138 39 10MB
English Pages 335 [360] Year 1972
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UNIVERSITY
of ALASKA
ELMER E. RASMUSON LIBRARY
Digitized by the Internet Archive In 2021 with funding from Kahle/Austin Foundation
https://archive.org/details/saturatedmodelthO000sack
Saturated Model Theory
MATHEMATICS LECTURE NOTE SERIES
Published
J. Frank Adams E. Artin and J. Tate Michael F. Atiyah Jacob Barshay Hyman Bass Melvyn Berger and Marion Berger Armand Borel Raoul Bott Andrew Browder Gustave Choquet
Paul J. Cohen Eldon Dyer Robert Ellis Walter Feit John Fogarty William Fulton
Marvin Marvin Robin Robert
J. Greenberg J. Greenberg Hartshorne Hermann
Robert Hermann
Lectures on Lie Groups, 1969 Class Field Theory, 1967 K-Theory, 1967 Topics in Ring Theory, 1969 Algebraic K-Theory, 1968 Perspectives in Nonlinearity: An Introduction to Nonlinear Analysis, 1968 Linear Algebraic Groups, 1969 Lectures on K(X), 1969 Introduction to Function Algebras, 1969 Lectures on Analysis Volume I. Integration and Topological Vector Spaces, 1969 Volume II. Representation Theory, 1969 Volume III. Infinite Dimensional Measures and Problem Solutions, 1969 Set Theory and the Continuum Hypothesis, 1966 Cohomology Theories, 1969 Lectures on Topological Dynamics, 1969 Characters of Finite Groups, 1967 Invariant Theory, 1969 Algebraic Curves: An Introduction to Algebraic Geometry, 1969 Lectures on Algebraic Topology, 1967 Lectures on Forms in Many Variables, 1969 Foundations of Projective Geometry, 1967 Fourier Analysis on Groups and Partial Wave Analysis, 1969 Lectures in Mathematical Physics Volume I, 1970 Volume II, 1972
Robert Hermann J.F.P. Hudson
Lie Algebras and Quantum Mechanics, 1970
Piecewise Linear Topology, 1969
Irving Kaplansky Kenneth M. Kapp and Hans Schneider Joseph B. Keller and Stuart Antman (Editors) Irwin Kra Serge Lang Serge Lang Ottmar Loos
I.G. Macdonald
George W. Mackey Hideyuki Matsumura Richard K. Miller Andrew Ogg Richard S. Palais William Parry Donald Passman Walter Rudin David Russell Gerald E. Sacks Jean-Pierre Serre Jean-Pierre Serre Jean-Pierre Serre
Shlomo Sternberg Shlomo Sternberg Moss E. Sweedler
Rings of Operators, 1968 Completely O-Simple Semigroups: An Abstract Treatment of the Lattice of Congruences, 1969 Bifurcation Theory and Nonlinear Eigenvalue Problems, 1969
Automorphic Forms and Kleinian Groups, 1972 Algebraic Functions, 1965 Rapport sur la cohomologie des groupes, 1966 Symmetric Spaces Volume I. General Theory, 1969
Volume II. Compact Spaces and Classifications, 1969 Algebraic Geometry: Introduction to Schemes, 1968 Induced Representations of Groups and Quantum Mechanics, 1968 Commutative Algebra, 1969 Nonlinear Volterra Integral Equations, 1971 Modular Forms and Dirichlet Series, 1969 Foundations of Global Non-Linear Analysis, 1968 Entropy and Generators in Ergodic Theory, 1969 Permutation Groups, 1968 Function Theory in Polydiscs, 1969 Optimization Theory, 1970 Saturated Model Theory, 1972 Abelian /-Adic Representations and Elliptic Curves, 1968 Algébres de Lie semi-simples complexes, 1966 Lie Algebras and Lie Groups, 1965 Celestial Mechanics Part I, 1969 Celestial Mechanics Part II, 1969 Hopf Algebras, 1969
In Preparation: John L. Challifour
Generalized Functions and Fourier Analysis, 1972
a be! 5 we-2
@
Saturated Model Theory
GERALD E. SACKS Massachusetts Institute of Technology
1972 .
Ny e
W. A. Benjamin, Inc.
Prey
If
“n.)
totally
of
something
that
0
éj
(Corollary
sentence
result
uniqueness
apply
due
field
differential
implies
I am
to
characteristic
holds
DCFo
and influential
of
a general
to
theorem
differential
from
which
first
order
n
to
theories
all
of characteristic
showed has
rank
to L.
above
theory
theory
integer
0
of model
typical
in
exists
differ-
characteristic
compactness
of
of
applications
the
compactness
field
theory
of Morley
was
(One of the most
every
of
notable
algebra,
the
40 and 41 are devoted
applications are
is
of
Shelah
differential
theory,
credit
so
justly
it to
6
SATURATED
many
who
have
have
been
contributed
attached but
them
discovered
persons it
is
(not
rare
property mys
that
of
a historian
man,
this
in
of
early
sketched
in
proof
the
based
enough
by
derived
of
theory
to
realize
since
sole
will
not
the
many
me),
the
is
to
the
several
to
one
of
construce
necessary
to
that.
efforts
be
the
of
one
have
to
been
those
of
with
‘and
sections,
in
to
make
little
logic.
“sentences”
and
"logical
detail
taken
the
are
common-
consequence”
section
existence
7 to
are
make
theorem
the
(Tey
all. book
notes from
Institute
by
is
no
most
= It
fundamental
University
on
I hope
Names
that
known
idea
“structures”
properties
Yale
inviting
accessible
This
were
existence
sensical
readable
whom
precautions
the
of
certain
THEORY
Tarski.
Definitions
given
of
mind:
its
to
independently
of model
Alfred
book
morally
acs malice,
owes
Some
is
an
one
lenorance*
subject
all
subject.
parenthetically
theorems,
were
it
toes
MODEL
of
follows in
the
Fall
prepared
a course
closely
by
given
Technology
in
of
S. at
the
a course 1970,
Simpson, the
that
given
at
course
those
notes
Massachusetts
Spring
of
1969.
A
INTRODUCTION large
debt
falls
due
courses,
—
—
very
who
to
patient
Kreisel,
F.
Rowbottom
sort
students
that
proofs
all
MacIntyre
typist,
H.
and
them
S.
Cambridge,
in
never
both
but
rarely
complete
and
proved
to
and
to my
fellow
model
L.
Blum,
H.
M.
Shelah,
me,
fascinating
J.
whose
be
a
Keisler,
Morley,
contrary
Massachusetts
15, 1972
be
that
who
Lachlan,
opened
a truly
March
my
the
relentlessly
A.
explanations
to
of
insisted
among
G.
to
owed
Jane
theorists;
will,
fortunately
is
successfully correct,
ie
A.
