l generic solutions of f ( x ) over a r e isomorphic over
&. @
each f(x) irreducible over f(x) in over
@
belong t o
'63.
is a normal extension of
6?
in
d
if for
a , a l l o r none of the generic solutions of
An extension
if for each n > 0 and < b 1"'
C
@.
of
. ,b n >
E
B
n
is said to be atomic
, there
e x i s t s a finite
s y s t e m S of differential polynomial equalities and inequalities in n such that: b
variables over
,
. . . ,b n >
solutions of S a r e isomorphic over
Su.mmary Statement:
i s a solution of S, and all
a.
DCFO admits elimination of quantifiers.
is a substructure of a model of DCFO, i. e . Then among the models of DCF one denoted by
-
a.
0
extending
(Z is a model of DF
0'
there i s a unique prime
Every automorphism of every normal extension of
G can be extended to an automorphism of d-. fi of
Suppose
DCFo which extends
k , is
that every set of indiscernibles in
-
Ih, i s the unique model
atomic over
33
over
h!
&,
and has the property
i s countable.
The above summary statement applies to every theory T of algeb r a i c type: simply replace DCFO by T, and DF universal sentences t r u e in a l l models of T.
0
by the s e t W of
(Since T i s the model
completion of some universal theory, it must a l s o be the model
G. E. Sacks completion of W. ) Two further general properties of T not mentioned in the summary statement w e r e discussed a t the end of the third lecture The second of the two i m -
for the c a s e of a1,gebraically closed fields.
plies there i s an algorithm, f i r s t discovered by Seidenberg [8], for deciding whether o r not a finite s y s t e m S of differential polynomial equalities and inequalities in s e v e r a l variables over some differential
@.
field
has a solution in some extension of
Let
4
@.
be the rational numbers construed a s a differential field of L, Harrington [9] has
characteristic 0 (D i s n e c e s s a r i l y 0 on Q ). shown onto w
-
i s computable.
To be precise t h e r e i s a one-one map of Q
that t r a n s f o r m s f ,
.
and D into computable functions.
Har-
rington' s r e s u l t i s based on the finite basis t h e o r e m for differential ideals, and leaves open the following question.
Is there an algorithm
for deciding whether o r not an a r b i t r a r y formula in the language of DCFO defines an isolated point of SG ? The answer to the corresponding question for ACFO i s y e s , because there exists an effective method of factoring a polynomial into i t s irreducible factors. Suppose
c
to be minimal over model of T and
H(T),
@
& (C 53
k
8
'rf3
i s a model of
C@
if t h e r e i s no
such that
T.
fi
is said
@ Ce,
is a
e # 6 . ( F o r example the algebraic closure of a field
i s minimal over the field. ) If 7. 8 implies
and
T i s quasi-totally transcendental, then
i s minimal over
h?
if and only if
and every s e t of indiscernibles over
&
in
@
i s atomic over
is finite.
Rosenlicht [ll] and Shelah [12] independently observed that
Q
Kolchin [lo], is not
G. E. Sacks minimal over Q
.
Kolchin shows that the equation Dx = x3-x2 has an
infinite set of algebraically independent, hence indiscernible, solutions in
g.
His paper includes an illuminating exposition of the uniqueness
of the differential closure from the viewpoint of differential algebra. The definition of theory of algebraic type given in these lectures may be much too narrow.
In particular it does not encompass C.
Wood's theory of differentially closed fields of characteristic p (DCF ) P [13].
The language of the theory of perfect differential fields of charac-
teristic p (PDF ) i s that of DF augmented by a 1-place function symP 0 1 bol - (p-th root). The axioms of P D F a r e those for fields of characP P teristic p, the two axioms for D of DF and 0' 1
-
(x)[Dx = 0 -* (xp)' = x] 1 (x)[Dx # 0 -* xP = 01. Wood showed that PDF has a model completion, and defined DCF to P P be that model completion.
(She also observed that the theory obtained by
deleting the l a s t two axioms of PDF has no model completion. ) DCF P P i s not quasi-totally transcendental, but it does satisfy the density condition of 7 . 3, and so every perfect differential field of characteristic p has a prime differentially closed extension.
(The existence of the prime
extension was obtained independently by Wood [14] and Shelah [12]. Wood' s proof is simpler; it uses nothing more than the finite basis theor e m for differential ideals. ) DCF i s stable [12], hence perfect difP ferential fields of characteristic p do have unique prime differentially
G. E. Sacks
closed extensions by virtue of a r e s u l t of Shelah [15] which holds for a l l stable theories that satisfy the density condition of 7. 3. Perhaps quasi-total transcendentality i s not the best choice for the final clause of the definition of theory of algebraic type.
A less restric-
tive choice would be the density condition of 7. 3 coupled with quasi-stability.
N(T) a n d e a c h
Call T quasi-stable if for each f : & - . @ in
isolated p
6
-
1 s&, cardinality (Sf) p
<_ (cardinality
real closed fields i s quasi-stable, but not stable.
W
)
.
The theory of
At this writing it i s
still possible that the density condition of 7. 3 suffices for uniqueness of prime model extensions, a doubtful consummation devoutly to be wished. Grazie dell1ascolto, e buon giorno.
References [l] L. Blum, Generalized Algebraic Structures: A Model Theoretic Approach, Ph. D. Thesis, Massachusetts Institute of Technology, 1968.
[2] G. E. Sacks, Saturated Model Theory, Benjamin, Reading, Massachusetts, 1972. [3] N. Jacobson, Basic Algebra
I, Friedman,
San Francisco, 1974.
[4] G. E. Sacks, Effective bounds on Morley rank, to appear. [5] S. Shelah, Uniqueness and characterization of p r i m e models over s e t s for totally transcendental f i r s t - o r d e r theories, Jour. Symb. Log. , 37 (1972), 107-113.
[6] J. F. Ritt, Differential Algebra, Amer. Math. Soc. Colloquium P u b l . , 33 (1950). [7] A. Robinson, On the concept of a differentially closed field, Bull. Res. Council I s r a e l , Sect. F , 8 (1959), 113-128.
[8] A. Seidenberg, An elimination theory for differential algebra, Univ. California Publ. Math. , 3 (1956), 31-65.
G. E. Sacks [9] L. Harrington, Recursively presentable prime models, Jour. of Symb. Log. , 39 (19741, 305-309. [lo] E. R. Kolchin, Constrained extensions of differential fields, Advances in Mathematics, 12 (1974), 141-170.
[Ill M. Rosenlicht, The nonminimality of the differential closure, Pacific Jour. of Math. , to appear [12] S. Shelah, Differentially closed fields, I s r a e l Jour. of Math,, 16 (1973), 314-328. [13] C. Wood. The model theory of differential fields of characteristic p # 0, Proc. Amer. Math. Soc., 40 (1973), 577-584. [14] C. Wood, P r i m e model extensions for differential fields of characteristic p # 0, Jour. of Symb. Log., 39 (1974), 469-477. [15] S. Shelah, Stability and the Number of Non-Isomorphic Models, North-Holland, Amsterdam, in preparation. Harvard University Massachusetts Institute of Technology Acknowledgement: The author i s grateful to the Centro Internazionale Maternatico Estivo for inviting him to give a course in their Summer Institute, and to the Institute for Advanced Study (Princeton) and the National Science Foundation for partially supporting him during the preparation of these lectures.
CENTRO 1NTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
CONSTRUCTIONS IN MODEL THEORY
H. J. KEISLER
Cr;rso tenuto a Bressanone dal 20 a1 28 giugno 1975
CONSTRUCTIONS I N MODEL THEORY
H
. Jerome K e i a l e r
. CONTENTS
Introduction Lecture 1 ,
The method o f diagram8
Lecture 2.
R e c u r s i v e l y s a t u r a t e d modelo
Lecture 3
.
L e c t u r e 4.
Expansions and p a r t i a l i somorphi umu S o f t model t h e o r y and Lindstrom's theorem
Lecture 5.
Basic model t h e o r y and Robinson f o r c i n g
L e c t u r e 6.
Model t h e o r y f o r l a
Lecture 7.
Model t h e o r y f o r d'mc,
Lecture 8.
I n f i n i t a r y s o f t model t h e o r y
,,
H . J . Keisler
CONSTRUCTIONS IN MODEL THEORY
H. Jerome Keisler University of Wisconsin
In these lectures we survey several fundamental methods of constructing models,
ke examine which logics
admit which constructions, and illustrate their use with examples.
The lectures begin with methods available only
in classical first order logic and proceed to methods available in the infinitary logics
d,
-
and
&,.
We
shall focus on three recent developments which have become more pro~inentin model theory since the publication of the book Chana and Keisler
-7
They are recursively satura-
ted aodels ( L e c t u r e s 2 and 3 ) . aodel theoretic forcing (Lectures 5 and 6 ) , and soft model theory (Lectures 4 and 8). Not all methods of constructing models are discussed here; for instance, ultraproducts and admissible sets have been left out.
For backaround reading we suggest the books of
Chang and Keisler
7 , Keisler
, and
Morley
29. This
work was supported in part by NSF Grant number I\PS 1'1 -
(
3~'1 tji
1
Our notation is standard, but we mention here a few features which may need explanation. infinite cardinals, and 1x1
always denote
.
is the cardinal of X
We
sometimes use vector notation for finite sequences, -3
x = (xl
,...,xn) .
A language
L is a set of relation,
function, and constant symbols, and generates a first order predicate logic with equality. and
~t
.
The notation
,
A for L , the n-tuple A
theory T
J/ L
)I is the maximum of 1 L]
p
means that in the model
satisfies the formula
is a set of sentences of L
.
T
q
($1
.
is consistent
if it has a model, and finitely satisfiable if every flnite subset of
T has a model.
set of formulas model
A
for L
Given a subset B submodel of
,
.
Given a sublanguage Lo C, L and a
&[L~ denotes the reduct of & to Lo of the universe A
.
A with universe B
tion symbol of L
( A _ ~ L ~ )I U*-
j7(v)
We use similar terminology for a
interpreted by
of A_
If U
&
, &I0
is a unary rela-
in A_ , the model
is called a relativized reduct of &
The opposite of a reduct is an expansion, finite expansion L1 = notation fil = ( & , P I ,
P
1P
is the
of
to LO
.
In case of a
L
, we
...,Pn) for an expansion of
use the
A_ to L1
.
.
LECTUaE 1.
THE METHOD OF DIAGRAMS
The method of diagrams was invented by Henkin and A.
Robinson around 1950. Both men used the method for
a new proof of the ~Pc;delcompleteness theorem and gave other applications.
The idea of the method is to study
models by adding new constant symbols to a language. DEFINITION. A
, the diagram
Given a model
language for A_
4
for L with universe
is the expansion LA
of L
forned by adding a new constant symbol ca to L for each a6 A LA
. The diagram expansion of
such that each ca
ly, given a subset X for
+
A
is the model
is interpreted by A
, the
language
are defined in the natural way.
diagram of 4 expansion of A_
a
+
.
AA
for
More general-
and model
The elementary
is the complete theory Th(hA)
of the diagram
.
In model theory the most useful consequence of the completeness theorem is the compactness theorem.
As a
warmup for these lectures we give a direct proof of the compactness theorem using diagrams.
AX
T be a set of sentences of
COMPACTNESS THEOREM.
Jf every finite subset of T has a model, then T &as
L.
a model.
P
Proof; Let C
of
K
il
.
Ll1 = K
Form LC by adding a set
new constant symbols, and let ,
K
be an
Form an enumeration of the set of all sentences of L C * increasing chain T,
,~
i
k
of , sets of sentences of LC
with the following properties: (1)
To=T.
(2)
Fewer than
(3)
T,
(4)
If (P,c
5
occur in T x
.
is finitely satisfiable.
, then
is consistent with T,
If then
constant symbols from C
K
is of the form Y ( c ) ~ < + , for
Let T
h
=LicKT,.
33'ltz)
some c 6 C Then T k
4,
6
T,+l
.
and is cnnsiatent with Tq
. contains T and is
maximal finitely satisfiable in LC
.
Let A
be the set of
all equivalence classes of constants ' c , 6 C under the relation cdd There is a model
&
iff
(c=a) G T~
with universe A
atomic sentence Y of LC
.
.
such that for each
A
%
(6)
p-q
if and only if
~ 6 % .
By induction using ( 5 ) at the quantifier step, it can be shown that (6) holds for every sentence cP of LC
&
, and
is a model of T K
the reduct
4
.
Thus
is a model of
r.4
The compactness theorem has a great variety of applications. We give two examples concerning the important notion of a stable theory. Let X C A
DEF!NITION. over X by
a
and a
is the set of all formulas V(v)
&
in the model
.
A
-stable if for every model power at most
K
, A
A
has at most
than
and an X < I;
over X K
of a
in LX satisfied
of
different types over X,
K
Let L n
CA
e
, h
t
,
be -- countable.
T is
-
K -stable
.
Proof: Assume T is not model
The
and every set X
If a wepry T in L -i_s &)-stable h
.
theory T is said to be
(Morley 22- ) .
THEOREM 1.1.
for al!,
A
6
if both
types over X
of power
A
types over X
K
.
-stable. , ih
Then there is a
such that & has more
Call a formula
q:(v)
of L2( rich
ci'(v) and its negation belong to more than
.
We may form a binary tree
of rich formulas over X where the arrows represent logical implication and formulas on different branches are contradictory.
Let Y be the set of constants in X which occur in Then Y is countable and the branches of the
this tree.
ur
tree may be extended to 2 over 9
.
By the compactness 'theorem there is a model
which has 2LL
types over Y, so T
THEOREM 1.2.
A
different consistent types
( lV?oiniey 2 2 )
is not
. Suppose
such that for some infinite X
GA
c2 -stable.
K
,T
REMARK.
is not
$
T has a model
-and formula G (u,v)
the relation { ( a , b ) : i ~ n , ~ , bb) c ~ j linearly orders X for all
51
,
. Tan
k-stable.
If the variables u, v in
V(u,v)
are
replaced by n-tuples, the converse also holds and gives a ilur fo J L - I U h 2'1
fundamental characterization~oftheorles unstable in all k Proof: -
We use the fact that for each k
linear ordering (Y,<)
or power greater than
there is a K
with a
.
H. J. Keisler
whose axioms are T
TI
Let TI be the theory in Ly
together with
y
1
'j
.
of power K
dense subset Z
:
x
-
7
Y
j.
is finitely satisfiable, hence has a model
Since Z exists z
is dense in Y r
Z with x
Therefore x
and
has more than
LECTURE 2.
h
, if
x < y
in Y
-.z i y , whence
-
AY
then there
Z
, I -1 '(
q (x,z)
y hfive different types over Z types over Z
, SO
T
is not
.
L
(y,z)
.
, and A
-stable.
'4
RECURSIVELY SATURATED XOiIELS.
Historically, saturated models have been a central tool in model theory, and gave a uniform method for proving a large number of results, However, saturated models cannot in general be shown to exist unless one assumes the continuum hypothesis os the existence of inaccessible cardinals, and most theories have no countable saturated models.
In the next
two lectures we discuss a pleasant recent discovery of Barwise
and Schlipf
recursively saturated models.
