The

com

ple

xnum

bers

and

com

ple

xexponentiation

Why

Infinitary

Logic

isnecess

ary

!

John

T.Bald

win

Depart

ment

ofM

ath

em

atics,

Sta

tist

ics

and

Com

pute

r

Scie

nce

Univ

ers

ity

ofIllinois

at

Chic

ago

ww

w.m

ath

.uic

.edu/

jbald

win

/Bogple

05.p

df

August

21,2005

TO

PIC

S

1.

Intr

oduction

-lo

gic

and

math

em

atics

2.

Modelth

eory

ofth

ecom

ple

xfield

3.

Quasim

inim

alexcellence

4.

Psu

eudoexponentiation

5.

Ispsu

edoexponentiation

genuin

e?

6.

Fro

mm

odelth

eory

tonum

ber

theory

SELF

CO

NSCIO

US

MAT

HEM

AT

ICS

•A

signatu

reL

isa

collection

of

rela

tion

and

function

sym

bols.

•A

stru

ctu

refo

rth

at

signatu

re(L

-str

uctu

re)

isa

set

with

an

inte

rpre

tation

for

each

ofth

ose

sym

bols.

•T

he

firs

tord

erla

nguage

(Lω

,ω)ass

ocia

ted

with

Lis

the

least

set

offo

rmula

sconta

inin

gth

eato

mic

L-form

ula

s

and

clo

sed

underfinite

Boole

an

opera

tions

and

quan-

tification

over

finitely

many

indiv

iduals.

•T

he

Lω1,ω

language

ass

ocia

ted

with

Lis

the

least

setof

form

ula

sconta

inin

gth

eato

mic

L-form

ula

sand

clo

sed

under

counta

ble

Boole

an

opera

tions

and

quantifica-

tion

over

finitely

many

indiv

iduals.

ModelT

heory

and

Num

ber

Theory

Priorto

1980:

Use

ofbasic

modelth

eore

tic

notions:

com

-

pactn

ess

,quantifier

elim

ination:

Ax-K

ochen-E

rshov

Late

r:In

cre

asing

use

of

sophisticate

dfirs

tord

er

model

theory

:st

ability

theory

;Shela

h’s

ort

hogonality

calc

ulu

s;

o-m

inim

ality

:Hru

shovsk

i’s

pro

of

of

geom

etr

icM

ord

ell-

Lang.

ALG

EBRAIC

ALLY

CLO

SED

FIE

LD

S

Fundam

enta

lst

ructu

reofAlg

ebra

icG

eom

etr

y

Axio

ms

forfield

soffixed

chara

cte

rist

icand

(∀a1,.

..a

n)(∃y

)Σa

iyi=

0

MO

DEL

THEO

RET

ICFUND

AM

ENTAL

The

theory

Tp

of

alg

ebra

ically

clo

sed

field

sof

fixed

char-

acte

rist

ichas

exactly

one

modelin

each

uncounta

ble

car-

din

ality

.(S

tein

itz)

That

is,

Tp

iscate

goricalin

each

uncounta

ble

card

inality

Cate

goricalStr

uctu

res

I.(C

,=)

IIa.

(C,+

,=)

vecto

rsp

aces

over

Q.

IIb.

(C,×

,=)

MO

RLEY’S

THEO

REM

Theore

m1

Ifa

counta

ble

firs

tord

erth

eory

iscate

gorical

inone

uncounta

ble

card

inalit

iscate

goricalin

all

uncount-

able

card

inals.

Morley

Rank

‘...w

hat

makes

his

paper

sem

inalare

its

new

techniq

ues,

whic

hin

volv

ea

syst

em

atic

study

ofSto

ne

spacesofBoole

an

alg

ebra

sofdefinable

sets

,called

type

spaces.

Forth

eth

eo-

ries

underconsidera

tion,th

ese

type

spaces

adm

ita

Canto

r

Bendix

son

analy

sis,

yie

ldin

gth

ekey

notionsofM

orley

rank

and

ω-s

tability.’

Citation

award

ing

Mic

haelM

orley

the

2003

Ste

ele

prize

for

sem

inalpaper.

DECID

ABIL

ITY

Coro

llary

2T

he

setofse

nte

ncestr

ue

inalg

ebra

icallyclo

sed

field

sofa

fixed

chara

cte

rist

icis

decid

able

.

LIN

DST

RO

MS’S

LIT

TLE

THEO

REM

Theore

m3

IfT

is∀∃

-axio

matizable

and

cate

gorical

in

som

ein

finite

card

inality

then

Tis

modelcom

ple

te.

CO

RO

LLARIE

S

Coro

llary

4T

he

theory

of

alg

ebra

ically

clo

sed

field

sad-

mits

elim

ination

ofquantifiers

.

