Saturated Model Theory [1 ed.] 0805383808, 9780805383805

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UNIVERSITY

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ELMER E. RASMUSON LIBRARY

Digitized by the Internet Archive In 2021 with funding from Kahle/Austin Foundation

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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

tracer

ofa

Barwise,

sets,

Pa,De sThesiss

Jour.

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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