Robinson,
generous
to my
subject.
initial
1.
Ordinals
Ordinaleszeare
each
ordinal
ordinals; seu,
is
Cardinals
cardinal
is
one-to-one The
an
denoted
equal
thus
{B|B
are
denoted
ordinal
it
with Ane
«
.
Kr
A
set
w
.
be
Card is
is
A
can
the
put A
successor
by
with
in
into
least
O
is the
“Ks
any
one-to-one
cardinal
is
an
O
nor
a successor.
8
empty
put
lesser
into ordinal.
order
are:
correspondence of
greater
ordinal
neither
be
cardinality
a limit
lesser
S a set has cardinality
finite
denotes
is
6s is.=. 3) 2
increasing
the
65s
aly
is
‘%
aVs5
of
it
@+i1.
if
Bs
cannot
Wyaees
is
countable
set
oon
2
c@. then. are
(i)
gf
@
then
@>
and
are
elementary,
if
2)
But
then
elementary. is
an
g
(41)
f
of (ii).
é
.f:
is
@k
&
and
and:
“ef.
elementary.
Suppose
F(gfe,,..-.gfa.)
mentary.
@
f
THEORY
Suppose
ievelementary.
Proof
is
och
6.1.
If
elementary,
Then
Ma
MODEL
@
F(a,5-.-28,)
pessince
(ef
F(fa,,...,fa,)
1
evele-
>. Since
.¢
2 elementary
substructure
of
& ,
in
symbols
a
ks
then
conveys
(If
formula
with is
a
is
@
if
elementary.
of
set.
atomic
the
same
for
extensions
formula
of
the
of
@.
of
ali
m>O
and
H
has
no
Oe
atomic true
in
group, as
models An
form
(Ey, )...{Ey,,)# ; where
is
diagram
sentences
The
T
monomorphism
The
information
@.)
inspired
A theory
every
the
was
is a multiplicative
table the
(9.2).
lb? ,;, is
negations
multiplication coincide
completeness
(A.
sentences
D@?
Completeness
Nullstellensatz
complete
cd,
Model
quantifiers.
the
of
D7
existential
Proposition
existential Ss
MODEL
SATURATED
36
C=
Ay
formula,
3
then
Proof.
H
If “G(x,,...2%,)
@ | G(cy5...sc,)
fle
Let
(By). where
eine
G(ge,,
2
G(x,5...s%
(By )H(e, be
has
no
+ 3 EC
2)
is an
and
)
be
Rye Fy eo
quantifiers.
THEORY
90)
;
Then
Cl- H(c,5---s0,2d,5--2d) for
some
dy s+--5d,
5 and so
€.C-,
«But:
then
H(ge,s---»8C,28d,5---284,) £f G(ge,s..-sg¢,)
Theorem 8.2
.
9,
a
(A. Robinson).
T
is model
complete iff (i) iff (44) Gy)
model
T UW Db?
ads
Of,
(ii)
Proof.
ak with
slat a5. the
leeiby
a
complete
theory
for
every
1.
For
existential
is
each formula
formula
G
such
Suppose
T
WS
Br
intent
and
1 of
F
showing
is
there
is an
that
T |. F
model
complete
be models
of
By = Bo
in
G. and
9150) Dg order
to
MODEL
COMPLETENESS
conclude
from
B, (i=1,2)
7.1
Si that
T VU D¢?
is a model
morphism
f,:@>
8, .
is model
complete.
is
complete.
of
D? , there
f,
is
Since
is a mono-
elementary
since
T
Consequently
8, [== F(f,a,....f,a,)
iff Now
suppose
monomorphism
F(X, 5++.5X
(ii)
between
rm)
Suppose
of
g: @> T
.
#
isa
Let
in the language
is an existential
F(X,,-.-»%,)
@k
(ii)
implies
Finally,
.
Let
F(x)
38
SATURATED
Assume Si
for the
sake
*is.consistent.
Clearly
of a reductio By 7.1
S
aca F M(e).
follows
from (i) that
character
of
|
Tu
MODEL
THEORY
ad absurdum
has
a model
that
qd.
forteome
ch € AVE
D?@}/ F(c)
.
Tt
The finitary
implies
Th Q(¢s a5---58,) > F(c) ; where
Q(c,»@)....53,)
finitely
many
are
individual
can
be
where and
sentences.of
constants
replaced
TE
Q
Q(x
Dl”
not
xX)....2%,)
>
Since
¢c,a
in
T
pe ee
,
they
Thus
F(x)
quantifiers.
, where
.
occurring
by variables.
has no
@|= K(c)
is the conjunction of
3
So
K(x)
Tl} K(x) > F(x)
is the
existential
formula
(Ex,)...(EX,,) Q(x» X19+++9Xp)
But the definition of The
exist
inconsistency
existential
S
entails of
formulas
S
@|= ~K(c).
means
there
K, (x),...2K, (x)
that
TEL K, (x) > F(x) - (
a model is
Q
an
intuitive
part
of
the
following.
the
that
"infinite
feeling
proof
of
A natural
"infinite
Such
Let
model
such that
has
be
disjunction"
imply
F
.
disjunction"
a reduction
©
{c, | i¢I}
each
Exercise and
the
equi-
is
to
termed
phenomenon.
and
for
last
the
Exerelse-o.5.2
Suppose
the
disjunction.
a compactness
acquire
formulas
reduces
41s logically
formula.
to
upon
choice all
G(x)
Then
-O
Theorem
9.1
(A. Robinson).
ACF
is model
complete.
Proof,
Get
i:
algebraically
closed
g:
and
ad ——>
card
Gine
ay
card
isomorphism
k:
a>
2
besa.
fields.
By 7.3
h: 8 ——
By > card
6
ay aS
fe) as
Ho
monomorphism
B, .
If
such
there
sot
exists
such that there
that
exists
an
MODEL
then
COMPLETENESS
OF ACF
Ke
By ’
hl
©
is
assume
Gere
elementary
f,g
and
are
(respectively
B,) Over
Let
uncountable;
vks
UW
that
k
extended tively
“be
‘one-one
B,) is the
that
fields
of
two
the
same
are
isomorphic,
ing
sin
cannot the
be
theory
tablished method
to
of
real
in
card
onto.
U =
U
ay
for
so cardV.
| Extend?
6
closure
turns
cardinality
there
is
setting.)
prove
the
closed
section
involving
Let
k
to
Kk
can
al
be
@, (respecof
@(U)
2a
a eneral used
base
tye ond.
uncountable,
so
maps.