They have most
of the advantages of saturated models but are easier to find.
For simplicity we assume that L is a finite language. We shall require only an intuitive knowledge of recursive sets.
The main facts we need are that there are only
countably many recursive sets, and whenever we write down a scheme to describe a set of formulas, the set is recursive,
A
DEFINITION. a model every finite set X 5 A
,
for L is L-saturated if for
every set of formulas
Ly which is consistent with Th(kX)
-A
Y-(v)
of
is satisfiable in
&.
is said to be recursively saturated if for every finite
set X -- A
, every
recursive set of formulas Ct (v) which is is satisfiable in
consistent with Th(kX) Obviously, every saturated.
. -saturated model 2
, but
.
is recursively
Every consistent theory has an
nodel of power at most
&
L L
-saturated
not every consistent theory
has a countable - -saturated model.
A
counterexample is the
theory of real closed ordered fields. THEOREM 2.1.
Every cogsis-tent theory T has a finite
or countable - recursively -saturated model. We leave the proof to the reader.
The proof is like the
proof of the compactness theorem, and depends on the fact that for each finite X
,there
are anly
~ o i i q t & h J ymarry
.
recursive sets of formulas in LX
We now develop some properties of recursively saturated models.
It is easy to see that reducts and relativized
reducts of recursively saturated models are recursively saturated. DEFINITION.
A model
A_
a ,b
is said to be u-homop;eneous
4
if for any two n-tuples for all c € A
such that
E
there exists dt- A (A,Z,c)
THEOREM 2.2.
( Schlipf
saturated model- is -- - -. -- - woof: Let
-
z (e,%,d)
LL
x
=
?r
.
(4.a) 5
such that
. -Every -- .-. recursively - -
-homogeneous. - -..-
~a,;,~? .
we must find
A
which satisfies the following set of formulas 2 (v) in
,
Y ( i , v ) :
+[<,")
a C o . - w * l ~ ~o f
'5. (v) is recursive, and is consistent with T h ( f & ) q(?,u),
for each
(1) *(2)
(1)
&P~(G,c)
(2)
g(
I=
3 u ~(Q,u)
(3)
&
P
3v ~ / G , v )
Therefore
'=7(3)
.
(Barwise),
A
&:
L j because
.
2 (v) is satisfiable in Ax
DEFINITION.
(A_,b),
.4
I s said to be res~lendent
H. J. Keisler
if for every finite X 5 A expansion Ltr(
of LX
,
and every sentence 9 in an
if
then some expansion
of
is consistent with Th(&)
&
satisfies cF
.
This notion is analogous to recursive saturation but instead of a recursive set of formulas
V(v)
we have
a single sentence CP with extra relation or function symbols.
Using a classical result of Kleene in recursion
theory, Barwise shoned that every resplendent model is recursively saturated.
In general, an uncountable recur-
sively saturated model need not be resplendent. We shall now prove that every countable recursively saturated model is resplendent, THEOREM 2.3. saturated mode2
(Barwise).
Every countable recursively
is _~esp&e$de_n_t.
Proof: We work in
the language Li and use an
argument similar to the compactness proof. be a list of all sentences of L
and let 4 Xe
be a sentence of
form a sequence y',
-
, y., , y ,,. . .
with the following properties. (1)
I=
Y,,+,
, Let X
q,,
Let 00, el,...
be finite
consistent with ~h(% ). of sentences of LA
H. J . Keisler
.
is consistent with Th(&)
(4) 1f en is consistent with Th(&)
Yn+pen
I=
5
en
If
.
is of the form
~h(&)
yq ,
and
Y/, , then
and
dve(v)
then b
and is consistent with
Yn+l-7
for some at A.
€)(a)
Recursive saturation is needed for property ( 5 ) . assume the hypotheses of (5). We have finite Y of
.
Then
'1 '
2 (v)
satisfies
=
yZn
NOW
-
e(a)
and
( 5 ) that
.
4&(r.l
By
such that a is con-
q l , , so we take
.
let T =
a model of T
.
in L y , It follows that e(a)
v
sistent with Th(&)
(1)
of formulas
is consistent with Th(ky)
recursive saturation there exists a & A
!n+i
Z(v)
There is a recursive set
t,,
for some
which has exactly the same consequences as Y,,'
Ly
in L
, (96L?
Given
LA
J v:, v;, Y,,
-
-
}
.
has an expansion
Therefore
A'
~t fol1.0~~ from
hi
to Li
satisfies (F
.
which is
-/
H. J. Keisler
LECTURE 3.
MPANSIONS AND PARTIAL ISOMORPHISMS.
A simple but powerful method in model theory is the To prove a new theorem for a given
method of expansions.
, use
language L
.
language L'
an old theorem applied to an expanded
We shall illustrate this method using
recursively saturated models in an expanded language.
For
simplicity we assume that the original language L has predicate symbols only, and is finite, L = DEFINITION.
Let
and
with disjoint universes.
B
f P I ,...,Prj
.
be two models for L
The model pair
(A,g) is the
model for the expanded language L(U,V) =
{ PI,.
..,Pr,U,~f
given by (&,B) =
Thus both
4
A (A L'B,P1 I I i3P ,~
and
B
..., PrA L
B ~
, PA,
are relativized reducts of
~ 8)
(A,@)
We shall use the fact that any model pair (A_,B)
.
. is
elementarily equivalent to a countable recursively saturated model pair
(C,g)
.
First we introduce the notion of a
partial isomorphism, which is due to Karp 11 and is important in infinitary logic.
DEFINITION.
partial isomorphism I:& gp B
A
relation I on the set of finite sequences elements of A
x
01 0
(i)
(ii) If
(2,g)
is a of
B such that:
. I
(~,6)and ,
then
satisfy the
same atomic sentences. (iii) If 6 I
,
such that 2'
-A
y
P
6
then for all cc A
, and
I d
there exists dc B
vice versa.
g means that A and g are isomorphic, and
g means that
A
and B are partially isomorphic.
We shall see later $hat 4 but weaker than A 4 8 THSOREE
P
.
3.1. (Karp 1
1
.
-B
g
is stronger than A_
Ir
A_
and 9 are cotantab*
and partially isomorphic, then they are isomorphic. The proof, which is a back and forth construction, is left to the reader,
THEOREM 3.2. model pair
-
Thus if
A
&,B)
(Schlipf 7
.
A L' B and-t_h_e
is recursively then 4 .- -.- -- - --. - saturated, ---- . are also countabLe, then
A Y2
TPg
.
.
Proof: This is essentially a generalization of Theorem 2.2.
We skip the details but remark that the partial
isomorphism I is given by
- -= (g,b) .
-.
a I
if and only if
d
(&a)
j
We shall now give some applications of model pairs of recursively saturated models.
In each proof S e l ~ w ,
(C,D) will denote a countable recursively saturated model elementarily equivalent to a given model p i r ROBINSON CONSISTENCY THEOREM.
. Then
-
the language L1 L' L2 Proof.
Th(A)
Suppose A and g L2 such thst Lo = L1 L2
are models for languages_ L1 gn& and A/LO Z E ~ I L ~
.
(A_,g)
Th(El)
L
isconsistent in
.
The notion of s mod-elpair can be extended in
the mtural way to models fox. different languages. Form the model pair
(A,g)
and let
(C,D) be an elementarily
equivalent countable recursively saturated model pair. Then C I L ~ g E Flake D_
r ~ ,~so by
Theollem 3.2, CrLO 2
into a model for Ll L:L2
symbols of L2
- L1
.
by interpreting the
by their images under the isomorphism.
The resulting model is a model of Th(&) EXAMPLE 1, (Tarski 3 1 ordered --.--
:-D~L~
.
L/
Th(g)
.
The theory of real closed
fields is complete, -
Proof, Let A_ an8 g be real closed ordered fields.
Let
(g,D) be an elementarily equivalent countable recur-
sively saturated pair.
It suffices to show that
C g pD_
.
Using recursive saturation, it can be shown that the following relation I is a partial isomorphism.
& I % if and only if for any polynomial P(;) rational coefficients, ) : ( P EXAMPLE 2.
-7 o
"0
iff p(s)
( presdburger 2 9 )
with
, j
. The followinn axiomsT
characterize the theory -- --- - -- complete -
of the inte~erswith
order and additi-on,i. e . the model
(z,+,-,O,
-, 0
I,< }
.
1.
Ape-liangroup axioms with +,
2.
Axipms for linear order.
4.
1 is the_ least element - -greater than 0 -2
.
.
the axiom 5. For each integer n , ---- - --positive --.-. bx (
njx
v
where n/x m~
...
nJx+l L
3 y(y
+
... + 6
Proof. Let as before.
A, g
L n)x+(n-1) y = x)
.
1
be models of T, and
form
(C,g)
Using recyrsive saturation, one can check that
the relation I defined as follows is a partial isomorphism from C to
2.
I
if and only if for each linear function P(;)
with integer coefficients, and each positive integer n
P(;)
-)
0
I
n) : ( P
if and only if
~('i;) 7
if and only if nip(;)
0
.
LYNDON HOMOMORPHISM THEOFEM.
(Lyndon 2 0 ) .
A
is true in g
every positive sentence true in there exist models
C
homomorvhic image of C
& a -
2 5 B such*
,
, -i Suppose
.
Txn
Q La.
.
In the above theorem, a positive sentence is a sentence whose only connectives are A and phism
is a mapping f of C
atomic formula satisfied by
g
in
.
t,
.
A homomor-
into D such that every in
2 is satisfied
by
2 is a honomorPhic ima~eof 2 if there is a
homomorphism of C onto g Proof.
Let
.
(2,D) be countable, recursively satu-
rated, and elementarily equivalent to that Q is a homomorphic image of C
, .
We
Let
.
show mean
2 -
that every positive sentence true in (2,a) is true in -L
(Il,b).
fg
Usiw recursive saturation, I is a partial
homomorphism in the natural sense. By a back and forth argument there is a homomorphism of C onto Q
.
j
LECTURE 4. SOFT MODEL THEORY AND LINDSTROM'S THEOREM
Llndstrom's theorem shows, roughly, that clarsslcal first ooder logic is the strongest logic which has the
properties we used in the first three lectures, namely the compactness theorem, the Gwenheim-~kolemtheorems, and the method of expansions.
In order to state such a theorem,
we need a general notLon of a Logic.
Lindstrom's theorem is
the beginning of a subject called soft model theory, which studies logics from an abstract viewpoint.
It has been
developed recently by Baruise, Peferman, Friedman, Makowsk?, Shelah, Stavi, and others.
DEFINITION. An abstract logic is a pair of classes
( x * ,/=*)
with the following properties.
of sentences and k *
is the satisfaction relation.
(i) Occurrence ~ro~erty.FOP each associated a finite language Lq
. A_
occurring In
9
and models A_
4
VC't
, called the
z* there is
set of symbols
/=*p is a relation between sentences
for languages L 3 L
(ii) Exmnsion provert~. If A_ expansion of
* is the class
9
b * c ~and , B
to a larger language, then
B
*
Is an (&'
.
b* y,
then
k* cC,
and &
(iii) Isomor~hism~ro~erty.If A 2 g
. *
(iv) Closure ~roverty.
contains all atomic sen-
tences, is closed under the connectives A
,
V
,
1
, and
the
usual rules of satisfaction for atomic formulas and.:onnectives hold for
.
P*
(v) Quantifier vrovert~. For each constant symbol c
i-
L
[f
there are sentences
set of symbols L q)
- {c )
3 vc y
b'vCV and
with the
such that
A
k*
\dvcq iff for all a € A
,
(&,a)
A_
b*
3vcY
,
(L,a) P * q
iff for some a € A
b*
q ,
.
(vi) Relativization ~ropertx. For each sentence 4 in
,i*
and relation R(v,;)
Lq, , there is
with 8 ,
A
a new sentence y' '(v*~)
, read
relativieed to A
such that whenever g = A_/ {a & A : R(a,b) (A,R,T;)
1
R(V,<)
is a submodel of
,tQ(e
Before stating any theorems we list some of the most widely studied examples of abstract logics.
Each axample
is described by adding one or more rules for building formulas to the rules for classical first order logic, and
, 1,
taking for
*
the class of formulas which are sentences
(i,e. have no free variables) and have only finitely many nonLogicsll symbols.
i d _ This .
EXAMPLE 1.
zQl4.
M*MPLE 2.
is classical first order logic.
Add the rule:
countable set of formulas, then /1$ and
.
EXAMPLE 3. f m ,
formulas, then EXAMPLE
2
3.
8
is a formula and U
d U 4 and 3U
@
and
K
Q )~
-
I
is a countabld
are formulas.
. Add the rule: if
elements a 6 A
1 {a,b)
is a set of
If @
is a countable set of variables, then
f(
X WO.
EXAMPLE6.
relation
8
are formulas.
5.
then (Qvouv)+
if
are formulas.
~8
formula, ( Q k v )y) is a formula. & at least
is a
zQLUl . Add t)e rules: if' @
set of formulas, then A
EXAMPLE
V9
-
~ ' 4are formulas.
Add the rule:
and
$
if
such that
Add the rule:
is a formula. :
/;l (QKv)
(*,a,b) P'V
P is a iff there are
(&,a) if
cC:
.
is a formula,
Y ( Q ~ ~ U V4) iff the is a well ordering].
We next introduce sevenal properties of abstract logics which are motivated by basic results for DEFINITION.
An abstract logic
( $-
*, k * )
.
fc,,.
is said to
be countably comm&
if for every countable set T
< z* ,
if every finite subset of T has a model then T has a model. The ~'dwenheimnumber of ( nal
such that every sentence of
k
has a model of power at most The Hanf number of J
b, /=*)
(
K
power at least h
is
is the least cardinal which has a model of
has models of arbitrarily large power. has the Karp prouertx if any two models
%*, b*)
(
l* which has a model
.
z*, h*)
such that every sentence of
is the least cardi-
which are partially isomorphic are elementarily equivalent with respect to
(
1*,
/= *) ; in symbols,
2
-P
-B implies
The classical downward and upward ~owenheim-~kolem theorems show that number w
.
exist if
L*
Note: -
L:h
.z 1 -B
i,
L;
has ~gwenheimnumber e and Hanf
The Lowenheim number and Hanf number need not is a proper class, for example
If L
X,,
.
has Eunction or Constant symbols, then
is defined to mean that the correspnnding purely
relational models are partially isomorphic b j I. The following table summrizes the behavior of our exaqples,
H. J. Keisler
countably compact
Logic
2,L'd 1 4
-
cll
v? 0
L(Lu) (Qw,) t
THEOREM 4.1.
K (
w
(
L*,
*I
Let
A'
and l e t
YeJ
c
Yr J
zU)
ver/ jar qe
1.10
b7 0
(2
luck
Y 'i
y r j
wI
1,
no (Hanf
LW
11o ye 1
041
I
7). I
f
very
%*
luC-yT
is a s e t , then t h e (
*
*
exist.
PC-Z*, l e t @
.