Coro

llary

5(Tars

ki,

Chevalley)

The

pro

jection

ofa

con-

stru

ctible

set

isconst

ructible

.

ST

RO

NG

LY

MIN

IMAL

I

Definitio

n6

Mis

stro

ngly

min

imalif

every

firs

tord

erde-

finable

subse

tofany

ele

menta

ryexte

nsion

M′ o

fM

isfinite

or

cofinite.

Every

stro

ngly

min

imalse

tis

cate

goricalin

all

uncounta

ble

powers

.

The

com

ple

xfield

isst

rongly

min

imal.

GO

DEL

PHENO

MENA

Itfo

llow

sfrom

Godels

work

inth

e30’s

that:

1.

The

collection

ofse

nte

nces

true

in(Z

,+,·,

0,1

)is

un-

decid

able

.

2.

There

are

definable

subse

tsof

(Z,+

,·,0,1

)w

hic

hre

-

quire

arb

itra

rily

many

altern

ationsofquantifiers

.(W

ild)

CO

MPLEX

EXPO

NENT

IAT

ION

Consider

the

stru

ctu

re(C

,+,·,

ex,0

,1).

Itis

Godelian.

The

inte

gers

are

defined

as{a

:ea

=1}.

The

firs

tord

er

theory

isundecid

able

and

‘wild’.

ZIL

BER’S

INSIG

HT

Maybe

Zis

the

sourc

eofall

the

diffi

culty.

Fix

Zby

addin

g

the

axio

m:

(∀x)e

x=

1→

∨ n∈Z

x=

2nπ.

REPRIS

E

The

firs

tord

er

theory

of

the

com

ple

xfield

iscate

gorical

and

adm

its

quantifier

elim

ination.

Modelth

eore

tic

appro

aches

base

don

Shela

h’s

theory

of

ort

hogonality

have

led

toadvances

such

as

Hru

shovsk

i’s

pro

ofofth

egeom

etr

icM

ord

ell-L

ang

conje

ctu

re.

The

firs

tord

erth

eory

ofcom

ple

xexponentiation

ism

odel

theore

tically

intr

acta

ble

;

We

now

explo

rein

finitary

appro

aches.

GEO

MET

RIE

S

Definitio

n.

Apre

geom

etr

yis

ase

tG

togeth

er

with

a

dependence

rela

tion

cl:P(

G)→P(

G)

satisf

yin

gth

efo

llow

ing

axio

ms.

A1.

cl(X

)=

⋃ {cl(X

′ ):

X′ ⊆

fin

X}

A2.

X⊆

cl(X

)

A3.

cl(c

l(X

))=

cl(X

)

A4.

Ifa∈

cl(X

b)and

a6∈

cl(X

),th

en

b∈

cl(X

a).

Ifpoin

tsare

clo

sed

the

stru

ctu

reis

called

ageom

etr

y.

CLASSSIF

YIN

GG

EO

MET

RIE

S

Geom

etr

ies

are

cla

ssifi

ed

as:

triv

ial,

locally

modula

r,non-

locally

modula

r.

Zilber

had

conje

ctu

red

that

each

non-locally

modula

rge-

om

etr

yofa

stro

ngly

min

imalse

twas

‘ess

entially’th

ege-

om

etr

yofan

alg

ebra

ically

clo

sed

field

.

We

willst

udy

Hru

shovsk

i’s

const

ruction

whic

hgave

coun-

tere

xam

ple

sto

this

conje

ctu

reand

maybe

much

more

.

ST

RO

NG

LY

MIN

IMAL

II

a∈

acl(B)

ifφ(a

,b)

and

φ(x

,b)

has

only

finitely

many

solu

-

tions.

Acom

ple

teth

eory

Tis

stro

ngly

min

imalif

and

only

ifit

has

infinite

models

and

1.

alg

ebra

icclo

sure

induces

apre

geom

etr

yon

models

of

T;

2.

any

bijection

betw

een

acl-b

ase

sfo

rm

odels

of

Texte

nds

toan

isom

orp

hism

ofth

em

odels

QUASIM

INIM

ALIT

YI

TrialD

efinitio

nM

is‘q

uasim

inim

al’

ifevery

firs

tord

er

(Lω1,ω

?)

definable

subse

tof

Mis

counta

ble

or

cocount-

able

.

a∈

acl′ (

X)

ifth

ere

isa

firs

tord

erfo

rmula

with

counta

bly

many

solu

tions

over

Xw

hic

his

satisfi

ed

by

a.

Exerc

ise

?If

fta

kes

Xto

Yis

an

ele

menta

ryisom

orp

hism

,

fexte

nds

toan

ele

menta

ryisom

orp

hism

from

acl′ (

X)

to

acl′ (

Y).