6, , since
argument
any
to
is infinite.
anid
algebraic
@(V)).
above
safe
a(v)
k,: @; ——>
(respectively
fact
4 2
PertNe we dent
1
to
The
inclusion
consequently
Ee (U) > so
[tis
a transcendence
V) be
is
by 6.1.
h
(respectively
B al
Ay
17
by
saturated
on
the
rather
special
algebraically and
little in model
fields.
closed
characteristic
chance
of
particulan
applyat
completeness That
a virtually structures.
will
be
universal
of es-
4e
SATURATED
Corollary 9.2 (D. Hilbert). finite
system
in
several
@.
if
@,
then
Ole.
(66
of
variables S
has S
is
that a
a
for
wette
a
has
PYoots* There
polynomial
a
every
@
is
an
closure
formula
of
F
in
is
language
a
extending
type
70f
Le
1d.
such
solution
algebraically
elementary
@.
field
algebraic.closure
Shas
every
inequations
in the
field
the
the
®@2>@: 9.1
algebraic
the
in
By
of
TsSvot
H
.
extension
and
‘ber the+-similerity
field
H
some
THEORY
S bea
coefficients
solution
sentence
Let
equations
solution.in
Leticct:
Bl
A
with
MODEL
in
closed
extension
of
the
2
universal
existential
if
it
form
(x,)-..(x,) (Ey)... (By,)¢ " where
m>O,n>0O,
A theory
T
is
and
G
universal
a theory
W
member
of
a universal
ACF is
oo.
is no
is
a universal
accident
when
no
quantifiers.
existential
exists
W
such
has
that
T=W
there
and
every
existential
existential viewed
if
in
theory; the
light
sentence. that of
fact
9.1
and
MODEL
COMPLETENESS
Proposition) model
complete,
is
A
of
T
formula
the
H
has
no
(Q,x,)
The
a prenex
of
of
alternations
existential.
said
to
be
prenex
normal
’
denotes normal
occurring
an alternation
orrer
is
quantifiers,
i (lO
is
and
universal T=W
K
which
has for
G
K of
has
is two
the
same
Then
(Ex,)...(Ex,)K GOn.=
G
“Then.
universal
F)
, and
sh aie
We a logieas
existential
sentence
W.
universal
(x, ) airelie (x,,)H
where
of
> K.
(Ex, )...(Ex,)K . WE OP:
TEG
conjunction
each
WkEG
%)
0
requires
10.
The erect
Direct
direct
whose
of
Structures
limit
operation
is
e.g.
saturated
models
structures,
theories,
Systems
existence
is
not
needed of
to w-stable
immediate
from
s
Hct A
D
with
+s J 6D
i oe
and
58
SATURATED Ee
Ty
is
Ty
is
model
and
T
bute
TL
complete.
satisfy is
A,
not
Theorem are
model
the
chain
completion
@
T
be
an
showing
@
,
that
of
T
If
1,
then
T, =
arbitrary
is
of structures
then
TS
complete,
and
T,
of
Ty
T,
model
a model
THEORY
T.,
is model
(A..Robinson). of
of
happen
a model
Let
of
can
(ii), T,
{a | nn
a certain Bo
the
.
same
property
i
finite
B 17 3o
of
as
for
of
all
a "back-andLet
by
that
a,=b,=C,
induction
On:
7b
fb, =C,
and
some
=
exists
Cc, = fe, | i
Mel
Weu
(jen
iis
goa]
a
& ~d
Proof.
individual S
realized
Sal ( that
aan
yey
“7
bean
sma —
exists )
vey) 2
p-.
CO
a
is
ainrinice
To prove
constants
there in
CnC
8>dQ@ _ such that
is realized pmax(w
then
realized
» card
in
#
every
and
Y)
(1) let
c,,...s¢,
not mentioned
in
be
T(+.49¢,) T(
wen?
()
yey
Beare
can
be
ae, 2
regarded
as
ssacsumed:
every
| F(x, 2--25%,) €P). an
that:
extension
a. = vy
of
“for
ie Geran. coetOty
(Ex,)...(EX,)F(x,5...5%,) for
by
wy ers
cardkd.sx
Carine
i>) Let
such
There
Dice be
diagram
property.
sort
ea
ean
diagram
shown.
Then
~T
‘is model
a is
derived
[Kol] @*.
for is
from
model
a
criterion
completeness.
replaced
by
an
ultra=
ELIMINATION power
of
For
the
more
Theorem
QUANTIFIERS
#8.
algebra, seems
OF
elimination fields.
of
germane
the
the
of
The
directness
in
section
40
to
of
differentially
find
of
closed
of
direct
method
various
17.2
will
fields
extension
for
completeness
axioms
and
theories
be
of
exploited
for
of
to
ultrapower.
for
simple
theory
saturated
model
quantifiers
model
notion
most
completeness,
89
of
partially
than
provides
establishing
RCF
applications
notion
17.2
FOR
the
theory
characteristic
O
B(c)
B(c)
denotes
is the
universe
least
T
is
following
Let
T
language
such
that
every
model
completion
# 3 i.e.
&(c)
(L. Blum).
and
to*some
of
of
whose
{c}
same
universal,
extended. model
BU
17.2
in the
extension
substructure
contains
Theorem theories
a simple
of
T
of iff
model
of
T*
Then
.
every
and
T
T*
Tc can
T*
diagram
T*
is of
the the
B,B(c) ET.
X
\
Be |= T* ,
\
\
,
be
sort
pe
be
p* is (card @)*saturated.
90
SATURATED
can be completed Proof. of
extends aS
T* U DS
Gl.
#(c)
{F(x)
T*
Let.
‘be a model
by Now
property.
| pep
some’
“b*
Then
of
i
6*s
following
>
1s by
>a, limit K
.
The
eis rsealized
diagram
stronger
property:
sort
px le T+ |
*
B* is (card @)"-saturated. Let
{c, lS:ahie
ordinals”
consecutive
a
as shown.
= Cx (cg) » and
Catt
T*
Be Cees.
\ h\
chain
Vp.
given
the
\
the
the
has
can be completed
of
ac = b*
has
\
Define
implies.
T*
&
model
pe S, (t* U De)
3 set
T*
the
E F(c)}
and
of
assume
diagram
2
is
Let
is complete,
ine.
every
assume
.
pervLalwsaturation
THEORY
shown.~
First
eonplLetion
that
as
MODEL
C-f[B]
bye
= {c, | ee
EGATesPeRos O
C, = U {c, (o8< When: uses
of
Rieter
“hs
Co
8
the
given
es)
sis -derined diagram
property. ete
Saebes
a modell
Om
seb
ends
Wet
Cu
no
aes
ELIMINATION
OF
and
be models
£2868
QUANTIFIERS
(card @)*-saturated
of
FOR
T*
RCF
91
. -By 16.4
C* >+@.
there
is a
If the diagram
ax
can
be
completed
complete,
and
completion
T
TcT*
.
ate
SIIC Dt ab
Dl
shown,
then
T*
U D@
T*
is
the
consequently,
of So
as
the
.