Let
has a r b i t r a r i l y l a r g e models, and
be
.otherwise.
has ~b;uenheimnumber and Hanf number
THEOREM 4.2. Lowenhelm number
1'0,
l'b D
the supremum of t h e cardinals of models of Then
Y eJ
v)
For each consistent sentence
if Q
cu
!?gfl e
be the l e a s t cardinal of a model of
U
(Karp prop,--
r? 0
Ldwenhelm number and Hanf number of Proof. -
Hanf number
'-V
Ye3
A,, y 0,
f b4,,L,
~Gwenheim number
(Barwise Lt,
, then
.3 1.
If ( f * ,
h * ) w.
i t has the Karp property.
'be t h e s e t of f i n i t e sequences of elements of F:A8 x A + A 8
be the function
A
Define B'
and G analogously, Since A_
B
2
-P
we can
make A n B be the set of constants and form the expanded model
(,B,A,B,F,GI).
By the closure, quantifier, and
relativization properties there is a sentence
y'cl*
which states that
Sfnce the ~owenheimnumber is w , y' has a countable model (A+,F&,A~,B;,FO,GO,IO)'
+A
and
F&
are countable and
partially isomorphic, hence isomorphic. and +B
p
*
7
,$
, contradicting the
But
/= * (4,
isomorphism property.
-/
We conclude that the Karp property holds.
(71 LINDSTROM'S THE0REM.q XKL
&
( L , , r r t ~t r i m
Log_icU.w~gch has ~Genheirnnumber
is the ~ l ! ,abstract and one of the following:
LL
(i) Countable compactness; ( ii)
Hanf number
ci ,
~ r o ~ f We . prove the case (1).
Let
abstract logic with ~owenheimnumber compactness, We must show that equivalent ko some
'+(.A ' L L ' ~
A_ b a y ,
c d
fi*,k*) be an
and countable 9% J *
every
i.e. for all A_ iff A _ / =
is
,
y .
It is sufficient to consider a finite language L with only relation symbols.
Given two models 4 and
B
for L
and
H. J. Keisler 2
3
m-tuples a and b
we define a In b inductively as
-.
-3
follows.
2
; and
a IO b means that
atomic formulas.
a
2
,
b satisfy the same
-a
. z
b holds if and only if: a
(1)
for all c C A
(2)
vice versa.
there exists d G' B
. There is a finite set
& Zn _B means $ I, 0
9,
satisfy the same sentences of
*
Let some n
,
(3)
-A
for then
yn
.
L v G L.
Zn8 and 4 /=+ y
In b,d,
of and
.
It suffices to show that for
implies
2
,L*9
,
y must be equivalent to a Boolean combination of
Suppose ( 3 ) fails for all n
Ln -zn En A, k*+
( '-;I
Gn
1 zn 8 if and only if
sentences of i , , such that
-B
with ;,c
.
and
9
Choose models &,En
En k* TG;
Me can now form an expanded model (G,E,R,S,c
~ ,5
such that for each n
kn
=
,...1
,
~ l l a: ~(a,n)i , 8, = Qllb
and there is a sentence 8 6 all n
.
*
: ~(b,n)),
stating that (4) holds for
By countable compactness, 8 has a model (G1,Q',B',S',
w',
$I,...)
3
H. J. Keisler suoh that (a1,5
&i
I)
,
gi
has a nonstandard element
b*lg ,
is a partial isomorphism between (
k *,
*)
Then
m-tuples given by
$ and
.
But
has the Karp property by Theorem 4.2, so we have
a contradiction. sentence in
.
EHgi.
and
It follows that the relation between
H
We conclude that
Ywd.
-I
($
is equivalent to some
LECTURE 5.
BASIC MODEL THEORY AND ROBINSON FORCING.
In this section we present an alternative development of model theory which is more appropriate for the study of algebra.
We concentrate on basic formulas, i. e. atomic
formulas and their negations.
Much of model theory can be
developed in either of two parallel ways, emphasizing basic or arbitrary formulas.
Basic model theory seems more special,
but it is often more general because one can make every formula equivalent to a basic formula as follows. relation symbol R(;)
to the language L
( - ( < ) , expand each theory of
r3
R();
+
P):(
, and
Add a new
for each formula
L by adding the axioms
uniquely expand each model for L
to
satisfy these axioms. The basic analogue of the elementary diagram of the ordinary diagram of A_, denoted by
5
is
D(A).D(A) is defined
as the set of all basic sentences true in
.
In basic
model theory we concentrate on inductive theories rather than arbitrary theories.
A theory
if it has a set of v 3
T is said to be inductive
axioms.
It is easy to see that the
union of any increasinq chain of models of an inductive theory
T
is again a model to T
.
The notion of a K -stable theory
has the following basic analogue. Given a set X (: A
DEFINITION. the basic type f o Cf'(v) in LX
oxr X
satisfied by
to by&sicallg set X < A
a
X
a
and an element a E A ,
the set of all basic formulas
in
.
A_X
k - a b l e if for every model 4
of power at most
K
,4
T is said
A theory
of T and
has at most K
basic
Cypes over X , The analogues of Theorems 1.1 and 1.2 hold for basic stability and have similar proofs.
Thus every basically
-stable theory is basically ,<-stablefor all
lc
.
Basic
stability is a more natural notion in algebra and it is easier to find examples. Examples of basically
L ,
-stable theories: Fields,
abelian groups, integral domains, differential fields of characteristic zero. basically Examples of theories which are/stable in some but not all k
(in fact for exactly those
r<
with
k- = k L c ):
Differential fields of characteristic p with a pth root Su t c l c i a n t i ? symbol, R-modules where R is a fixed non-trivial countable
A
H . J . Keisler
Examples of theories which are basically unstable in : Linear order, ordered fields, fields
all cardinals
with valuation, groups, division rings, boolean algebras. We shall now develop Robinson's notion of finite f o r In the next lecture we generalize forcing to the
C f“f-19,
infinitary logic
f,, I
. We shall see then that forciq is
the key constuction which gives
Xu,
a well behaved model I
theory. Let L be a countable language with an infinite set C
of constant symbols (there may be other constant symbols
outside C).
Hereafter we let T be a consistent inductive
theory in which no c DEFINITION.
A
C
occurs,
condition for T is a finite set of
basic sentences which is consistent with T denote conditions for T DEFINITION. by
The relation p forces 4 p /hT (CI
p and arbitrary sentences
p 11-7 cp p
I/- 3
is atomic, p [ t ' f '
...
of L
iff p
, denoted
, is between conditions
. It is defined by:
iff ( e 6 . p .
iff there 1s no condition q u Q(u)
p, q,
.
p / / - q ~ or if necessary
~f 'V
.
q(c)
2 p with
for some c E C
q)f4J,
.
H. J. Keisler
In this definition we take
7
,V
We say that p weakly forces 9
,and 3 as fundamental.
,
P j=
Wcp
, if
P lt-77 P/
(i) Lf- p G.q and F ) / - ~ P , then qlk-9
LEMMA 5.1.
(ii) If cqc P tu!!??? pJI'C4, ( iii)
exists r 2 q
p 1kw
if and only if for all q 2 p there
with -r It-
q,
To construct models by forcing we introduce the notion of a generic set.
The following theory is due to Robinson 1 8 .
DEFINITION. A generic set for T
is a set G of
basic sentences such that each finite p
I. G
for T and for each sentence Cp of L
, there
which either forces cP or forces
.
some p C G forces LEMMA 5.2,
set G -
7
@
.
G
is a condition is a p c f G
cP means that
For every condition p there is a ~eneric
containing p
.
GENERIC MODEL THEOREM. For each generic set G for
T -there is a unique (up to isomorphism) - model A(G)
for L
such that (i) Each a 6 A(G)
is the interpretation of some
(11) For each sentence CC' DEFINITION.
4
L
,
A(G)/=~
is a T-$eneric model if &
some generic set G for T
. I
iif GI~P,
&(@)
for
.
The proof of the Generic Nodel Theorem is by an induction on the complexity of formulas similar to the proof of the compactness theorem. (1) p l / - w q if and only if
THEOREM 5.2.
in all T-xeneric models &(G) such that p C G
.
(11) Every T-generic model is a model of T
Model Theorem.
(ii) depends on our assumption that the
3
axioms of T are 2
d
Consider an axiom of the form
, that is,
CI
3; 737'~(:,:).
, then some
, so some
. Then
7
y%'(t?,;)
hence forces 1 3
gome
.
-
rails in some b(G)
tent with T
.
(1) is an easy consequence of the Qeneric
Proof.
vx3?4'(%,$)
holds
q
p
If this axiom
forces 3 5; '73 9 ~ ( z , ? ) , 4
for some c
. But
p
2 p must contain rp (z,:)
forces
3
is consisfor
y (P(z,;) , a coni!radiotion. 4
The following theoremsgive a feeling for the nature bf the T-generic models. DEFINITION.
A
model A_
of T is existentially closed
in T if every existential sentence which holds in some
extension of &A
.
which is a model of T holds in &A
If the class of all existentially closed models in T is characterized by some theory T* model companion of -
in L
,
T*
is called the
T,
Examples: The theory of algebraically closed fields is the model companion of the theory of fields.
The theory
of real closed (ordered) fields is the model dompanion of the theory of formally real (ordered) fields. The theory of griups does not have a model companion. For a survey of this extensive subject see f ! ; ~ ~ c h C ; e / ad d Wlleelel-
THEOREM 5.3. ---tiall~closedin
THEOREM 5.4.
[c)
Every generic model for T is existen-
T.
If
T has a model companion T*
,then
the class of a11 T-generic models is equal to the class of
-
all countable models of TW
.
The proof of 5,3 is straightforward, The proof of 5,4 uses 5.3 and the fact that TW is model-com~lete,that is, if is a model of T* and A+(=.w To prove 5.4, let the diagram of then
G 11- ip
.
A,
then T*
U
D(&)
cQ
A be a countable model of T*, let and show by induction that if
. G be
A+, )= cq
It then follows that G is generic j
bAzd(GI.
LECTUBE 6. NODEL THEORY FOR
h, 1 ,.
When we go beyond classical first order logic, LinBstrom8s theorem shows that we must do without either countable compactness or the downward ~Swenheim-~kolem theorem.
The logic A,,,
is not countably compact but still
has a well behaved model theory.
This is becauae the
forcing construction is available as a kind of substitute for the compactness theorem. Another logio with a well behaved model theory, which we shall not have time to develop here, is the logic
$-( Q 1.
9
This logic is interes-
ting becauae it is countably compact but does not have ~owenheimnumber
(J
. Kim Bruce 6
that the model theory of
L (Q,
)
has recently show% can also be Besad on a
forcing construction. Historically, there are two separate lines of development of the forcing construction in
i y ,, the model exis-
tence theorem of Smullyan and Makkai, and Robinson forcing. The relation between the two approaches 8nd the generalization of forcing to L y ~is explained in Keisler /Y
.
xu,,
is like
kb,
but allows the conjunction and
disjunction of countable sets of formulas. preceding lecture we regard 7
,
We assume in this lecture that L an infinite set C
, and
v
As in the
3 as fundamental.
ie countable and contains
of constant symbols.
Denote by
Low I
the set of all sentences 'P of are in L
of cp
S
of
such that the eymbols c C C
and only finitely many
DEFINITION. subset
4zW
By a forcing
LbIu
base
occur in
we mean a countable
which Contains all atomic sentences of
L, and is closed under negations and under substitution instances of subformulas,where free variables are replaced
.
by constants in C DEFINITION. property on S subsets
p
(il) If (iii) If
some
c6 C
be a forcing base.
S
we mean a nonempty set
p 6 P
p / = 9 and
v
If 3 u
.
and
g
~
V(U)
q
cp
(p€S
then q a P then
.
pulp)
pthen p v l ~ ] € P 6 p
By a forcinq
P of consistent f$fe
C S such that:
( i ) If
(iv)
Let
f P.
for some ~ t @ .
then p w { ? ( c ) ) E
P for
.
The elements p 6 P are called ~0~dibi0ns of P By (I), the empty condition # always belongs to 3 DEFINITION. The relation p forces CP
PI/-@
, is
or p
arbitrary sentences
If Q
($6
.
.
, in symbols
between conditions p and
LCIjLC). It is defined as follows.
is atomic, then p
iff
cp6 p
.
pil-7Viff thereisno q 2 p in P with q / 1 ^ 9 . p / t 3 u Y(u) iff p 1)-'F(c) PI,-V$
iff p
it 9 ,
P weakly forces
for some c 6 C
for some
p I ~ ~ (, Pif
~
6
.
.
$
P )t77@.
When we take S to be the set of all basic sentences of L and P the set of all finite subsets of S consis; tent with T
, we
obtain Robinson forcing.
More generally,
the set of all finite subsets of a forcing base S satisfiable in a class of models K
is a forcing property.
We
shall use still other forcing properties later in this lecture. The basic results on forcing are now familiar. LEMMA 6.1.
(i)
g
(ii) kf Y C p then P I/-" (iii) p
/kWg
p)tLP
p C q
then
q/k-@,
LC,,
iff for all q 2 p there exists
r2q
such that rll-4'. DEFINITION. A subset G G S is said to be peneric
for P if: (i) Each finite p ( ii)
cG
.
is a condition in P
For each (?6 S there exists p G G
such that
pltrq or pIi-7(4 We write G IF
CP ii. p 1)- C4 for some P
G G
.
LEMMA 6.2, Each condition p L P _belongs to some generic set.
GENERIC MODEL THEOREM. For every generic set G P there I s a unique (up to isomorph*)
(i) Each element of A(G) some c € C
model
We call
&(G)
@G S
,
such
A(G)
is the interpretation of
.
(ii) For each
lor
A _ ( G ) / = ~ iff
G 2kq
.
the peneric mod21 generated by G
.
We now give a variety of applications of forcing in
2 Li,LCL These applications use the following countability lemma.
LEMMA 6.3.
Each countable set of sentences of
can be extended to a (countable) forcing base.
L
(7''
There are
2
Lci
different sentences in L W, w
.
The
lemma depends on the fact that L has countably many symbole and each single sentence has countably many subformulas.
has Lgwenheim number Proof, Let
cl-)
be a consistent sentence in L LL
V
(LC
.
Let S be a forcing base containing i k and let P be the set of all consistent finite subsets of
S
.
By Lemma 6.2
and the Generic Model Theorem, model A ( G ) some c
holds in some generic . . Since every element is an interpretation of
in the countable set C
OMITTING TYPES THEOBEM. on a forcing b ~ s e S
--
m,(c) C p
L -
S
.
A(G)
Let.
and --for each -
P, there exists n
sw-tmt du
is countable.
P be a -f orctng property m,n\-c and- c 6 C
let
,
i
Suppose the& for each
A, -has
,
p
m rn
;iL
c
C- C
,~
.
U :Y~(C)) ? ~l-i
VnyIrm(u)
a model, Proof.
Using the definitlon of forcing, the above
sentence is weakly farced by the empty condition, and thus holds in every generic model for P.
-I
The following result is proved using
CG,
applications
An analogous theorem was
of the Umitting Types rneorem. earlier proved for
L,,
(Keisler 1 3 1.