QUASIM

INIM

ALIT

YII

QUASIM

INIM

AL

EXCELLENCE

Acla

ss(K

,cl)

isquasim

inim

alexcellentif

itadm

itsa

com

-

bin

ato

rialgeom

etr

yw

hic

hsa

tisfi

es

on

each

M∈

K

there

isa

uniq

ue

type

ofa

basis,

ate

chnic

alhom

ogeneity

conditio

n:

ℵ 0-h

om

ogeneity

over∅

and

over

models.

and

the

‘excellence

conditio

n’w

hic

hfo

llow

s.

Inth

efo

llow

ing

definitio

nit

isess

entialth

at⊂

be

under-

stood

as

pro

per

subse

t.

Definitio

n7

1.

Forany

Y,cl−

(Y)=

⋃ X⊂Y

cl(

X).

2.

We

call

C(t

he

unio

nof)

an

n-d

imensionalcl-in

dependent

syst

em

ifC

=cl−

(Z)

and

Zis

an

independent

set

of

card

inality

n.

n-A

MALG

AM

AT

ION

M{1

,3}

// X

M{1}

//

55 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j jM{1

,2}

44

M{3}

OO

// M{2

,3}

OO

M∅

OO

55 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j// M{2}

OO

44 j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j j

4-e

xcellence

M{0

,2,3}

// X

M{0

,2}

ff M MM M

M MM M

M M

// M{0

,1,2}

77

M{2

,3}

//66 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

M{1

,2,3}

55

M{0}

//

OO

xxqqqq

qqqq

qqM{0

,1}

OO

''NNNNNNNNNNN

M{2}

eeJ JJ J

J JJ J

J

55 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l// M

{1,2}

55 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k

88 q q q q q q q q q q

M{0

,3}

OO

// M{0

,1,3}

OO

M∅OO

//

yy tttt

tttt

tt

55 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l lM{1}

55 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k

&&MMMMMMMMMM

OO

M{3}

OO

66 l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l// M

{1,3}

XX

55 k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k

P(n)

powers

et

of

n,le

tp−

(n):=

P(n)−{n}.

Let

say

tpqf(

X/C)

isdefined

over

the

finite

C0

conta

ined

inC

ifit

isdete

rmin

ed

by

its

rest

riction

toC

0.

[Quasim

inim

alExcellence]Let

G⊆

H∈

Kw

ith

Gem

pty

or

inK

.Suppose

Z⊂

H−

Gis

an

n-d

imensionalin

dependent

syst

em

,C

=cl−

(Z),

and

Xis

afinite

subse

tofcl(

Z).

Then

there

isa

finite

C0

conta

ined

inC

such

that

tpqf(

X/C)

is

defined

over

C0.

EXCELLENCE

IMPLIE

SCAT

EG

ORIC

ITY

Excellence

implies

by

adirect

lim

itarg

um

ent:

Lem

ma

8An

isom

orp

hism

betw

een

independent

Xand

Y

exte

nds

toan

isom

orp

hism

ofcl(

X)

and

cl(

Y).

This

giv

es

cate

goricity

inall

uncounta

ble

powers

ifth

e

clo

sure

ofeach

finite

set

iscounta

ble

.

CAT

EG

ORIC

ITY

Theore

mSuppose

the

quasim

inim

alexcellent

(I-IV)

cla

ss

Kis

axio

matized

by

ase

nte

nce

Σof

Lω1,ω

,and

the

rela

-

tions

y∈

cl(

x1,.

..x

n)

are

Lω1,ω

-definable

.

Then,fo

rany

infinite

κth

ere

isa

uniq

ue

stru

ctu

rein

Kof

card

inality

κw

hic

hsa

tisfi

esth

ecounta

ble

clo

sure

pro

pert

y.

NO

TE

BENE:T

he

cate

goricalcla

sscould

be

axio

matized

inL

ω1,ω

(Q).

But,

the

cate

goricity

resu

ltdoes

not

depend

on

any

such

axio

matization.

DIM

ENSIO

NFUNCT

IONS

Let

K0

be

acla

ssofsu

bst

uctu

res

clo

sed

undersu

bm

odel.

Apre

dim

ension

isa

function

δm

appin

gfinite

subse

tsof

mem

bers

of

Kin

toth

ein

tegers

such

that:

δ(X

Y)≤

δ(X

)+

δ(Y

)−

δ(X

∩Y

).

For

each

N∈

Kand

finite

X⊆

N,th

edim

ension

of

Xin

Nis

dN(X

)=

min{δ

(X′ )

:X⊆

X′ ⊆

ωN}.

The

dim

ension

function

d:{X

:X⊆ f

inG}→

N

satisfi

es

the

axio

ms:

D1.

d(X

Y)+

d(X

∩Y

)≤

d(X

)+

d(Y

)

D2.