2
is:a
stronger
model
diagram
vor
is model
T ,
since
property
supplies
CX
“ea
Ss
cone
B To
and
see
that
the
model
theory
h
stronger
is
elementary,
diagram
property
complete.
0
The theory
of ordered
of
relation
fields symbol
(TF)
.
(rr)
sis) be
the
amal-
Woe
fe
that
K(T)
Thus
with
les
reviery)
Ss
theory.
174
SATURATED Proposition .29 2.
variant
functor
Proof,
To
see
Fix.
DF
to
is
see
vor
i?
DPF some»
contains
‘as limit
point,;and-so
Veca
Deo.
(Fe) “+x
admits
following
ByP2oce joxeubohsy
Vis
in
wuchytney.
so that *
DFG
the
DEC
y=
contains with
diagram
be
“has
can
INEGI)
“OCLONSS Suppose
tO™-
“DS,
{7,5 £45}
DIZ.
(Fe) ty
(ret) ~+Fry
Defy
¥
to
DF
-such
that
(v*rr)~!
Pad
THE is
MORLEY at
deg
x
DERIVATIVE
most
,
is
n.
The
called
Proposition Tanked.
PointPory
Lio least
the
degree
29.5
| Ry
such
of
(Degree
send
n,
denoted
by
x.
Rule).
Zancies
If
be,
x
isa
them
deg x = ) {deg y | Ffy = x & rank.y = rank Proof.
Choose Ls
“
Ldes x)
so
the
safer
thatthe
Sinces%
amalgamation, as
C
a6)
to
assume
cardinality
sacmitcurittrations
following
diagram
can
x}.
ranigix=
(Fg )
of.
-1
with
be
completed
shown.
Then
deg x = card Proposition
such that aamite
(Fhg) “tx = y {deg Viv hry 29.6.
D°’R@ = 0
£1 tracions.
Suppose
for‘some -Lnen
DFR
a
is an ordinal
@ew'.
If
=)
2nlal,
@)
age)
atone
% ep SG Nec
O
e
180
SATURATED Proof;
There
ge: B>@. fe
an F
a
By 29.2
On
is totally.
totally
degree
are
rule
no
in
then
the
the
study
pre-image
of
x
Suppose
f:
@¢->
and
x € F@.
can
7 The=desreecruLesimplies:.i1f = degex=i1
same
rank
central
a unique
and
degree
to the proof
Exereise
substructure be
the
complete
contravariant
Assume
S
exists
is
x
.
in
F@
This
of Theorem T
theory.
Let
is
such
the
fact
-is
a complete,
S: *(T) > ¥
transcendental.
a@
5
‘
mentioned
ordinal
of
last
35.6.
Suppose
belongs
in Ci eee Show
that
for
there
all
» 0S@ = 0°.
Exercise is
as
functor
totally
a countable
Tex(T)
S
29.7.
pre-image
8
have
Cola.
has
.
If
rank
of
A
exists
rank
x
x
if there
vA.
higher
then
than
Mend
@e«X,DFe=0.
helpful
says
PAC,
tvans@enicntal
transcendental,
rules
rank
atl i
THEORY
wre = (D’re)(D*rr)
such that for all
is
The
exist
MODEL
totally
29.8.
Let
S: K(ACFy)
transcendental.
> NM .
Show
50.
Autonomous
Throughout
that
admits
direct
amalgamation,
of
of
in
is
definition and
the
value
“”*
to
X*
its
be
seen
most
rivative
for
many
of
S@
equals
a-th
of
are
7 enna.
¥& .
on
%**
The
section
@'s,
the about
because
its
In
section
this
,
a
EL Nem
the
sie
derivative
under
Minow.) by
with
contravariant
given
only
theorem.
that
a
cumbersome,
inspired
in
is
Morley
all
category
filtrations
the
CONG
extreme
a
Fixe
depends
is
and
is
limits.
conditions
Skolem-L&Bwenheim at
»
over
OlmGm
®
*>**
practice
D° Fa
SUDCALeCLOLYs of
D Fa
ranges
next
of
F:
preserves
computation
,
section
limits
and
PuUnCTOraALbat,
F@
this
Subcategories
which
the
small
full
MeCUcin On
downward
going
reduction
will
31,
it
the
a-th
where
a-th
Morley
be
will
de-
Cantor-Bendixson
Lee
SATURATED
derivative From
F¥:
#% >
Ke)
S@
for
all
now
on
XK*
is
the
restriction
WW
Ke
CViCIVia
of
is
de vautonomous
his
SKa>
THEORY
‘a a
full
if.
subcategory
of
F:
of
K>¥W
(DF) * = DGF*)»
X
to
«for
Na
PFOPOSLULON.
then
MODEL
507."
(D°F)* = D’(F*) Proot,.
Lt
MK”
Ts
for all
-By induction
on.
7autonomous,
a Gs
“Let
©7icr
xe
es:
(D'ttp) ag = D((D°F)@) = D((D°F)#@) = Dt (P*)¢@ . suppose (D’F)* = D’(F*) for all a .
exists
contains
¥c%
sig
there
is
universe i:
if
U,V;
MODEL
psy
and
x
BOUNDS
ON
splits
in
D's
splits
in
D’sy*
OvieT
RANK
for
some
for
%# «€ C(Ch)
some
such
that
Direct
{Yr 33
Let vs
is
{v,}
By 29.2
finitely
ex
generated
be thesseteot-all
finitely
by set
inclusion.
in
Psy,
POL
there
ween
exists
Cehiccl Wie
Proof.
is defined
Then
eeLae.
Y
.
# = 1im v, -
ane
dew (T)
a universal
Let
UG Ce
2.
I ye=eUalU
3,
for
«=
Se intinite
domain
%>d€@_=
such
card @.
ordinal
Us ,
tovchnoose
on
A chain
6.
Y=
a,
each
{u, | 8
generated
3
over
in
Y
D°sy
oy! Chae whose
~
with
the
for
Let
some
U,,,
universe
following
# € (DT)
property:
4 then
if
x
x
isplite
be the least member of
contains
splits
(T)
ain
190
SATURATED
MODEL
THEORY
Vvox los Ve: x} é: U, U CWo Assume
card
Us =
The cardinality generated} is
an
ax als ie
in
of
order
the
Str
point
least
Rielela
to
see
{vy|re U, av
is,at.most
isolated
be
K
«.. of
such
Fix
Usay
=
{k
is finitely
“% .
D’sy @,.
card
for
Suppose
some
Choose
a
x
3° Let
a basic
open
woke
{x} = D“sr n Nee: Then
N,, i Ny
generated,
wnen
the
is at most in
D’sr
the
number
GO
Us
number
for of
some
Lemma l%.
with
its
(Ye
basic
open
at
Let
most
%
Morley,
Cantor-Bendixson
not
be
or
tsarini tery
subsets
of
generated need
clam
the
.Since®
is
finitely
nen
7°
the number
a
31.5.
y
of
w 3 hence
THe) Crea
fon
xs
x's w
.