TWO CARDINAL THEOREM. has -
unary --
A
model -
-B
? 3 using compactness.
by Vaught
relations U,V , a d a sentence
Suppose L
9 G L,,
\u& 1 4 I ~hi,Then B I = = w and- \v.
such that e 5
such tha:
B \U-1
,has a
has a mode:
We shall omit the proof, The Two Lardinal 1-heorem
can be used to prove the following extension of Theorem 1.2 to
Lcc
. We call a sentence
r
every model
A_
of iF
Y ' i L,,_
and every set
most ,
,4
over X
(in the original sense of
has elements of st most
THEOREM 6.4.
t n
I
-stable if for
X < A
of power at
different types
k
L,.,,
).
hjgL is
y
If a sentence
6
-stable,
(L
.
is ii-stable for all K ( ri) Suppose 9 is ot h -st ble Proof. / Form a sentence p fn ai; expansion of L
such that if (C_,U,V,..)&s amodel of y 9 then C k y y - and the elements of V
.,
has a model
a model
-B
has
all have different types over U (&,U,V,
U V , . . ) LLI
I
...
) with
with
types over U'
, so
/U1l = d CF
.
s 1 U 1ilV1
,
, and
Then
,
]V1l = b1
is not &.)-stable.
has
.
Hence
f
NON-WELLODERING THEOREM.
(Lopez-ISscobar
/9
bor
5 ~ W I
Let < be a binary relation)i>
L and suppose CP
E
Lki_
has a model
A
in which < is a well ordering of type
at least w,
.
Then
has a model g in which < is not
(O
a well ordering. Proof. We may assume
(ol$))cA
.
Let dr
,r
rational,
be new constant symbols added\to .L., and let S be a
.
forcing base containing 3
,..., rn ... :r n
P = p(c,.drl
r l q r2 A
,
K
d
, there
)
are
Define P as follows.
Let
be a finite subset of S with Then p G P if and only if for all
; €
A'
and ordinals
..., 0,
6, ,
<
ol
such that d s p . , @ , + I 5 fi,) 2
(Asat PI,-*,
..
pn)k
P is a forcing property and V E P
.
= G
model
In B
, b n - , +6<,3,,, P
, so
CV has a generic
, the interpretation of the dr8s
are ordered like the rationals, so < is not a well ordering in g .
4
INTERPOLATION THEOREM. L'
L* = L
L'w,w
=
.
LY,,LU
Lf
p
,
(Lopez-Escobar 1 8
)
. &&
are sentences in the languages
res~ectivelpand if /= g-+3, , then there
-i s a sentence proof. -
of
L
Assume no such
be forcing bases containing
Let
p C P
pn C -
SN
with
i f and only i f
, p'
7yl,
a d n kt3-K
such t h a t /= ( e - 6
a?
(P
exists. and
Y
in
Let LlvJ,
p = p l u pn where
is consistent w i t h c p ,
and t h e r e i s no 196 L
S'
p1
P
S"
and
Ln
S*
4.W
,
,g , ~ * b - i e ,
with
i s a forcing property i n a s l i g h t l y more general
sense, and ( P A 7 V holds i n a generic model
&(G)
for
P
.
This c o n t r a d i c t s the hypothesis Note. -
In
f w,
t h e I n t e r p o l a t i o n Theore& was f i r s t
proved independently by Craig and Robinson, and i s equivalent t o t h e Robinson Consistency Theorem.
-
pM i s consistent
"'1
Then
g
and
However, i n Wlw
t h e analogue of t h e Robinson Consistency Theorem i s f a l s e even though t h e I n t e r p o l a t i o n theorem i s true.
LECTURE 7. MODEL THEORY FOB
Lma .
In this lecture we discuss some model theoretic constructions which are important in classical model theory and are available even in
.
I,,
The language
j,,
allows conjunctions and disjunctions over arbitrary sets of formulas.
The forcing construction is no longer available
because it depends on a sentence having countably many subformulas.
The constructions still available are elemen-
tary chains, Skolem functions, and indiscernibles. We fix a set L set of all formulas U:
of symbols and let L,, of
LwcG
be the
such that cP has finitely
many free variables and the symbols of 'P belong to some finite LO C L
.
The notion of a subformula of CF Is
defined in the natural way.
Since L,p,
is a proper
class, we need a way of choosing nice subsets of ,,L
DEFINITION. A frament of L , , Lww
is a subset
5
at'
closed under subformulas, finitary connectives and
quantifiers, and substitution of terms for free variables. Clearly, each subset of ,,L of ,,,L
.
generates a fragment
H. J . Keisler DEFINITION.
Let
be a fragment of
S
an
S-elementary submodel of
AG
g
YE S
and f o r every
whenever
, is
A is
.tihen each
a
in
A
we have
22
(Tarski and Vaught
t h e union of an S-elementary chain
, 7(
A,
i s an S-elementary submodel of
A
Ap
8
c: /; c
A_
(A_,&)P@
&,AS
a n S-elementary chain i f
ELEMENTARY C H A I N THEOREM. If --
a
B , if
A_ -CS
(*El,%)bc~
,d A
and
symbols
is
.
i f and only i f
&,
B , in
.
LOO-
A
i
),
,
.
The proof i s by induction on the complexity of formu-
l a s , and depends on t h e f a c t t h a t each subformula has f i n i t e + ly many f r e e variables. The elementary c h a i n theorem i s o f t e n used together For example, i t i s needed i n the proof
w i t h o t h e r methods.
of t h e D o Cardinal Theorem i n t h e l a s t lecture. W e now introduce Skolem functions i n DEFINITION.
Let S
Skolem exwns_i_on of formula to
S
be a fragment of
e,
,
i s constructed as followe.
Y ( u , " s ) & S add a new function symbol
L , forming
generated by
S
.
L,,
For each
Fqic)
L'
.
.
Then i t e r a t e t h e procedure, forming
Let
S'
be t h e fragment of
The
L'
...
L, ~ l L, ~ , The unions L* =
,... .
S , S1 , S2
7
onLn
UnS"
and S* =
S-Skolem exwnsions of L and S
.
are called the
The S-Skolem t h e m
is the theory with the axioms
T~kolem in L*md
,
3urp(u,?) -s P ( F y G ) , C )
PC
Notice that each axiom of TSkolem belongs to S* Of
T~kolem are called S-Skolen.models
s*
.
.
Models
.
We state three classical application8 of Skolem functions. SKOLEM EXPANSION THEOREM.
(i) Every model A f01' L
has an expansion to a n S-Skolem model &++ (ii)
TSkolem
If A+
, ehen
submodel of
.
is an S-Skolem model, i.e. a model of
every submodel
A* , B+
<S, 8_*
B* C
A,*
is an S*-elementary
.
DOWNWARD LOMENHEIN-SKOLEN THEOREM. ( Hanf &/ ) ff S has Dower Y
, then every
submodel of Dower at most
K
model- & has an S-elementary
.
DEFINITION. Given a fragment S an8 model A_ ThS(A_) ($ S
, the S-theory of A , be which hold in 4
let
the set of all sentences
.
PATCHING THEOREM. L& fragment S*
,
T* be a theory in the Skolem
of L* such that for each finite subset
Lo< L
there is an So-Skolem model
~'nS;;
.
gp whose
Then T* has an S-Skolem model
Proof.
Build A_W
A+
out of terms in L*
that an atomic sentence holds in A_W
S; theo~yis
. in such a way
if and only if it is
. 7'
in T*
We now turn to the notion of indiscernibles, They were F3 introduced by Ehrenfeucht and Mostowsk~for ZwcU , and have been an important tool in both finitary and infinitary logic. DEFINITION.
Let X
linear order on X we say that
(x, <)
.
be a subset of A
is S-indiscernible in A_ xl <
...
< xn
EHFZNFEUCHT-MOSTOWSKI THEOREM.
if any two
and y1 <
from ( ~ , c satisfy ) the same formulas of S
C 4 ) *L
a
Given a fragment S and a model 4,
increasing n-tuples
L
and <
... < yn .
in A_
Every theory T & I
which has an infinite model has a Skolem model with a 4
infinite set of LEI*, -indiscernibles. The Ehrenfeucht-Mostowski Theorem Is a special case of the following theorem for infinitary logic. STRETCHING THEOREM. of L@
&d
i
L&
cQ bdlon~to a fragment S
. The following are equivalent. '
has models of arbitrarily large cardinality.
has an S-Skolem model &+ with an infinite
( ii)
set of S+-indiscernibles. Proof. -
Assume (11) and let (x,<)
set of S*-inBiscernibles in b_+
< Y, <)
let
.
be an infinite
Given any cardinal K
.
be a linearly ordered set of power K
the expansion L a of L*
cY for each y generated by S*
6 Y
by adding a new constant symbol
, and
. Let
Form
let
S
be the fragment of L 1
TI be the theory in S t such that
, c 6 T' if and only if (A+,xlj...,X n ) y1 yn satisfies 'f' for all xi < < x, in x By the c
...
.
Patching Theorem, T o has a model Bf is a model of CP of power at least (i)
.
.
K
the reduct g
Thus (ii) implies
The converse is harder and also uses the Patching
Theorem, along with the
~rdos- ado
NON-WELLORDERING THEOREM FOR
<
theorem on infinite par-
d
titions.
4_e_t
, and
Am_
(Lopez-Escobar
be a binary relation symbol in L
and for each ordinal
4
,
.
Then
)
Suppose T E LOOc;
has a model in which
well ordering of type at least d in which
.
17
Y
L
a
has a model
< is not a well ordering.
Proof. Let S be a fragment containing
.
has
.
I-I. J. Keisler
arbitrarily large models, so by the Stretching Theorem CCI has a Skolem model &* with an infinite set of S*-indiscernibles.
Using the proof of the Stretching Theorem with Y
B+
ordered like the rationals, we obtain a model
in which
< is not a well ordering. d Morley used indiscernibles to compute the Hanf number of
d,_
and several related logics.
The cardinals
1,
are
I
defined recursively as follows: for limit THEOREM 7.1,
-Proof.
orle ley 23
)
.
The Hanf number of
TO show that the Hanf number is at least
we need an example of a sentence (Pa of model of power at least large models, Let
Jd
d
.
The first step is to construct a
B~
This is easily done by induction on 4
sentence Vd describes an* 6
has a
but does not have arbitrarily
sentence which has a well ordered model
4
iq,,which
Id ,
of order type
.
The final
structure in which every set
has rank less that d , To show that the Hanf number is at most q6 I , ,
I
, suppose
has a model of power at least l WLet , . S be a
countable fragment of
containing @
L ,d,,
.
similar to the Non-wellordering Theorem for shorn that
4
has an S-Skolem model
set of S*-indiscernibles.
A*
By a proof
tq,it can be
with an infinite
By the Stretching Theorem,
9
has arbitrarily large models.
LECTURE 8. INFINITARY SOFT MODEL THEORY.
In this lecture we return to soft model theory in the light of the methods developed in the preceding two lectures.
Am-
We shall give an analogue of Lindstromts theorem for
due to Barwise 3 , which shows that d , , is a 'batural* logic. We intraduce an abstract Skolem expansion property Then which has applications even beyond , % ,
.
we discuss the open problem of characterizing largest logic.
A,,
as a
In our discussion we shall examine the
relationship between several familiar properties of in the abstract setting.
La.,
It should be mentioned that
Makowsky, Shelah, and Stavi 21 and others have begun
a
more ambitious project for soft model theory, to find new "naturaln logics.
For b r e v i t y we denote an a b s t r a c t l o g i c
J'.
by
If 4 is a model f o r
iq. $f: L T
L*
if
C L and g p * q j
i s weaker than
f:
L O
and
fOt-L*
4
g* ,
i 3 such cp
.
REMARK.
f
f
3odels a s
5
we c a l l
Lo a
ky
y 6 f
G
8
on
suhlogic of
.
Lo
xo
has e x a c t l y t h e same
0
G f-* implies t h a t
implies t h a t f o r some
96
~ h ~ ~ =( 4 )
i f t h e r e i s a mapping
,fY,
t h a t each
O
let
agrees with
b O
L O
.
,
L
(L*, jcr)
L*,
,(*
5
Lo 5
,f ):
x05
c*
f 0
*
5
.
Here i s a l i s t of p r o p e r t i e s which hold a t l e a s t f o r every l o g i c
1 * G f,,
KARP PROPERTY.
If 4 'Xp
2
4z
then
iF
ordered model we mean a model
By a
-A
,
8 (Lecture 4). -
A
i n which
i s a w e l l ordering. 13CW WELLORDERING PROPERTY.
WeX with
Any
w e l l order-
ed models of a r b i t r a r i l y l a r g e order type has a non well ordered model.
%' class
i s s a i d t o be l o c a l l y bounded i f f o r each L
fP : L p g I) i s a set.
bounded, but each 'P 6 sublogic
Jw,
Note t h a t
v%,_
,
the
i s not l o c a l l y
belongs t o a l o c a l l y bounded
. .
SKOLFM EXPm S I O N PROPERTY. (SEP)
Each ~ p 6
)L.
belongs
t o a l o c a l l y bounded s u b l o g i c properties.
For e a c h
for
t'
L+
K(L) iff
= ~ 'L~ j+ A/LO
(iii)
(iv) L
0
(A:
2 L~ ~ i n i t e f , ? f i r 1
6 K ( L ~ )f o r a l l f i n i t e
4 for
L , i.e.
(patching).
- T
.
g G K(L)
.
cl z0
T
can be expanded t o a
a model
Let
Lo
i s a set of c o n s t a n t s .
X L
t h e r e i s a Skolem model
Th 0 ( ~ 0 C) T
Th
+
: L
Each model
Skolem model f o r
.
c a l l e d Skolem models
L+
(LX)+ = ( L ' ) ~ where
(ii)
L~
K(L) f o r
L+
, such t h a t :
L
(i)
-A
with t h e following
t h e r e i s a s s o c i a t e d a language
L
and a c l a s s of models
c LA
O
If f o r e v e r y f i n i t e
for
A
-0
L
with
0
A
, t h e n t h e r e i s a Skolem nodel
for
with
L
.
THEOREM8.1.
-F T o o-~ .
If
4
-B
. fP
4
4
-P
-B .
f
L,,
P
B
-
implies
( ~ a r v i s e3 ) .
iff A
1:A 4 2 where a16
tken
L.,
THEOREH 8.2.
-
=p 3
L/
(4.i) E ( ) . If LP by i n d u c t i o n on Y' t h a t I iff
-A
(Karp 1 1 ) .
2 ,,
t h e n it can b e shown (
)b
(2,;)kq;
iff
a the strongest
logic
w l t h t h e Karp and Non w e l l o r d e r i n g p r o p e r t i e s . The proof i s l i k e t h e proof o f Lindstrom's Theorem and uses Theorelp 8.1.
T h i s theorem . c h a r a c t e r i z e s
By t h e r e s u l t s of Lecture 7
.
.,
.
has t h e SEP.
The SEP i s fundamental and q u i t e powerful.
Fuhrken 34
e s s e n t i a l l y showed t h a t
.
has t h e SEP I n f a c t , it can K be shown t h a t every sublogic of t h e enormous l o g i c
.( (Q* : r
a cardinal)
f(g )
has t h e SEP
.