Consequently, adjoined
more
than
a universal
analysis
analysis;
of
i.e.
Sr
isolated
its he
of
«.
domain
Su
coincides
for
each
Qs
DS at rroots, a sy ,
Es By
Clearly
Anduction +1
=D
one O+1
‘Sye Ora week
Q
os)
= Suppose
Su.
Suppose
D sy =
BOUNDS
ON
RANK
Q+1 OID) Su
Dene
Clearly,”
point
in
the
“xX. -e D’su
of
follows in
LoL
D’sy,
there
D°st.
finitely
hope -
is a
by inclusion.
#e
an
nou
Then
isolated
split
D’su
be the
Dy
»
must
that
D Sfx
splits
in The.
be
dt
f,: 7, CU of
of
Y%
5
SLOSS
and
of all
directed
by 20128
D’st,x
D'st,x
DSi,
Te
splits
system
so that
D'sr,
x
gr
does
Solis
Ds
ke
and
there
isolated
x
direct
D’sy, Then
because
does
not
Slngiaal pe Ble Deane
fy: ay ue.) depend
on
the
Note: choice
that of
EC)
Coroltary transcendental model
32.4,
Tf)
and
@qeX(T)
Ice
(OnGro ern
Teiretsaquasi=tovatly. , then
@
has
a prime
extension.
POO
Suppose
extension ta, D> (0)
of
boil
a principal
qc
@
of 52.55
&
is an atomic
over
é Bo
S(T
q
q@) if for every
realizes
Ol)
© Lie
uce
abomMmlCrOVierG@as
Wissatomictovermr..
UNeCn
Ca
Se
acomlicnover
ive Proof.
a principal
suppose
n-type
F(bbys+--sbi where
of
KCy 9-02-90?
ec?
Sea Gis U D8)
generated
» X19+++9X)) ¢ BY
.
The
-,
realizes
by
; completeness
of
PRIME MODEL EXTENSIONS
203
Tost) D(a(b,,.--50,,))
implies
realizes
n-type
a principal
Sa(r Eat generated
by
F(b,-++5D, realize
SGN U D@)
generated
AD
realizes
a
S(T ne
of
Dia (beat tbe) ))
A074 6
U D@)
5 Xqo+++9X))
a principal
by
et
m-type
of
G(Yy +++ 5Yp)
a principal
generated
(=
.
then
n-type
of
by
(By) fre GB) ICs poe sar) & F(Y}9+++9Vn > X1o+-+9%,)]. Theorem
of
S@
Let
are
&
(Seto
be
32.6.
dense
in
a prime
enough
isolated
for every
model
to
&
woineCun
find
proof of 32.2,
Thus
ne realizes
atomic
ey an
over
Gi}
extension
Si
prime
model
extension
#=
oy pend.
isolated be
he
atomic
model
AHA
be
an
the
Bie
S@
the
points
@eXx(T)
.
of
@.
Then
Over.
6s)
tums
FOMLCmONeC Vane Cae
PLOOt.
Let
Suppose
beiy = be(d of
every
of
constructed
U {5, | B< yy)
point
since
extension
SB. member
@.
in the
3 oy
a
» where
5) : Feay of
fea
ad
is is
6
204
SATURATED
of the form
t(d,)
language
T U Be
of
dncuctwen
on
~6)
for some term
that)
“is atomic
Seah
Cues)
2COMa. Ce OViEI
=... ,
and
Cnr
G5
als
meee
©
(SUDDOSC
Prooi ) Suppose
Canis (C 26103)
anGUcTION HOM
on
sas tixed
Gacc1on, On!
mm
Let
5(
Gta qeé
GG
Clearly ates
=the
F(x,y)
q-,
5.end-an
“Ga.
eon
36c8 et
is)
proposition
proposition:
to 52.5.
*)
and
is proved
proceeds,
be the
l-type
"e q 2)
To
by
principal
\ H(x,y)
the
language
ee where “Tally
;
of
HiGcny)s 7s
and
ais
PRIME MODEL EXTENSIONS TU
205
DBL F(bsy)
Let
Suppose
C
be
@ =U
> H(y)
a prime
said
Morley
Of
be
a Morley
prime
52255
1S
over
Vis Morley
generated
prime
of
SC,
fer
extension
The
prime
@.
Go =a,
¢,(c,) » and
1-type
@).
of
c, SUV CLE, CAS
of
@
model
(or
extension
Morley:
PrOpOs itLOn CG
extension
{c, | 8 rank
keyes (0) =.
Thus
degree
in
rank
sor
O
by
sequence
induction
lis
Po =
ei
a.
3.
Assume
on
fa, .8
qé¢&
S@
in
minimal such
of
tnat:
“SG
Et
q
be
the
T
1
cate-
of
the
isa deg
image
deleting of
the of
all
T.
A
pe
S,T (= SZ)
pr=
1. «
ACFo
has
generator.
none
of
be a sequence
which
ea finitely
occur
in
of individual
the
language
cenerated=extension
of
of st
eS
TAU {P(c) osc) for
define
language
and)"
fe, | i
implies
member
G(x) one
. -G(xa))&
of
ise the
F(x)r,
of
Goa, eb) formulas
THE
BALDWIN-LACHLAN
must
have
Since.
only
tnere:
THEOREM
finitely
issonly
p , such’that
were
integer
an
many
one,’
namely
n
ete realizations
'q
ef positive:
F(x) such
€
q.
that
in
rank,
Suppose
for
@,
every
there
aeA,
either
G(5c,78), had
at
G(x) mo
most’
or
G(x)
isjan but)
ave
not
that
the
existence
ability
tins
O10.
FACTR
CNA.
write
Avssuch.
of
4AM
.«°
Then
since
4-7
2"
such
“(either that
either
at most
Ltetotiows
G(x, y) & F(x)
for
each
n
there
that)
H(xja)
has
at
Jeast
many,
realizations:
in
%@s.
of
H(x,y)
S&
to en,
n*, Note
is a consequence
"approximate"
abalitys
the
derivedat
split-
roms
cine
1G H(x)]
constants
occurring
in
, H(x)
2g
SATURATED
belong,
tone
tru
(c) as
in
in
(b)
, the
aie
G, (x)
individual
«Belong
individual
THEORY
3
Cpe cee
W(x)"
some
A USB:
MODEL
to
constants
8T"U"
constant
has the
same
occurring
tejrU Me UPB, from
form
Cos
{ce} UC
rand
occurs
in
H(x) Let SEEING
A
be
the
ee) imemiloenes
NUS CerCC) Lemma
that
set
ra
all
Henkin
JN WU) IB WU @
.
axioms
Wwe
Ix
ae
SUT Cus c.f] Cat ene}
39.7.
is
(See P(ax))
ware
of
Suppose
infinite
T
is
whenever
“hasta Vaughtian
a theory
@E
T ~)
paiesiff
such
‘Then
oT Uekeeis
consistent.