THEOREM 8 . 3 . (Barwise and ~ e i s l e r ) . ~ _ rf * .-has t h e SEP,
The n a t u r a l analogue of t h e Stretching Theorem
(i)
.
holds in (ii)
A"
has t h e Hon Wellordering m o p e r ; ~
The proof i s l i k e t h e case f o r
f,,
COROLLARY 8.4.
/, ,,
.
i n Lecture 7.
i s the strongest logic u i t h the
Karp Property and t h e SEP. When
K=
In, t h e l o g i c s
can be characterized
k,
i n a way analogous t o t h e above c h a r a c t e r i z a t i o n s of X,, ( ~ a r w i s e3 ) .
X,,
is t h e sublogic of
a l l sentences CP with fewer than K
X,',
c o n s i s t i n g of
subformulas.
The more i n t e r e s t i n g problem of c h a r a c t e r i z i n g
_
w l LV
a s a strongest l o g i c remains open. so much i s known about p r o p e r t i e s which hold f o r LO, -NON
.
This i s s u r p r i s i n g since Here a r e some a b s t r a c t
A
WELLORDERING PROPERTY.
ordered model of order type a t l e a s t non well ordered model.
2.
IP V e i l i has a w e l l Li',
, then
(F
has a
bycp 4 Y.', t h e e i s a
INTERPOLATIOH PROPBRTY. If
sentence 8 s u c h t h a t L E L O L and k*LP+a, b * ~ - r i / ~ , 4 ' v ( I-* (p means t h a t 4 C ' q f o r a l l medals _A f o r L )
.
OMITTING TYPES PROPEPTY. l o c a l l y bounded
Every
~QEZ* be1 ongs
to a
f O CZ* such t h a t f o r each countable
L
and e x t r a constant symbol c $ L , t h e following holds; Let
and
be a consistent s e t of sentences oP L
T C_ ,ZO
Tmncp, m , n c k ) , be sentences of L I.$
t h a t f o r each sentence
~
with T , a n d e a c h
PI,
there exists n
i s consistent with
T
.
6
Suppose
c).
of~ L 0u i c ) which i s consistent suchthat
Then T has a model
YAY,
such t h a t
.
( V m < w ) o d a ~ ) ( 3 n < ~ ~ ) ( ~ , a4 ',) j = *
To t h i s l i s t we may add t h e Karp and SEP h ' o p e r t i e s ,
~gwenheirnnumber
~3
, and Hanf number 2 '
Kunen has r e c e n t l y given an example (assuming t h e continuum hypothesis) of a l o g i c
)L
1
which i s properly stronger
than
LLr3 4 1
and has a l l t h e above properties.
that
a '"1"
i s not a n a t u r a l l o g i c a f t e r a l l , and t h e r e a l
This suggests
problem i s t o f i n d t h e strongest l o g i c which behaves l i k e
f
y, .
Kunenls example adds a new i n f i n i t e connective t o
zwqru . we ~ 0 n C l ~ dbye s t a t i n g
some r e s u l t s which give
r e l a t i o n s h i p s between t h e above p r o p e r t i e s .
THEOREM 8.5.
a&
a i.* has
( ~ a r w i s e3).
t h e I n t e r p o l a t i o n Property, then
nuer
cL,
.
number i 2 ~ 3 ,-non
and Hanf number 5
Wellordering Property.
f o r any i n f i n i t e o r d i n a l d THEOREM 8.7.
Lw-has
~gwenheirn
Lf .f' has ~Gwenheirn
( ~ a r w i s e3 ) .
TIIEOREM 8.6.
t h e Karp F O p e r t y
1 ,
1
t=
d'
has t h e
( I n f a c t , t h i s r e s u l t holds
i n place of
w, ).
(Krynicki and onyszkiewicz)
t& Omitting T w e s Property, then
z
,f
.
IP ,f
has --
has ~Garenheimnumber w
.
The following r e s u l t i s new. THEORE14 8.8.
ASSUrue
t h e Axiom of C o n s t r u c t i b i l i t y ,
or a t l e a s t h a t t h e r e i s a subset -- r C u such t h a t uncountably many subsets of
ct,
gre c o n s t r u c t i b l e from
has t h e pmitting Types Property, then (
.^, -non Wellordering Property.
f ahas
the
r
; If --
L"
.
K.J.Barwise. Absolute lo6ics and LmCC) AHL 4 (1972) pp. 309-340. K.J.Barwise. Back and forth through infinitary logic. pp. 5-34 in Morley 24. E.J.Barwise. Axioms for abstract model theory. AML 7 (1974), 221-265. K.J.Barwise and H.K.Kunen. Hanf numbers for fragments of ,,L Israel J. Math. 10 (1971). pp. 306-320. Recursively saturated and K,J.Barwise and J.Schlipf. resplendent models. To appear. K. Bruce. Model-theoretic forcing and L( 4 ) Thesis, U. of Wisconsin, 1975. C.C.Chang and H.J.Kelsler. Model Theory. NorthHolland 1973. .A. Ehrenfeucht and-A. Mostowski. Models of axiomatic theories admitting automorphisms. Fund Math. 43 ( 1956) pp. 50-68. W. Eanf. Incompaetness in languages with infinitely long expressions. Fund. Math. 53 (19641, pp. 309-324. J. Hirschffeld and W.H.Whee1.r. Forcrng, a r i t h m e t i c a n d
.
.
d ~ v l ~ ~rrngi o a
T,
appcrkr
C. Karp. Finite quantifier equivalence. Pp. 407-412 in The Theory of Modbls, ed. by Addison, Henkin;and Tarski, North-Holland 1965. C. Xarp, Langua es with expressions-ofinfinite length. North-Holland 19f4. H.J.Keisler. Model theory for infinitary logic. North-Holland 1971. H.J,Kaislsr. Forcing and the omitting types theorem. Pp. 96-133 in Morley 24. H.J.Keisler. L ~ g i cwith the quantifier *there exist uncountably manyn. AML 1 (1970), pp. 1-97. M.Makkai. Preservation theorems for logic with denumerable conjunctions and disjunctions. JSL 34 (1969), PP. 437-459. P.Lindstrom. On extensions of elementary logic, Theoria 35 (1969), pp. 1-11. E. Lopez-Escobar. An Interpolation theorem for denumerably long sentences. Fund. Math. 58 (19651, 253-277.
H. J . Keisler
E. Lopez-Escobar. On definable well orderings. Fund. Math. 58 (19661, pp. 13-21. R. Lyndon. Properties preeerved under homomorphiem. Pacific J. Math. 9 (1959). PP. 143-19. J.AeMakowsk8, S. Shelah, and J. Stavi. a-logics and generalized quantifiers. To appear. *H.Morley. Categoricity in power. TAMS 114 ( 19651, PP. 514-538. M.Eorley. Omitting classes of elements. Pp. 265-273 in The Theory of Models, ed. by Addison, Henkin and Tarski, North-Holland 1965. M. Morley. Studies in model theory. Math. Assn. of Amer. 1973. M. Presburger. Warsaw 1930, A. Robinson. A result on consistency and its application to the theory of definition. Indag. Math, 18 (1956), PP. 47-58. A. Robinson, Introduction t o model theory and to the aetamathematics of algebra, North-Holland 1963. A. Robinson. Forcing in model theory. Symp, Math. 5 ( 1971) , PP. 69-62, S. Shelah. Stability, the finite cover property, and superstability. AML 3 (1971), pp. 271-362. J. Schlipf. Some hyperelementary aspects of model theory. Thesis, U. of Wisconsin, 1975. A. Tarski-and J.C.C.McKinsey. A decision method for elementary algebra and geometry. Rand Carp. 1948. A. Tarski and R. Vaught. Arithmetical extensions of relational systems. Comp. Math. 13 (1957), pp. 81-102, ,M.Morley and B. Vau ht. Homogeneous universal models. Math. Scand. 11 (19 21, pp. 37-57.
%
Skolem-type normal f v r m s for a f i r s t - o r d e r language with a generalized quantifier. Fund. Math. 54 (l964), rp. 2 9 1 - 3 0 2 .
G. Fuhrken.
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
SH - FORMULAS UND GENERALIZED EXPONENTIAL
M. SERVI
Corso tenuto a Bressanone dal 20 a1 28 giugno 1975
M. Servi
MARIO
S E R V I
SH-Pcrmulas and Generalized ~ x p o n e n t i a l ( L e c t u r e d e l i v e r e d on June 25, 1975)
There a r e two i d e a s underlying t h i s t a l k of mine. The f i r s t one i s concerned w i t h t h e i n t e r p r e t a t i o n of a c l a s s of f i r a t o r d e r formulas ( c a l l e d SH-formulas,
since they a r e
a c l a s s of HORN formulas i n SKOW open form) i n a categor y s a t i s f y i n g a v e r y weak requirement, namely t h e s o l e e x i s t e n c e of f i n i t e products. This i d e a was developed some y e a r s ago and l a t e r I w i l l devote some of my t i m e t o d i s c u s s
it. O f c o u r s e , it i s f a s h i o n a b l e t o i n t e r p r e t every formula
.....
(even of h i g h e r order and multi-sorted)
i n a t o p o e , b u t to-
p o i have a v e r y r i c h s t r u c t u r e and s a t i s f y many more prop e r t i e s t h a n j u s t having f i n i t e products. I might mention t h a t C. MARCHIFJI a t a r t e d working on t h e comparison between t h e two d e f i n i t i o n s of t r u t h and w i l l soon p u b l i s h a paper on t h e s u b j e c t ( s e e [I, 21)
.
I n o r d e r t o e x p l a i n t h e second i d e a , I have t o r e c a l l t h e w e l l known f a c t t h a t t h e s e t - t h e o r e t i c e x p o n e n t i a t i o n
artesian
power) s p l i t s i n t o t h r e e non-equivalent
f o r g e n e r a l c a t e g o r i e s : 1) t h e Hom-bifnnctor
-C,
om:
0
2) ,
concepts
(any c a t e g o r y
2) t h e LAWVERE e x p o n e n t i a l ( C a r -
M . Servi
I;,
t e s i a n closed cstegory
Exp:
C) .. and
go%:
3) t h e
S-fold product of a n o b j e c t by i t s e l f (when such e x i s t s , S an o b j e c t of t h e c a t e g o r y g of s e t s ) , h e r e a f t e r i n d i c a t e d
n:gO~gC!g.-
by
-
Nor, t h e i d e a 18 t o f i n d a s u i t a b l e
g e n e r a l i z a t i o n of t h e s e t h r e e concepts which w i l l behave l i k e them a s f a r as t h e & r u t h of SH's i s concerned. I n a p r e v i o u s work of mine ([5]) I i n t r o d u c e d a b i f u n c t o r
8
G:
--i,
C- ,-
s u i t a b l e catkgoriee) s a t i s f y i n g
hypotheses which h o l d f o r b o t h Horn and Zxp, b u t f a i l e d t o a s a special case, since
include
D s=
is not, i n general,
t h e category of s e t s . Let us summarize a l l t h i s a s f o l l o w s : 0
>
(-1
(-) :
- -s --,s; 80% s-o x _C - +8
I-
1)
Ham:
2)
Exp: z 0 x g ----,
s t a r t i n g case
t h e t h r e e g e n e r a l i z a t i o n s of 0 ) ;
?J:=S O I-C --, -C
3
G:
4)
(?):
- 2-
Dog
EOxD - - -+
g- , _C, -
introduced i n [5] ; wanted.
(?) : EOAD -- -d
The b i f u n c t o r
C- of
c a s e 5 ) , although more
g e n e r a l t h a n t h e p r e v i o u s ones, should s a t i s f y enough assumptions a s t o y i e l d t h e u s u a l r e s u l t which h o l d s f o r d i r e c t powers:
..
I
.
.
.
......
-
Let (n be any 2 - s t r u c t u r e ( i n I)) and l e t SE. Then
H
H
be any
i s t r u e i n & iff it i s "uniformly t r u e n i n
M . Servi the
G(X, d3) I
s
(induced s t r u c t u r e s ) w i t h X E Ob
(in
2)
E. - In for-
iff
Now, a few words of e x p l a n a t i o n about t h e i r s a t i s f a c t i o n . I w i l l n o t s p e l l out how
SH-formulas and
SH's a r e made
( s e e [4] f o r d e t a i l s ) , but I w i l l g i v e a g e n e r a l i d e a , p o i n t i n g o u t t h e main f e a t u r e s . F i r s t of a l l , I w i l l confine myself t o a f i r s t o r d e r language
2
w i t h no f u n c t i o n nor
c o n s t a n t symbols. The expanded language expanding
2
g* , obtained
by
w i t h dummy f u n c t i o n aymbols, w i l l have t h e
a u x i l i a r y purpose of t a k i n g c a r e of e x i s t e n t i a l q u a n t i f i e r s . A s it i s customary when d e a l i n g with i n t e r p r e t a t i o n s of f o r -
mulas i n a c a t e g o r y , we a r e i n t e r e s t e d i n o r d e r e d p a i r s (H, n)
-H
a formula and 2 a n a t u r a l number
- rather
than
i n formulas themselves. The n a t u r a l number component w i l l be t a k e n c a r e of by i n c o r p o r a t i n g it i n s i d e t h e formula, so t h a t an
SH w i l l n o t be d i r e c t l y r e c o g n i z a b l e ae a f i r s t
o r d e r formula of t h e o r d i n a r y kind. The main d i f f e r e n c e , though, w i l l be juglt t h e f o l l o w i n g minor change: t h e supply of v a r i a b l e s w i l l be a t r i a n g u l a r a r r a y
They rill be i n t e p p r e t e d as p r o j e o t i o n s , E:: being t h e kth
n-place
+A
p r o j e c t i o n on A, t h e u n i v e r s e of
interpretation. The n a t u r a l nunber aesociated with an called its (
t
,
.
M.A t
t y p i c a l atomic
with terms
ti,
w i l l have t h e form
SH
..., ta
dummy f u n c t i o n eymbole. An elementary form
H,A
... n
% j H ,
r a n k ) , and a g e n e r a l ary
SH's
,
with
SH w i l l be
SH
O$r,
b u i l t up with w i l l be of t h e
Hk atomic ( f i x e d
w i l l be a conjunction of element-
SH
of t h e same rank.
Now f o r t r u t h . F i r s t of a l l , an g - s t r u c t u r e , i n a category
g - with f i n i t e producta, rill be t h e usual t h i n g ,
i. e. a p a i r
= (A,
c a r r i e r o r universe) and
)
,
Y
with
A
an object of
g
(the
the "structure" (realizing the
p r e d i c a t e symbols), bearing i n mind t h a t an wary r e l a t i o n u on A i s j u s t a eubobject R >-> A~ (no c h a r a c t e r i s t i c functions are available)
. Similarly f o r
f,*
.
A n I'*-struc-
a=(A, y") rill a l s o be c a l l e d an .$-interwe-
ture
t a t i o n , and it w i l l be eaid t o be a s s o c i a t e d with a struc-
a , if
ture
i s t h e reduct t o
of
a*.