PLoOOn,
Veuentien
parr
(Seay ac. value assume
Thus pas,
of C.
Deen
€
for
Ln —=c
suppose
By
ce
been
evaluated.
axiom Bo
aLat
ێ A
to UB.
nislee:
for
all
to
faa ;
The
in
1ciou
i.
occurring
)c
Cua
on Re
evaluated
= Choose:
attached CHa
B, - B
constant
ESS)
Evaluate
by induction
berany has
Bo
(Tl, F(x) )° .°
hj every individual
Henkin Cc hi
“First
Let ii
chee jg lon alae
is
ag a model
\e
Now
So; that
LS
m*
an
a nonprincipal
and
Choose
an
is
ois ta sninimal
x
4
there
ap
OMe
al xP
(Caution: of
x
whose
does p-th
not
denote
power
must
the
usual
p-th
root
x
3 the
last
axiom
be
HH
artificially
without
Show
it
DF,
DF,
theory
of
Peristic It
fixes
would
has a model
function
lack
0p (DcF.,) ) 5, is
not
known
xP
a model
completion
differentially
of characteristic prime
the
closed
Dx ea Ou
completion. )
(known as the
fields
of
charac-
(Cf, 41.2) and, 41.32) if
every
differential
p (as defined
differentially
when
closed
above)
extension.
has
field
a
41,
The
Morley's
apprLied
rank
directly
closed
fields
because
40.2
structure
to
machinery
the
theory
of
implies
ssendeeh
4. eae
DCF,
can
be
differentially
0
(DCF))
is complete
;
and
sub-
41.1
(L. Blum).
polynomial
“ais 2
hy €
Seneric
S@
be
the
Suppose
@|= DF os
over
of
@
solution
1-type
of
(xx)
realized
by
neers
(i) most.
degree
complete.
.n
OVEN. Df
and
is a differential
order
Closure
of characteristic
Proposition
f(x)
Differential
The Cantor-Bendixson
rank of
q,
is at
7:n>.
(19) fore oN
slf lin
wand
ed
.cohtainsrsolutions tajeOm)
end
505
in)
Rex).
or
Dxe= is
Dos
j
306
SATURATED
then
the
Cantor-Bendixson
Proof. AnHoOuUct LON
TON
(1). nm
‘sue
over
Both:
b
and
andeuh(X)
all
y=) O of
p
so
is
by
Bendixson
formula whose Tess
d@ .
rank
h(x)
only
(ii)
The
Ona
Aen
a
ie
ene
andependent
=m
orm
.
is
are
“Then
by:
3; ete
ee
2,
Then
rv,
n
5)
CtChnoose
n
,;
rank
since
subset
implies
1-type
some
than
and-.1-types
Dx =0
om
of
the Cantor-
an open
the
lov eri
of
less
at most”
of 40.1
are
everyereal—
solution
independent
be
4) p*4
order
or
h(x)
p -€7S@7
Cantor-Bendixson
defines
specified
i?
of
of—
h(x)
Consequently
proof
algebraically
pte
.
of
are
by
of
pm
generic
the
n
members n.,
er
of order
solution
“Suppose,
the
= 0
than
lagSha Wh
h(x)
solutions
polynomial
than
completely
by:
generic
is
is an
a generic
induction
less
hy
(ii) are proved
there
belongs*to)
p
differential
and
is
over
of
THEORY
f
By 40.1(14)
feomorpniie.
ization
(4) and
Ax.
that
@
rank
MODEL
; 0-4
over
@
.
G
is
By Deex each
specified
algebraically
a, € @
is completely
S@
rank
-For
completely
are
of
of
that
the
so, that
specified
by:
THE
DIFFERENTIAL
CLOSURE
ie;
(x-a,,) = 03
(x-a,),
are
algebraically
an
has
-1
il-type
the of
induction n-1
.
dy
open
and
nn ,
Since
the
of
&
1-types
of
rank
includes
TENA
IE)
It
the
when
@|
is
widely
Lemma
as 3
only
than
believed,
the by
a
O
is rank
are
n=,
whose
and
but
yet
becomes
an
or
l1-types
not
of
defines
members
many
of 41.1(i)
28.3
so
of
noe
infinitely
inequality DCF
rank
formuila
less
Dae
By
Cantor-Bendixson
whose
membership “idle
the
@. rank
of
Cantor-Bendixson
subset
that
solution
Consequently
iS)
over
Cantor-Bendixson
generic
the
D(x~a,)>.--2D nae (xX-a, “
independent
same
a
SOG
of
proved,
an
equality
O
41.2
transcendental.
(L. Blum). The
DCF)
Morley
rank
is totally of
DCF
O
is
wt
Proof.
w-stable.
It
follows
So by 31.6
from
DCF
O
40.1
is
that
DCF
totally
is
transcen-
dental.
model
new
CG
ise
of,
DCFo
a
wiealweresel,
Geraci
cloetia,
atone4
oupyalo.4..
Gaye)
elt eurrices,
308
SATURATED
by 5135.
to
Of
ts)
16a,
that
wis
the
ak»
ae,
wide
ib follows:
THEORY rank
trom 42 VG)
rank of
by 41.2(1),.40,1(4)-
S@
is at least
and 40,1(144),,
it is
@)
Let
@,8
characteristic
and
O
be
differential
(6
Lise
@
if
extension
of
can
completed
as
entially
C@
.
closed be
the Cantor-Bendixson
Cantor-Bendixson
then
Mar
show
MODEL
prime
the
shown
fields
of
differentially,
following
whenever
C
diagram
is
differ-
closed
a x
a
Theorem 41.3 (L. Blum). fitéeldor
characteristics
entially
closed
field
By 41.2
Theorem
41.4,
over
ad.
prime
ditier—
and 32.4. Let
characteristic
ferentially
hasea
extension.
Proof.
of
Or
Every differential
closed
@ O
be a differential .
extensions
Any
two
of
@
prime are
dif-
isomorphic
THE
DIFFERENTIAL
PYOOUY For
s.Dyae ee
each
istic
0,
tially
closed
@
BndeD Oe. field
@’ be the unique
@
of
character-
prime
differen-
afforded
by 41.3
extension
of
@
is called
the
differential
closure
@.. SUDDOSS
92)
POmMCAC MEN i>
On
a
nomial
equalities
over
@
finite
with
(1)
toe
aS
AUOM1G
system
and
the
S
of
following
OMe
enn
»
(7a
in
eit
there
differential
inequalities
(2)>-Ald Oca
Ge
manG
exists
n
poly-
variables
properties:
is a solution
‘solutions+ef
‘S
are
of
Ss.