Note t h a t i n t e r p r e t a t i o n 6 s e r v e t h e only purpose of defining t r u t h , according t o t h e following d e f i n i t i o n : H
( i n _C)
-
i f t h e r e i s an
aR associated with a*, i n t u r n , w i l l be given
Wuth i n that "* i n terms of s a t i s f a c i o n , which I w i l l define e x p l i c i t l y 8uCh
'=T
M. Servi only f o r atomic
SH1s.
say
..., t m ) ,
Ho = P ( t i ,
andlet
Thus, l e t let
b e atomic of rank n ,
Ho
R >-!-+Aa
Wewillwrite
g: X - - - t l n .
i f t h e r e i s a morphism' X -+It
be
Q=H,[g]
Pa = P~ (in
p), -
euch t h a t t h e f o l l o w i n g
commutes:
The d e f i n i t i o n extends t o a r b i t r a r y way.
%H
ina ally,
and every
means
SH's H g
i n the usual
,
f o r every
X
g: x - A ~ .
L e t u s now i n t r o d u c e t h e
"generalized exponential".
-E , -Il, -5 be t h r e e c a t e g o r i e s , C, - - a r e n o t r e q u i r e d t o have a Let
-Q
with f i n i t e products. f i n i t e products,
11
b u t t h e y w i l l t u r n out t o have enough of them f o r t h e i n t e r p r e t a t i m s o f symbols i n $* t o make s e n s e . L e t G: gO*I) - -?
-
be a b i f unctor,
U 6 Ob _C, -
K:
2 4 E-
t h e f o l l o w i n g axioms hold: ( i ) t h e standard f u n c t o r
C- [u, -1: g
faithful;
a f u n c t o r such t h a t
-
g
i s f u l l and
( i i ) there i s a natural bijection
0-
:
gl-,
9-
i. e. a b i j e c t i o n :
~(-11
-
> = c[u, G(-,
-11 ,
M . Servi
n a t u r a l both i n X and Y, arrow i n
E, -
and t h e lower one i n
j$-
t h e upper
Y E Ob g, -
g; -
i s f u l l and f a i t h f u l and h a s a l e f & a d j o i n t ,
(iii) K
F
X E Ob
--+ K. It i s an eaay m a t t e r t o check t h a t , w i t h a s u i t a b l e
choice of
K
and
Exp
U and enough hypotheses on
and
Horn,
g, -
meet t h e s e requirements.
It t u r n s out t o be ueeful t o c o n s i d e r thk f u n c t o r
given by *
G
8(x) ( y ) = ~ ( y x). ,
In case
is
G
Horn, t h e n
i s t h e YOmEDA enbedding; t h e f o l l o w i n g theorem s t a t e s
6
that
t u r n s o u t t o be an enbedding i n t h i s g e n e r a l c a s e
as w e l l . TREORJ3lfi 1.
-&
THEOREM 2.
- &&
I-
, then
f o r everz
serves ves -
I-liinits
-
i s f u l l and f a i t h f u l . be a diagram schema. If X f Ob
E, -
G(x,
-)
K
pre-
preser-
1---l i m i t s . COROLLARY 1.
-
For every
X€Ob
E- ,
G(x,
)'
preserves
limits.
COROLLARY 2.
- 2
p r e s e r v e s f i n i t e products and monos.
M. Servi Mow, c o r o l l a r y 2 p e r m i t s us t o a s s o c i a t e , w i t h every structure
6
,y;)
= (;(B)
h* .
(B, \V )
=
(in
(in
a structure
2) -
=
and s i m i l a r l y f o r i n t e r p r e t a t i o n s
-
@O)
Moreover, theorem
1 g u a r a n t e e s t h a t every i n t e r p r e -
tation
;(a)*a s s o c i a t e d with
(
f o r a unique dL;( a s s o c i a t e d with d)
)
;(A,
~ ( L D ) i s of t h e form
.
This, i n t u r n ,
y i e l d s a "uniform f a m i l y of s t r u c t u r e s N :
63,
= G(x,
a) =
(G(x, B),
pi)
whose a s s o c i a t e d "uniform f a m i l i e s of i n t e r p r e t a t i o n s w
(X e Ob g) -
'h;
a r e i n a one-one correspondance with t h e
6(d3)* ( s and u l t i m a t e l y w i t h t h e
nets, a s s o c i a t e d with:A.
I am s o r r y , but I need one more d e f i n i t i o n :
I=X= t E : ,fix
H
(read:
(NX),
iff t h e r e i s a
f o r a l l X EOb
i s uniformlg t r u e i n t h e
H
g. -
L: , uniform,
-
fi
X
's, X E Ob E)
such t h a t
t--- H,
.a>
Te a r e now i n t h e p o s i t i o n t o s t a t e t h e main theorem: THEOSEE 3 . An
true i n the
SH
i e true i n
i f f i t i s uniformlg
Axt= p H 03
iff
-1 x t g h*
H.
I n o r d e r t o prove t h i s theorem I w i l l need s e v e r a l
lemmas, which, but f o r one, I w i l l s t a t e without p ~ o o f . '?@at will be proven w i l l g i v e a sample of t h e ( r a t h e r
elementary) t e c h n i q u e s involved. The complete p r o o f s can be found i n [ 6 ] .
LEUU 1.
$1, l e t
(&
g: Y--,C.
n
-Let
H,
4
be any i n t e r p r e t a t i o n
= C , )
be an atomic
SH of rank 2 and l e t
Then i f f f o r every
I n t h e f o l l o w i n g lemmas, l e t interpretation i n
-
and l e t
H,
--,Y,
x: U
&=
8, [gx]
.
(B,Y)be an
be an atomic SH of rank
n,
v i a the following n a t u r a l transformations: -
Then -
:
U
4 Z,
since
gx: U
G(x,
Bin, & g,:
using the following natural bijections: U --+G(x,
B ) ~
X
---t
(KB)~
M. Servi
- E\or
LEXMA 4.
g: X
through isomorphism
( D ) ~
-4
iff
[z]
F--IK
g) , get 9 : P X d B
( g ~ ) ~ -K- (, B ~ )and ad j u n c t i o n
~‘7
- Lemma 4 i s t h e
REMBRK.
(in
F-4K.
n
Then
Ig1
only p l a c e where a d j u n c t i o n
i a used.
-
cOROLL*RY 3 .
i s t r u e i n A* PROOF.
-
If
i s t r u e i n everx
H
6; ,
then
H
. Take
X = ICY
- Let
COROLLARY 4.
and apply lemma 2.
g: Z
-
G(x, B)",
Using t h e nota-
t i o n s of lemmas 3 and 4 , t h e f o l l o w i n g h o l d s : $H,[g]
m, f o r -
COXOLLARY 5. H is true i n
&',
~f
H i s t r u e i n A' and
H
elementary, say
g: Z ----+ G ( X , B ) ~be such t h a t
Then, f o r every
H being t r u e i n Ho
ig],
%R,[B~]-
x E ~b 2, then
x-
PROOF. Only f o r Let
eevergx: 0----+Z,
x: U --+ Z ,
8 ,
Hi~...~Hr-Hoe
H iA
...
A
Hr [g].
~~[ii,] by c o r o l l a r y 4.
it f o l l o r s t h a t
+=H,g[&]
and hence
again by c o r o l l a r y 4.
We can now prove theorem 3 , t h u s ending t h e p r e s e n t lecture.
M. Servi
Assume,
t=====H and l e t
4 c i a t e d with h such t h a t corollary 2,
an i n t e r p r e t a t i o n assoBy t h e remark f o l l o w i n g
@ H.
&X
y i e l d s a uniform f a m i l y
& I
x €6
by c o r o l l a r y 5. Hence ?onversely,
4
assume
remarks, t h e r e i s a G(x, A').
and /===
Ax
H
/== H, a l l X 1s .
@x associated with
h"
d$=
and l e t
Ih
H,
BX
-He
uniform f a m i l y such t h a t
6,=
a*be
be a
By t h e above such t h a t
Now c o r o l l a r y 3 y i e l d s t h e d e s i r e d r e s u l t .
REFZREBCES
. COSTE,
-1
Logique du
ier o r d r e dans l&s
topes
814-
mentaires. (Himeographed n o t e s ) . C.
':2]
SH, -
BiCHIBI, F u n t o r i che conservano e r i f l e t t o n o l e
-
A t t i Sem. E a t . e F i s . Univ. Modena 22 (1973).
~ 3 : 2 . Yd3CHIM1, Alcune q u e s t i o n i d i semantics c a t e g o r i a l e ( t o appear).
14) K. SERVI, Una q u e s t i o n e
d i t e o r i a d e i modelli n e l l e
-
c a t e g o r i e con p r o d o t t i f i n i t i , Matematiche ( ~ a t a n i a )26.
"
[5]
" ,
Su a l c u n i f u n t o r i che conservano l e SH, Riv.
-
Mat. Univ. Parma ( 3) 3 (1974)
[6]
"
" ,A
.
g e n e r a l i z a t i o n of t h e e x p o n e n t i a l f u n c t o r
i n connection with t h e SH-formulas ( t o appear)
.
CENTRO INTERNAZIONALE MATEMATICO ESI'IVO (C.I.M.E.)
TOPOLOG ICAL MODEL THEORY
Corso tenuto a Bressanone dal 20 a1 28 giugno 1975
J . A. Makowsky TOPOLOGICAL
MODEL
CONTENT
Introduction The lattice of closed sets The open set quantifier Product topologies The interior and
Chang's modal operator
A maximal logic for topological structures Uniformities, Proximities and closure spaces Decidability Some open problems
THEORY
TOPOLOGICAL MODEL
THEORY : A
SURVEY
Istituto Matematico Firenze 1I.Mathematisches Institut der Freien Universitdt
0.
Berlin
INTRODUCTION
Topological model theory is getting "en vogue". It stems from the fact that model theory has been very successful1 for algebraic structures, clarifying algebraic concepts (algebraic closure, Lefshetz principles etc)
and the concept of infinitesimals (non-standard analysis) and
recently is even invading "hard" algebra (Whitehead's conjecture). It was always considered unsatisfactory that topological spaces could not be treated within the model theory of first order languages. It was usually argued that topology is basically a second order concept and even in ROBINSON'S book "Non-standard analysis" [ 301 non-standard topologies are
non-standard models of the full structure over the topo-
logical space. Until recently, say 1970, few attempts were made to attack the problem. It was again ROBINSON who now proposed it as a problem [ 3 1 ] :
*
Partially supported by the Italian C.N.R
(G.N.S.A.G.)
J. A. Makowsky
"The task I which to specify here is of general nature: It is to develop TOPOLOGICAL MODEL THEORY in the direction exemplified above in any other direction "
([
31 1) or
His approach was to consider structures
21 equiped with a topology where all the functions of
and all the relations either open or closed.
%are continuous
PETRESCU [ 271 proved
compactness and L6wenheim-Skolem theorems for this approach and ROBINSON, in one of his last papers [ 321,studied continuity of definable predicates and Skolem functions. In this paper I try to give a survey of what has been done up to now in "any other direction", and I hope to show that there is a very good notion of first order topology.
This paper is an updated and extended
version of my talk given at the CIME 1975
.
The organization of the paper is more or less chronological. Very briefly the approach by HENSON,JOCKUSCH,RUBEL and TAKEUTI [ 101 will be discussed.
Their paper "First order topology" is full of topologically
interesting ideas, but it does not seem that their approach could fit into a program for model theoretic investigations. The more successful1 approach starts with SGRO's thesis [ 331, where a generalized quantifier (the open set quantifier) was introduced. There techniques developed by KEISLER to study the quantifier "there exist uncountably many" [I2 were exploited. These techniques were also successfully applied by the author and ZIEGLER [
231 to construct the logic L(1) which has a new
logical operation inspired by CHANG's paper "Modal model theoryn[ 3 ] to deal with the interior of a definable set. L(1) behaves essentially exactly as first order logic, but it is too weak to express many interesting topological concepts. What makes these and related approaches
J. A. Makowsky successfull was captured first by GARAVAGLIA [ 7 ] who discovered the general principle behind all this: The notion of topological invariance for two sorted languages. A syntactical characterization of invariant sentences, similar to LEVY'S approach to set theory via absoluteness, was first given by ZIEGLER [ 4 0 ] . virtually all local concepts in topology are definable with invariant sentences but their logic has still most of the features of firts order logic. More surprising even, it can be characterized as the strongest topological logic satisfying these properties exactly as in LINDSTR~M'Stheorem. Finally it turns out that the same program works for a treatement of uniform spaces and a glance is given also to proximity spaces and closure spaces. The paper concludes with a brief discussion of decidability questions and a list of problems. I said that topological model theory is getting "en vogue". Many of these ideas are in the air. My presentation is biased by my own work and the work I heard of personally. Priority questions occur whenever a new branch develops. But the main priority belongs to the late A.ROBINSON whoadvocated the undertaking of this project.
Without his encouragement,
directly and indirectly, few of us would have concentrated our efforts so much to contribute to the developement of topological model theory.
The CIME summerschool on model theory 1975 was dedicated to the memory of A.ROBINSON who personally has contributed much to the developement of mathematical logic in Italy and it is therefore a special pleasure for me to contribute this paper in this volume. But this survey could not have been written without the many discussions I had with A.Macintyre and J.Sgro,with P.Mangani,A.Marcja and S.Tulipani and with M.Ziegler and W.Rautenberg.
J. A. Makowsky
1.
THE LATTICE OF CLOSED SETS
The first serious attempt to study the model theory of topological spaces, or rather to introduce the notion of elementarily equivalent topological spaces, was suggested in a paper by RUBEL and TAKEUTI
[lo ]
entitled
Given a topological space X
HENSON, JOCKUSCH,
First order topology".
"
they consider the structure
where c(X) is the lattice of closed subsets of X and
C
=
is set inclusion+
Standard structures are structures which come from topological spaces, i.e. A
structures
with c(A)
cz A '
Q =
<
A'
,c >
such that there exist a top. space
up to lattice isomorphisms.
therefore is always a
A'
not first order.
complete lattice, a property which is obviously
Tbe question the four authors are mainly interested in is as follows: Given any space X, what other spaces Y are elementarily equivalent to X, i.e. for what spaces Y c(X) s c(Y) holds ? rems are due to SWE'IT and FLEISSNER
[
The following sample theo-
38] ,[39 ]
:
Let R be a topological space homeomorphic to the real line. THEOREM 1.1
If X is,a topological space elementarily equivalent to R then li) X is first countable and a-compact and its topology is induced by a dense, complete linear order without endpoints (ii) X has precisely continuum uany points and (iii) if X is locally separable then
X
is homeomorphic to R.
Let S be a topological space homeomorphic to a Suslin line,i.e. a dense.
linean.,complete order with the order topology and such that every family of disjoint intervals is countable. THEOREM 1.2
If S is a Suslin line then (i) S is
(
m,w
)-equivalent to R
(ii) it is provable in ZFC that there are TI-spaces X which are not Suslin lines and X is elementarily equivalent to R. Rany topologiqal properties oT\spaces are expressible within this framework,
SO
connectedness, most separation properties, inductive dimension
n, but topological compactness is not.
Still there are compact spaces
which are elementarily equivalent only to compact spaces. The main defect is that certain model theoretically important theorems fail in this context. For example the Skolem-Ldwenheim theorem fails and its generalizations or the behaviour of the Hanfnumbers is governed in part by large cardinal axioms
. Obviously the compactness theorem failsttoo.