Lsomorphic
C/aee
Theorem
41.5.
Proof,
By 4152;
It follows alcebvarcover
@
309
differential
let
and 41.4. of
CLOSURE
Af 211
Gest
“Vor
polynomial
over
ed
£(x)
is atomic 32)2rend
from 41.5 @-.
generic
isomorphic
@
Call
solutions @.
, and
&
jf
32.6.
@. "1a
that
-@
£(x%)
“arreducible
of
is
over
is differential
f(x)
a normal
over
over
@.
extension
for
each
differential
irreducible
over
@ , eltther
are of
all
50
SATURATED
or none
of the generic
belong to
6...
reducible
over
THEORY
of
f(x)
in
(It can happen that
f(x)
1s 4r-
@
and
solutions
MODEL
has
no
generic
@
solution
in
aa) Theorem extensions and
@
ones
ee
41.6.
of
@.
over
@
Proof. atomie
can
(
is minimal
over
@
@#8.
is
Gd=
9
It
,
open
the
has
shown
able
from into
2
trivial
that
Q
onto
32.9
of maximal
@
w
that
@>
of
is
transforms
style theorem
is
is
no dif-
over
occurs
when
rational
numbers
L.oHarrinston
has
a comput-
a one-one
+ ,
construction radical,
and
minimal
+
Harrington's
for
@
68> aq
attached.
there
functions.
a Henkin basis
i.e.
#6
automorphism
implies
interest
field
derivative
an
if there~is
when
the
to
of
2
such that
known
be normal
2 (= )
computable
finite
&
is
presentation;
combines the
not
case
where
Wit
since C.
@
isomorphism
extended
over
closed
An
be
and
any
and
ferentially
@.
@
Then
‘By 36.1,
over
@
Let
of
map
and
—D
argument
2
with
differential
THE
DIFFERENTIAL
polynomial
of
normal
not
to
@
totally
Exercise
OL
Let
characteristic
Exercise is
41.7.
extension
morphic
Byte
ideals.
Exercise field
CLOSURE
arortrarily
of
over
41.8.
@
.be a differential
O.
@
In
Suppose
@-.
‘Show
@&
is
a
6
sie siso—
@.
Show
DCF, (Exercise 40.5)
transcendental.
41.9.
Nish
Show
finite
S2
rank,
has
isolated
points
42,
The
most
promising
in
this
pook“is
ected
parvuacutar generated
Chest
of
Ux
Ans ow
iery Ti
then
: 0" -.)) been
GCH
total
one
“Asshortnproof
(to obtain
being
THEORY
a nondecreasing
discovered
circumvents
principal of
“is
MODEL
by Rosenthal
some
it via the
ror
ultra-
absoluteness
absoluteness
transcendentality.
of
44%,
[Al]
J.
AX
Several
and
S.
over local 83,
[A2]
J.
References
Kochen,
Diophantine
TieldsvIIT,
problems
Ann.cof Math.
(1966),
437-456. Ax
over
and
S.
local
Math.
Kochen,
fields
(1965
and
Diophantine
I and
1966),
II,
87,
problems
Amer.
Jour.
605-630
and
631-
648, [Bal]
JI.
Baldwin,
University;
[Barl1] K.
J.
ble
scimon
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Barwise,
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Pa,De sThesiss
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Infinitary
Symb.
Log.
logic
and
(1969),
admissi-
34, 226-
2544,
[B11]
L.
Blum,
tute
[Ch1]
GC.)
of
Ph.D.
Thesis,
Technology,
Change
andoH.
Dg
Massachusetts
Insti-
1968. J.
Kelslen,»
Theory
of
SATURATED
Models;
[EL]
YY.
“to
THEORY
appear,
Ershov,
mal
MODEL
Om
normal
the
elementary
fields,
Algebra
theory
or maxi
i Logica
(1965),
31-70.
[G81]
K.
GSdel,
The
Continuum
Consistency
Hypothesis,
of the
Prancetom
Generalized Univ.
Press,
1940.
[Kal]
I.
Kaplansky,
Duke
[Kel]
HJ
Math.
H.
S,
J.
Ae
Kochen,
[Mal]
A.
H.
to
appear.
W.
Marsh,
forkkni inivery
1971.
Math.
(1961),
of Math.
Heelaecinitam..
wuine
theory,
Lachlan,
(1962),
Rhee
to
and
of
elementary
23,
“in the
477-495.
theory
74,
of
221-261.
models
of
a
stable
theories,
appear.
A property
Ph.D.
valuations,
313-321.
Ultraproducts
Ann.
with
Ultraproducts
Indag.
superstable
[La2]
(1942),
Model “Mneory
Keisler,
models, [Lal]
fields
North-Holland,
classes, [Kol]
Jour.
se Kelelen,
Logic,
[Ke2]
Maximal
Thesis,
of
Dartmouth
College,
1966.
[Mol]
M.
Morley,
Jour. [P11]
Re
Symb
Platek,”
The
number
Joe,
of
countable
(2O70)seses
Phy Do. Thesis,
models,
2 hee
‘stanford
University.
SEVERAL
[had
REFERENCES
3.
U.
MeEChw
[Rol]
[Ro2]
[Rosl]
A.
Ritt,
[Sa2]
Le UD
Robinson,
eLosed
Tsvacl
(1959),
A.
Robinson,
BStivo,
Varenna
E.
.councwi
and
methods
of
or
model
Matematico
proof
of
symb.
Loe.,
%~orvappear.
Effective
G.
Sacks,
On
to
appear.
Seott,
the
Logic
and
The Theory
A.
Res.
a
bounds
theorem
of
on Morley
appear.
formulas
[Sel]
bull
differen-
A new
CO
D.
a
1968.
rank, E.
of
Internazionale
Jour.
Sacks,
61-050)
concept
Problems
Rosenthal,
Amex.
1is=128.
Centro
G.
the
fields.
theory,
J.
Alecbrayy
Ce Chom.
On
tilly
models,
[Scl]
Ditterential’
DOC
wnelah,
[Sal]
Dex
with
finite
of Models,
Seidenberg,
differential
An
number
of
denumerably
strings
of
long
quantifiers,
North-Holland,
elimination
algebra,
countable
Univ.
theory
1965.
for
of Calif.
(1956),
31-66. [Shl]
SS;
Shelah,
property Logie,
Stability,
and
the
finite
superstability,
40, appear:
Ann.
cover
of Math.
322
[Sh2]
[Val]
SATURATED S.
Shelah,
Bos
Gonjecture
languages,
to
R.
Denumerable
Vaught,
theories, Press
THEORY
uncountable
appear.