From the point of view of decidability the situation is even worse:?he set of sentences true in all T -spaces is at least as complicated as 1
second order number theory. For more the reader should definitely consult
[lo],
where he may find also an interesting problem list.
Although the research along these lines is very interesting and challenging it is clear that it does not give any reasonable first order language for topology. It rather resembles the investigations on the elementary theory of the lattice of degrees of unsolvability. To find a"reasonab1e"first order language for topology is the subject of the rest of this paper.
J . .4. Makowsky
2. THE OPEN SET QUANTIFIER
The first logic for topological structures which deserves the name "first order" was suggested by KEISLER and developed in SGRO's thesis
133 I. It is derived from a way of looking at generalized quantifiers. Let L(Q) be the set of finite formulas formed as usually but with an additional clause: (Q) If x is a variable and
.
I$
a formula so is Qx@ (x)
Models for L(Q) are structures of the form 91 = < 9h,q > where
91
0
is
an ordinary L-structure (for first order logic) and q is any family of
1
0 1. Such structures where extensively studied by KEISLER
subsets of: 91
1 1 2 1 and were there called weak models.
Satisfiability for L(Q) is
explained as usually with the additional clause (*Q) < no,q >
1-
QX 4(xIz) iff ( c
€1
3
~ :~
1
I= @ (c,a)1
E q
.
Various interpretations for L(Q) have.been considered restricting the choice of q. If q consists of the uncountable sets of
I
we get the
9lOI
quantifier "there exist uncountably many" which has been studied by many logicians. In SGRO1s [34 ] and [35 ] i. e. families of open sets on logical structures
.
1'
q varies over topologies,
91 01. Such structures are called topo-
In contrast to ROBINSON'S approach [ 321 no res-
trictions are made on the functions and relations of A
formula of L(Q) is topologically valid
.
%
if it is true in all topologi-
cal structures. SGRO gives the following axiomatization for the topologically valid formulas: Qx(x=x)
Qx(x#x) r( QX @
&
QX $
)i Qx( $&$
) r
VyQx @(X,Y)
+
Qx3y $(x,y)
together with all the valid formulas and rules (and schemas) of classical predicate calculus. A formula is called topologically derivable if it can be derived by a finite proof in the above system. Consistency is defined as usually.
THEOREM 2 . 1
(SGRO) (i) A set T of formulas of L(Q) is topologically consistent iff it has a topological model. (ii) The set of topologically valid formulas of L(Q) is recursively enumerable. (iii) A set T of formulas of L(Q) has a topological model iff every finite subset of T has a topological model. (iv) A set T of formulas of cardinality
K
-
w
which is
consistent has a topological model of cardinality topology has a basis of cardinality
K
whose
K.
(v) A set T of formulas of L(Q) is consistent with VXQ~(X#Y) iff T has a
0-dimensional normal topological model.
SGRO also gives an ultraproduct construction for topological models which satisfies an analogue of $os'slemma. Sofar this logic with the open set quantifier behaves very nicely in the sense that its model theory can be developed along the lines of classical model theory. But there are two defects. The expressive power of L(Q) does not satisfy any reasonable demands of a topologist and there is no interpolation or definability theorem for L(Q).
Among the few topologi-
cal concepts definable in L(Q) we have "TI-separation" and "discrete space" if equality is part of the logic, but already "T -separationw or 0
"Hausdorff"
, "the interior of
a definabl-.set is not empty" or "open
J. A. Makowsky in the product topology","continuity of definable functions" are not
.
definable in L (Q),cf MAKOWSKY-MARCJA [20 ] and MAKOWSKY-ZIEGLER [23 1. SGRO
first found a counterexample to CRAIG'S interpolation theorem
for L(Q) which he extended together with the author to a counterexample to BETH'S theorem using an idea of SHELAH (cf. MAKOWSKY-SHELAH [21
I).
3.PRODUCT TOPOLOGIES
Let us now consider n-ary quantifier symbols for every finite n. We denote these symbols by Qn and add a new formation rule to L (Q):
..
(Qn) If xl,x2,. ,xn are distinct variables and is
+
.
is a formula so
Q " X ~ X ~. .X 4. For brevity we write also QnG
We denote by L(@
<W
)
+.
the set of formulas where we have one n-ary quanti-
fier for every new and have formed finite formulas with the usual rules for predicate calculus and with (Qn)
...,x
these quantifiers is "+(xl,x2,
)
. The intended interpretation of is open in the product topology",
so structures are the same as in the preceeding section, i.e. topological structures, and satisfiability is defined by the clause: (*Qn)
< ofo,q>
I=
Q~G+(G,~) iff i
c
E 191 01:
'U
I= +(:,a)
1 is open
in the product topology induced by q. Similar approaches do exist for other interpretations than toplogies: MALITZ and MAGIDOR [I6 ] considered a product quantifier for cardinality logics and MAKOWSKY and TULIPANI 122 ] for the more general monotone quantifiers. In fact if q is monotone (which is not the case for open sets) there is a canonical way to define satisfiability for L ( ~ < by ~ )
J. A . Makowsky
(*Q~,Mo~) t91 olq >
I= ~~x $(xla)
iff there exists a B Eq such that
B~ c { G E 1 ~ ~ 9tl=$(,,i)}. 1 ~ :
For monotone quantifiersmuch of classical model theory can be adapted whereas the topological product quantifier seems less easy going. The use of the topological productquantifier is its greater expressive power: "Hausdorff" e. g. now is definable. SGRO
05 I has axiomatized
L($~) for the topological interpretation, but we wont give the axioms here since they are rather complicated.
He also proved compactness
and Lewenheim-Skolem theorems similarily as in the previous section. Again the interpolation and definability theorems fail. Applications exist for the axiomatization of topological groups and vectorspaces and similar results hold for extensions of L(Q< W )
by
infinitary conjunctions and disjunctions and by the cardinality quantifier "for uncountably many".
4. THE
INTERIOR AND
CHANG1'S MODAL OPERATOR
In his paper "Modal model theory" [ 3 ] CHANG logical operation
introduces a new type of a
. He replaces the impersonal "it i$ necessary that 6''
of classical modal logic by the personal
" x finds it necessary that $".
A similar approach should work for a statement like "x is in the interior of the set defined by 4 " . L(1)
So let us define the formal language L(1).
is, like L(Q), an extension of classical predicate calculus where
we have a new logical symbol I. The usual formation rules for finite formulas are extended by
J. A. Makowsky (I) If x is a variable and x occurs
free
Weak models for L(1)
4
a formula of L(1) so is 1x4.
in 1x6, even if x does not occur in
4.
are defined similarily as for L(Q), but with a
local touch: the family q is replaced by a system of neighborhoods N for every element of the structure. More precisely structures are of the form
a
universe
=<< A
el 0, Na, a E
A
> where 91
is a classical L-structure with
and N is a family of subsets of A. The additional clause in a
the definition of satisfiability now reads (*I)
511= I X $ ( X , ~ ) iff{b ~~ E A
:
81 I=$(b,Z) 1 E N~
.
In [ 3 ]CHANG studies the model theory for weak models for L ( 1 ) and also for a slight generalization. He suggests several interpretations for L(I) which all come from modal or deontic logic. A semantic for an interpretation which is related to the modal system S4 was proposed by MAKOWSKY and MARCJA in
[I9
] where a completeness theorem is proved.
We call a weak model for L(1) a topological model neighborhoods,i.e. the power set of
if X
A.
C
A and X
if the Nals are "real"
E Na so a E X and Na is a filter in
The model theory for L(1) for topological structures
was initiated by the author [
171
and further developed in MAKOWSKY and
ZIEGLER [23 1, where all the results of this section come from. It is obvious that topological models for L(Q) can be transformed canonically into topological models for L(1) and vice versa, therefore L(Q) is a sublogic of L(1) for the topological interpretation. L(1) is strictly stronger in L(Q)
, but
not too much
:
"T -space"is definable in L(1) but not 0
and "the interior of a definable set", but "T -space1'or "open 2
in the product topology" are still not definable in L(1).
In contrast
to the usual quantifiers one should note that 1 x 1 ~ 4 is not logically
J. A . Makowskv equivalent to
1~1x4.
A
formula of L(1) is topologically valid
if it
is true in all topological structures. The set of topologically valid sentences can be axiomatized by a Gentzen type system. If S and T are any finite sets of formulas we denote sequents by is valid iff
C s s ES
-r
II t tE T
S
+T
, where
a sequent
is valid. In addition to the usual rules
(cf. LYNDON [ i 5 ,pp 6 4 ff. 1) we add two rules for I I-introduction on the left side
St
:
$(x) T '
S, 1x4 (x)
and +
T
I-introduction on the right side
for any term c provided x does not occur free anywhere else than indicated. Provability and consistency are defined as usually for Gentzen systems. THEOREM 4.1
A
set of sentences T in L(1) is topologically consistent
iff it has a topological model. With similar methods as developed in SGRO [3b ] one proves the complete analogue to theorem 2 . 1 for L(I), one gets an ultraproduct construction and most positive results which hold for L(Q)
. The particular axiomati-
zation has furthermore two definite advantages
:
One can prove an "omit-
ting types theorem" via consistency properties and one can prove, by straightforeward inspection, the LYNDON interpolation theorem THEOREM 4 . 2
Let 4, J,
be formulas of L(1) such that
4+9
:
is
topologically valid. Then there exists a €I in L(1) which contains only those extralogical symbols that occur both in $ and J,
and such that both
$ + 0 and €I + J,
are topologi-
cally valid. Furthermore a relationsymbol occurs positively (negatively) in 0 only if it occurs positively (negatively) in both 4 and $
.
(Note that = is a logical symbol).
From this the usual BETH definability theorem and the following preservation theorem are derived. THEOREM 4.3
A formula of L (I) is preserved under continuous open homomorphisms iff it is topologically equivalent to a positive formula. (In the definition of positive formula no restriction on the I-operator are made).
For the proof of the next theorem a method similar to the method used in NEBRES [ 26 ] is applied. THEOREM 4.4
Let
and
6j
be two structures such that 21
0
is a sub-
is open in 6
. Then the following
(i) If a formula 4 of L(1) holds in 8
then it holds in 91
structure of Bo
and
A
are equivalent
(ii)
.
4 is topologically equivalent to a universal formula
of L(1) (where in the definition of universai formula no condition on the I-operator is made). Another advantage of L(1) over L(Q) with respect to topological structures is a "back and forth" characterization of elementary equivalence. In the game theoretic description the game is a localized version of a game for the cardinality quantifiers and uses the observation that this game gives a definition of partial isomorphisms which fits with the syntax if the quantifier is monotone (cf
. MAKOWSKY-TULIPANI
[ 22 1)
. The "back
and forth" technique can be used to get FEFERMAN-VAUGHT type theorems such as'lelementary equivalence is preserved under sums and products
in the topological sense:'
With the above mentioned "omitting types
theorem" and the "back and forth" technique one gets easily a characterization of H -categorical theories, exactly as in the ERNS-theorem 0
(ENGELER,RYLL-NARDZWESK1,SVENONIUS). Here a L(1)-theory T is
H 0-catego-
rical if it is complete and two countable topological models with the coarsest compatible topology are homeomorphic. This last result theorem
,
, and
most classical results of model theory as MORLEY's
can be obtained by a different method. L(1) can be reduced to
classical predicate calculus in the sense that for every topological model 2[ there is an expansion constants for every a€ A
* formulas $"of %I=
4 iff
91
*
of
uo
by definable relations and
and a translation of formulas 4 of L(1) into
predicate calculus in the expanded language such that U*I= 4
* .
Obviously L(1) can be extended further to ~(1")
as L ( Q ) in section 3
and in fact similar theorems hold. But for L ( I< W ) it is open wheter the interpolation and definability theorems hold. The reason why so many topological logics do exist lies in the fact that t there is a common extension L which can be characterized similarily as predicate calculus in LINDSTR~M' s theorem [I 4
5.
A
1.
MAXIMAL LOGIC FOR TOWMGICAL STRUCTURES
All the topological concepts which were definable in one of the topological logics treated so far have one thing in common: Their topological definition does not depend generally on open sets but only on the exis-
J. A. Dhakowsky tence of some set in a basis for the topology. A way to exploit this was first discovered by GARAVAGLIA [ 7
, 8 , 9 1:
Let L2 be the two-sorted
first order language appropriate for structures <
,a,e > where
an L-structure and a is a set of subsets of A = )
1.
P1 is
We call such a
structure topological if a is a topology. We call such a structure if a is closed under finite intersection. If f? is any family of
basic
subsets of
A
we define
B
to be( Us
:
s c f? 1 . We call a formula $
B-invariant if for every L -structure < 2
-
iff it holds in < U,f?,E>
,f?,E> =
5
4 holds in
. Note that B is a topology on
iff
satisfies the sentence Top which is the formula vx3x(xEX)&vxVX(xEx y E X
&
y E Y)))).
* W ( X E Y
*
91
3 Z ( x E 2 &vy(y € 2
-
GARAVAGLIA defined his topological logic to be the
set of B-invariant sentences which are consistent with
a.
t We now define the logic L which is independently due to GARAVAGLIA and ZIEGLER [40
1. We follow here ZIEGLER's approach. Whenever a theorem is
attributed to both GARAVAGLIA and Z I E G E R
, this means
that GARAVAGLIA
has proved the theorem first or independently for the B-invariant sentences.
t L is the smallest subset,of L above where the existential set 2
quantifier occurs only in the following form 3X(t E X t is a term in L2 and X does not occur positively in $.
&
$(X))
where
(X occurs posi-
tively in $ if a free occurence of X in $ is inside the scope of an even number of negation symbols. If X has no free occurence in $ then X does not occur positively.) Note that VX is defined as 4 X LEMMA 5.1
.
t (GARAVAGLIA,ZIEGLER) All the formulas of L are B-invariant.
For the rest of this section we consider only topological models.
J. A. Makowsky THEOREM 5.2 (GARAVAGLIA,ZIEGLER) t (i) A theory T in L has a topological model iff it is consistent with
Top.
t (ii) The set of L -sentences true in all topological models is recursively enumerable. t (iii) L satisfies the compactness theorem for sets of formulas of arbitrary cardinality. t (iv) L satisfies the downward Lciwenheim-Skolem theorem: A
set T of sentences in Lt with
has a model of cardinality <_ -
K
I
T
I
=
K
which has a model
with a basis of cardinality
There is a natural extension of the usual EHRENFEUCHP-FRAISSE characterization of elementary equivalence and of partial iso- (here homeo-) morphisms. In particular, in contrast to the similar characterization for t L(1) in section 4,the L -partial isomorphisms do converge to homeomorphisms in the sense of FLUM [ 6 ] THEOREM 5.3 (ZIEGLER)
and therefore we have
*
Let L be any logic for topological structures
which satisfies (i) L* does not distinguish homeomorphic structures (ii) L
*
satisfies the downward Lciwenheim-Skolem theorem
for countable sets of formulas
* * L
(iii) L Then
is countably compact. t is a sublogic of L
.
..." is
The definition of "any logic given in KEISLER'S
completely analogue to the one
paper in this volume [ I 3 ] adapted for many-sorted
structures. As a corollary one gets immediately:
J . A . Makowsky
COROLLARY 5.4
(ZIEGLER) A sentence 4
t
$
C,
$
of L is B-invariant iff 2 t for some sentence $ of L
.
There are several points which make us believe that L~ is a natural logic for topological structures
-
:
All the logics treated in sections 2-4 are sublogics of L ~ . Furthermore the concept of a continuous function or an open relation is expressible in L ~ Therefore . Lt also fits ROBINSON'S program for topological model theory.
-
t L is maximal with respect to compactness and the downward LowenheimSkolem property. Now these two properties are generally considered as a good substitute for a formal definition of "first order". So this suggests that no extension of L~ is"reasonab1y first order".
-
A
t good deal of classical model theory can be developed also for L
.
The rest of this section is devoted to examples of such model theoretic results. Definability theory.
Using the construction behind the proof of Lind-
strom's theorem or its analogue theorem 5.3 one can prove CRAIG'S interpolation theorem and from this one can derive BETH'S definability theorem and related results. These results also follow using the ultraproduct construction below. But a proof theoretic argument was not yet found. Ultraproducts
. An ultraproduct construction for the B-invariant senten-
ces was first given by GARAVAGLIA. He and ZIEGLER proved independently t that elementarily equivalent (in L ) structures have homeomorphic ultrapowers. As an application (via a result due to BANKSTON [ I
])
one gets
that the theory of perfect T -spaces is complete and another proof of 3
corollary 5.4.
J. A. Makowsky t Categoricity. We call a theory T K-categorical (in L ) if two models of T of cardinality
K
are homeomorphic.
THEOREM 5.5 (ZIEGLER) Any countable perfect T -space is homeomorphic 3
to the rational numbers with the standard topology, hence the theory of perfect T -spaces is -categorical. 3 Ho t For L it is still open wheter al'good omitting types theorem" holds to prove interesting categoricity theorems (as the ERNS- or MORLEY's theorem), but as long as there are no further natural examples of categorical topological theories, there is no interest in such theorems. Preservation theorems.
Using the previously mentioned back and forth
technique one easily gets the classical results that sums and products preserve elementary equivalence in L ~ . ZIEGLER also obtained syntactical characterizations of the sentences preserved under
topological homo-
morphisms and substructures etc. Decidability questions will be discussed in section 7, but let us mention that the set of B-invariant sentences in L
2
is not r.e.
Let us finish this section with one more general remark: As one would expect in a first order logic, many important topological concepts are not definable in L ~ among , those compactness, connectedness, separation above T3 etc. But similar remarks apply to group theory and classical model theory. The question arises what use topological model theory might have for topology ? In my personal oninion model theory might lead topologists to ask a question so far neglected in topology: How many different (compatible) topologies of one or the other kind one may find on a given structure. As far as I could find out it is open wheter every infinite group admits a T2-non-discrete topology ?
J. A. Makowsky
WDEWSKI
[28, 291 and HESSE [I 1 1 investigated this problem for
fields and almost abelian groups and WDEWSKI showed THEOREM 5.6 (PODEWSKI) If a group (field) of cardinality
K
admits
a compatible T -non-discrete topology then it admits 2
2K 2 -many different topologies.
As an application of the compactness theorem of Lt one gets THEORE4l 5.7
A group G admits a T -non-discrete compatible topology 2
iff every finitely generated subgroup of G does. It may very well be that an adaptation of SHELAH's theory of stable diagrams (cf. his forthcoming book
[36 1) to Lt may help to solve
problems in this direction. It might even be a source for almost unlimited inspiration in this direction.
6. UNIFORMITIES, PROXIMITIES AND CLOSURE SPACES
In the topological literature many other types of spaces exist which are in some way related to topological spaces. On one side there are weaker concepts, among which the closure spaces are the most interesting. On the other side there are stronger concepts which come mostly from analysis. Among these the uniform spaces and the proximity spaces are the best known. What all these spaces have in common is that they use different basic concepts which canonically induce topologies. In this section we shall give some indications how the previous approaches apply for the above spaces. Topological reference books are e.g CECH [ 2 ] and ENGELKING [
5 I.
J . A. Makowsky miform spaces. The basic notions for uniform spaces are uniformities and entourages. They play analogue roles as topology and neighborhood. The analogue to SGRO's approach in section 2 would be to consider a binary quantifiersymbol Uxy@ which reads "the set defined by @ is an entourage". Structures then were pairs < O(u> where 21 is an ordinary structure and u is a uniformity. As one easily observes most of the results of section 2 carry over but the author did not check the details. Anyhow this approach seems rather pointless: its expressive power is very weak (as for L(Q)) and,that it works,just repeats an already made experience.
More interesting, it seems, is the approach sketched in
section 5.
Structures now are triples of the form < ?I u,,& > where
a
is a classical structure, u is a uniformity on &1 and
2 nary relation on A x u with
:
w c u').
is a ter-
&(al,a2,v) iff the pair (al,a2) is in v.
If u s is an arbitrary family of subsets of set{Uw
&
we denote by ;'
the
With this a notion of B-invariance for L -formulas 2
can be defined exactly as in section 5.
u L is the sublogic of L2 (appro-
priate to the above structures) where existential quantification of members of u occur only in the form 3v$(v) where v does not occur positively in $.
With these definitions all the theorems of section 5 can be
proved for L ~ . A detailed exposition of this program may be found in STROBEL's thesis (Diplomarbeit) [37 1. Proximity spaces.
The basic notion of proximity spaces is nearness.
A typical example is the following relation between sets of a topological T -space T: If A,B are subsets of T we say that they are near iff 4
cl(A) fl cl(B) #
0. For the definition of an abstract proximity we refer
to ENGELKING [ 5 ] or NAIMPALLY-WARRACK 125 1. One way to develop a first
J. A. Makowsky order logic for proximity spaces is the following: Weak structures are pairs < u,ll> where a subfamily of P(A)
a i s an ordinary first order structure and ll is
2
.
The logic L(P) is obtained from predicate cal.culus
by adjunction of a binary quantifier symbol P and the formation rule (P) If $ and $
are formulas of L(P) and x,y are variables such that
.
(4) then Pxy ($:$) is a formula of L (P)
x (y) does not occur in $
Satisfiability is defined by the clause
(*PI A
< 9 1 , ~>
I= ~xy($:q) iff
(
$ ( a) , $ ( U $ ) En
.
ll is a LEADER-proximity,
weak structure is called an LE-structure if
a LO-structure if ll is a LODATO-proximity and a P-structure if proximity. (cf. NAIMPALLY-WARRACK [ 25 1)
. L(P)
ll is a
with these three inter-
pretations is studied in MAKOWSKY [ 1 ]. The logic L (P) is closely related to L(I), in fact in a certain sense they are equivalent. Consider the following translations of L(P)-formulas into L(1)-formulas:
Rl (Pxy($:$))
:=
32($1:
a2 (pxy( $ :$)
:=
32 (-I=-$
&-Iz-
/
foraulas into L(P) -formulas
$1:
)
and and the translation from L (I)-
&-IZ-$l:)
R (Iz 4)
:=
PXY (z=x:$I)' Y
.
THEOREM 6.1 (MAKOWSKY) If $ is a sentence in L (P) then (i) $ has an LE-model iff El($) has a topological model (ii) $ has an LO-model iff R2 (@)is consistent in L(1) with
R -separation Vxy (Ix(x#y)-Iy
(x#y))
.
(iii) $ has a P-model iff R2($) is consistent in L(I) with T -separation Vxy (Ix(x#y)-(x#y) 1
)
.
If $ is a sentence in L(1) then (iv) $
has a topological model iff L($) has an LE-model.
As a consequence all the results of section 4 carry over to L(P) ,in
.J. A . Makowsky
particular the interpolation theorem. Closure spaces. In his book [ 2 1 CECH observed that much of the theory of topological spaces may carried over to closure spaces, i.e. spaces with a closure operation cl satisfying the axioms cl0 =
0,Xc
clX
,
cl(X U Y) = clX U clY. It is easy to check that the approach via L(1) of section 4 is also good for closure spaces. Most results of section 4 carry over although no Gentzen system has been developed up to now. The only interest closure spaces have in this context is their role in a conjecture of JASKOWSKI which will be treated more closely in the next section.
7.
DECIDABILITY
In this section some decidability problems for topological spaces will be discussed.
Some of them are by now classical, but new proofs
can be given with the use of model theory.
We shall restrict ourselfs
to decidability of topological spaces omitting important work on topological groups and fields. In particular SGRO is working on a project on topological groups where he already got interesting results. For an elimination technique of the I-operation and its application to topological fields the reader may consult MAKOWSKY-MARCJA [ 20 1. Consider now the following language for topology: We have only setvariables X,Y,..
. and
the operations n,U,~,cl (intersec-
tion,union,complement and closure) and the constant
0. With
this we may
build finitary terms.Denote by T(c1) the set of so built terms. The equational calculus over T(c1) consists of the equations t =t' for terms from
J. A. Makowsky T(cl).Denote by E(c1) the set of equations true in all topological spaces and by C(c1) the set of equations true in all closure spaces with the obvious interpretation. THEOREM 7.1 (1) (MCKINSEY-JASKOWSKI) E (cl) is decidable. (ii) (MAKOWSKY) C (cl) is decidable. If we replace the closure cl by a unary operation
'
(derivative) we can
.
consider the corresponding sets T (der),E(der) and C (der) In a footnote of MCKINSEY [ 241 it is claimed that JASKOWSKI had proved the decidability of E(der) but no proof was ever published. Let us therefore call this last statement JASKOWSKI's conjecture. Denote by El(der) the set of equations over T(der) true in all T -spaces. 1 THEOREM 7.2 (i) (ZIEGLER) El (der) is decidable. (ii) (MAKOWSKY) C (der) is decidable. The point of interest here is that both theorem 7.1 and 7.2 can be proved via the model theory of L(1) observing that the sets of equations above are subsets of the monadic L(1)-logic without equality (for cl) or with equality (for der). For 7.1 and 7.2(ii) one proves the stronger result that if an equation has a model it has a finite model where the cardinalitydepends only on the lenghth of the term occuring in the equation.
.
(cf MAKOWSKY-ZIEGLER
23 1)
. 7.2 (1) depends on the following
THEOREM 7.3 (ZIEGLER) The theory of TI-spaces is (i) decidable in L(Q) (ii) decidable in L (I) (iii) undecidable in L~ whereas the theory of all topological spaces is t (iv) undecidable in L (Q),L(I) and L For L(Q) the result was also known to SGRO.
.
3 . A . Makowsky Using theorem 5 . 5 and the decidability of the monadic weak second order theory of two successor functions (DONER 14
1) one gets
THEOREM 7.4 (ZIEGLER) The theory of T3-spaces with a finite number of unary predicates is decidable in L~ whereas the theory of T
-
2
spaces is undecidable.
8. SOME OPEN PROBLEMS
In this section some research problems are presented. Again their choice is rather personal and biased by the authors own research. Most problems are not original :they are in the air or more precisely in the background of the many discussions I had with all the mathematicians mentioned in the introduction. For problems concerning the lattice of closed sets the reader is refered to the excellent problem list in [ 101. For L(Q) no particular problems are listed since most problems for L(I! below have their analogues for L(Q).
Concerning the product quantifier the main problem is the absence
of a general theory (or even orientation). Therefore we state Develop a general theory of product quantifiers. In particular, what properties of the unary quantifier carry over to the product quantifier for a suitable definition of the n product '(in the monotone case (*Q ,Man))
. Is it true that
if the unary quantifier is wiomatizable for a given interpretation so are the product quantifiers for the corresponding product interpretation? What other model theoretic
J . A. Makowsky
properties are preserved ? Concerning L(1) the problems are more precise.
In section 7 a connection
between L(1) and the squational calculus of the closure and derivative operations was sketched. PROBLEM 2
Prove or disprove JASKOWSKI's conjecture eventually using the model theory of L(1).
In section 4 two preservation theorems (4.3 and 4.4) were presented. PROBLEM 3
Characterize the formulas of L(1) which are preserved under inductive limits (in the category of topological spaces). What are the formulas preserved under reduced products ?
L(1) has virtually all the properties of predicate calculus. It also has the following (technically) important property
:
(DEF) Given a topological structure < u , q > let d be the family of the
L(1) -definable sets using parameters from
. Then < a ,q > and
< u,d> are elementarily equivalent in L (I) (with parameters). ZIEGLER now conjectures CONJECTURE 4 Given any logic L
*
for topological structures which satis-
*
fies the hypothesis of theorem 5.3 and (DEF). Then L is a sublogic of L(1). More problems on L(1) may be found in MAKOWSKY-ZIEGLER [ 231.
ZIEGLER showed that the T -spaces with one equivalence relation is unde3
t cidable in L PROBLEM 5
.
Is the theory of T3 -spaces with one equivalence relation t with closed equivalence classes decidable in L
t The T -spaces are undecidable in L 2
PROBLEM 6
?
.
u Are the uniform Hausdorff-spaces decidable in L
?
J. A. Makowsky L(1)
has a nice proof theory as shown in section 4.
PROBLEM 7
Develop a "nice" proof theory for Lt or L ~ .
PROBLEM 8
What kind of useful1 omitting types theorem do hold for t L or LU ? What kind of categoricity theorems are true in
Finally there two more very general problems PROBLEM 9
:
Extend the framework of abstract model theory (cf. KEISLER's paper in this volume [ 1 3 1 ) such that topological model theory and even other model theories (KRIPKE models e.g.1 fit nicely into it
.
PROBLEM lo What use for topology does topological model theory have
?
Added in January 1976: From MdKEE
( Zeitschrift f.math.logik und Grundlagen der Mathematik,
vo1.21 (1975) pp
405-408 , Infinitary logic and topological homeomor-
phisms) I can guess that most of the results described in section
5
were known to T. A. McKEE already in August 1974. I am glad that I could
- better late than never - give credit for his work.
REFERENCIES
P.Bankston, Notices AMS, Abstract 75T-G37 E-Cech, Topological spaces, Prag 1966 C.C.Chang, Modal model theory, Proceedings of the Cambridge summer school in mathematical logic, Springer Lecture Notes in mathematics vol. 337 pp. 599-617 J.E.Doner, Notices AMS, Abstract 65T-468 R.Engelking, Outline of general topology, Warsaw 1968 J.Flum, First order logic and its extensions, Proceedings of the Logic conference,Kiel 1974, Springer Lecture Notes in Mathematics vol 499 7.
(to appear)
S.Garavaglia, Notices AMS, Abstract 75T-E36
8.
-
I
do
, Abstract
75T-E61
9.
-
I
do
, Abstract
75T-E79
lo.
C.W.Henson,C.G.Jockusch,L.A.Rubel
and G-Takeuti, First order
topology, preprint 1975 G.Hesse, Zur Topologisierbarkeit von verallgemeinerten abelschen Gruppen, Hannover 1974, preprint J.Keisler, Logic with the quantifier "there exist uncountably many". Ann.math.logic vol.l.1
-
(1969) pp. 1-93
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