Infinitistic
ooL,
for
MODEL
BOD 521:
models Methods,
of
complete Pergamon
AY
Gases
Ves
ep
epee Cars
Wa
Wy
card
Notation
Index
Oi grere
Co
9 Wo s+.
A
CD COM Coe COR
LO LO bah abba: cise We Le 2 aes
B29)
SATURATED
14 15 aS. L5 eG 16 16 16 16 16 16
(Ex,)
uy
Wis? hay
ag
(x; )
18
TT
el
MW
ae il
A
23 el 25 30 30
2>
AN 53
MODEL
THEORY
NOTATION
INDEX
7
B25 >
a” (Xx)
ee
ST
72
B(c)
89
ACFy
106
n(T)
118
2
126
W aa
144.
126
145
156 aksye 158
159 io 59
FC tx, tu)
159
x(T) a(b)
165
162
165 166
sf
LOT
ax
169
170
326
SATURATED
D7
:
slings
D
LTD
DFf
a
DF
175
rank deg
es
x x
LG |
179
pbk
182
Des
187
v(Y)
188
V(¥y>--+27,)
188
hen
191
dr
LOST
ph
256
g
261
SB
261
Aidan Bie
"26%
p-dim
264
dim
292
@
Dx 5 DFy
295
ond i (x)
296
DCF,
298
DCP
304
a
BIOS)
dl
510
MODEL
THEORY
45,
Subject
Index
absoluteness admits
direct
admits
filtrations
admits
filtrations
admits
inverse
158
limits
Ox with
amalgamation
ale(f
159
limits
algebraic
over
dq
210
aleebraic
over
x
262
algebraically
closed
210
algebraically
independent
262
antisymmetric
over
227 243
A-rank
202
atomic
extension
atomic
formula
atomic
model
120
atomic
over
202
Al
Dek
328
SATURATED
MODEL
a,
atom
182
autonomous
cardinality Canc Ga MeaiaiGiys
Oi
ie
167,
Cantor-Bendixson
derivative
169
Cantor-Bendixson
rank
170
category
156
complete
30
connected
227,
over
consequence
25
relation
26
consistent constant
Li
term
contravariant countable
direct
countability
definable
he,
functor system
69
proviso
227
relation
aA)
degree
degree dense
184
of
Morley
sequence
184
subcategory
diagram
completed
diagram
of
230
as
shown
55
@
differential
algebraic
2
differential
closure
509
THEORY
SUBJECT
INDEX
differential
Dag transcendental
differentially
dimension
closed
of
aot
298
field
@
292
direct
limit
47
direct
system
46
direct
system
directed
(in a category)
46
set
distinguished downward
LD
element
2
55
Skolem-L&wenheim
elementary
extension
24
elementary
monomorphism
2D)
elementary
partial
elementary
substructure
a4
equivalent
21
elementarily
elimination
of
automorphism
110
54
quantifiers
endomorphism
ils,
finitary
26
finite
character
basis
197
property
188
finitely
generated
finitely
generated
over
finitely
generated
extension
first
order
formula
language
188
Y of
261 16 17
330
SATURATED
free variable full
160
function
i
:
symbol
generator
generic
of
12
16
principal
n-type
97
solution
296
homogeneous
LO
identity
LEip
map
immediately inclusion
extendible
110
map
aL
indiscernible indiscernible
individual
yey over
228
constant
inverse
limit
inverse
system
16 .
159
158
irreducible
309
isolated
169
point
isomorphic
THEORY
18
subcategory
function
MODEL
over
@
165,198
isomorphism
15
K-categorical
99
K-dense
ate
K-saturated
76
SUBJECT
INDEX
104
K-stable
iialiaiae
ongielabarsil
Limit
point
169
map
156
minimal
generator
26.)
minimal
model
124
minimal
model
minimal
over
extension
209 209
model
complete
2D
model
completion
57
14.
monomorphism Morley
derivative
Morley
prime
Morley
rank
Morley
sequence
model of
AEE extension
205, 191.
T
250
Morleyization
256
naming
158
relation
nonprincipal
minimal
generator
276
normal
extension
206,309
normal
over
206
n-place
function
2
n-place
relation
arab
O52
SATURATED
n-type
‘
THEORY
Ve
object omit
MODEL
156
an n-type
;
order
indiscernible
order
indiscernible
.
96 ONY
over
228
ordinal
8
ordinal
recursive
in
T
195
w-consistent
145
w-logic
144
w-stable
104
w-structure
144.
p-base
263
p-dimension
264
p(mod
)-dimension
Th
p(mod
q)-dimension
ear,
preserves
limits
prime
model
prime
model
prime,
120
extension
proper
elementary
primitive
symbol
principal
finitely
Of principal
160
aT n-type
198
extension
266
16 generated
extension
2(o o7
SUBJECT
LD)
INDEX
quantifierless
formula
quasi-totally
rank
(Morley)
rank
Oi
real
closed
18
transcendental
194
177
Cl 9 cite
286
22):
92
field
ex
realize realize
pe
realize
qe
srt
the
same
n-types
114 aa
relation
relation
16
symbol
Mes te tl
Ono
arr
to
K
128 136
R-saturated
satisfy
21
saturated
76
sentence
18
Similarity simple
extension
singular
Skolem
type
89
cardinal
hull
Skolemization special
ual
model
splits structure
5D NS 108 188 a:
3544
SATURATED MODEL THEORY
subcategory
160
substructure
AS
substructure
complete
substructure
proviso
successor
63 228
ordinal
8
term
16
theory
30
totally
transcendental
functor
180
totally
transcendental
theory
193
two-cardinal
type
126
underlie
@
Z
universal
closure
universal
domain
universal
existential
universal
formula
universal
property
universal
structure
universal
theory
ew.
188
theory
4e
Yd of
direct
limit
158 letles
dy
universe
upward
19
aii
Skolem-L&wenheim
valid Vaughtian
Ah ab
pair-for
T
Biss
SUBJECT
INDEX
Vaughtian pair for weakly
Z-group
saturated
22D
(T, F(x))
282 119
g4
Saturated Model Theory
intended for the graduate level logic course, this book gives an account of the principal constructions of pure, first order model theory. It discusses such topics as: saturated model completeness
criteria
for elimination
of quantifiers
and
model
rank and degree of element types two-cardinal theorems
existence and uniqueness of prime model extensions of substructures of models of totally transcendental theories homogeneity of models of w,-categorical theories
Gerald E. Sacks
received his Ph.D. in mathematics from Cornell University in 1961. In addition to model theory, the author’s main interests are generalized recursion theory and set theory. Professor Sacks is the author of more than 30 research papers as well as the book, ““Degrees of Unsolvability.”
W. A. BENJAMIN, INC. Advanced Book Program Reading, Massachusetts 01867
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