X⊆

Y⇒

d(X

)≤

d(Y

).

THE

GEO

MET

RY

Definitio

n9

For

A,b

conta

ined

M,

b∈

cl(

A)

ifdM

(bA)=

dM

(A).

Natu

rally

we

can

exte

nd

toclo

sure

sofin

finite

sets

by

im-

posing

finite

chara

cte

r.If

dsa

tsfies:

D3

d(X

)≤|X|.

we

get

afu

llcom

bin

ato

rial(p

re)-

geom

etr

yw

ith

exchange.

ZIL

BER’S

PRO

GRAM

FO

R(C

,+,·,

exp)

Goal:

Realize

(C,+

,·,exp)

as

am

odel

of

an

Lω1,ω

(Q)-

sente

nce

discovere

dby

the

Hru

shovsk

iconst

ruction.

A.Expand

(C,+

,·)by

aunary

function

whic

hbehaves

like

exponentiation

using

aHru

shovsk

i-like

dim

ension

function.

Pro

ve

som

eL

ω1,ω

(Q)-

sente

nce

Σis

cate

gorical

and

has

quantifier

elim

ination.

B.Pro

ve

(C,+

,·,exp)

isa

modelofth

ese

nte

nce

Σfo

und

inO

bje

ctive

A.

THE

AXIO

MS

L={+

,·,E

,0,1}

(K,+

,·,E

)|=

Σif

Kis

an

alg

ebra

ically

clo

sed

field

ofchara

cte

rist

ic0.

Eis

ahom

om

orp

hism

from

(K,+

)onto

(Kx,·)

and

there

isν∈

Ktr

ansc

endenta

loverQ

with

kerE

=νZ.

Eis

apse

udo-e

xponential

Kis

stro

ngly

exponentially

alg

ebra

ically

clo

sed.

PSEUD

O-E

XPO

NENT

IAT

ION

Eis

apseudo-e

xponentialif

forany

nlinearly

independent

ele

ments

overQ,{z

1,.

..z n}

td(z

1,.

..zn,E

(z1),

...E

(zn))≥

n.

Schanuelconje

ctu

red

thattr

ue

exponentiation

satisfi

esth

is

equation.

ABST

RACT

SCHANUEL

Fora

finite

subse

tX

ofan

alg

ebra

ically

clo

sed

field

kw

ith

apart

ialexponentialfu

nction.

Let

δ(X

)=

td(X

∪E(X

))−

ld(X

).

Apply

the

Hru

shovsk

iconst

ruction

toth

ecollection

of(k

,E)

with

δ(X

)≥

0fo

rall

finite

X⊂

k.

That

is,

those

whic

h

satisf

yth

eabst

ract

Schanuelconditio

n.

The

resu

ltis

aquasim

inim

alexcellent

cla

ss.

ALG

EBRA

FO

RO

BJECT

IVE

A

Conje

ctu

reon

inte

rsection

ofto

ri

Giv

en

avariety

W⊆Cn

+k

defined

overQ,

and

acose

t

T⊆

(C∗ )

nofa

toru

s.

An

infinite

irre

ducib

lecom

ponent

Sof

W(b

)∩

Tis

aty

picalif

dfS−

dim

T>

dfW

(b)−

n.

Theore

m10

There

isa

finite

set

Aofnonzero

ele

ments

ofZn

,so

that

ifS

isan

aty

pic

alcom

ponent

of

W∩

Tth

en

for

som

em∈

Aand

som

eγ

fromC,

every

ele

ment

of

S

satisfi

es

xm

=γ.

Using

the

true

CIT

,th

eabst

ract

Schanuelconditio

nbe-

com

es

afirs

tord

er

pro

pert

y.

Repla

cin

gC

by

ase

mia

lgebra

icvariety

giv

es

the

conje

c-

ture

dfu

llCIT

,w

hic

him

plies

Manin

-Mum

ford

and

more

.

CHO

OSIN

GRO

OT

S

Definitio

n11

Am

ultip

licatively

clo

sed

div

isib

lesu

bgro

up

ass

ocia

ted

with

a∈C∗

,is

achoice

ofa

multip

licative

sub-

gro

up

isom

orp

hic

toQ

conta

inin

ga

.

Definitio

n12

b1 m 1∈

bQ 1,.

..b

1 m `∈

bQ `⊂C∗

,dete

rmin

eth

eiso-

morp

hism

type

of

bQ 1,.

..bQ `⊂C∗

over

Fif

giv

en

subgro

ups

ofth

efo

rmcQ 1

,...

cQ `⊂C∗

and

φm

such

that

φm

:F(b

1 m 1..

.b1 m `)→

F(c

1 m 1..

.c1 m `)

isa

field

isom

orp

hism

itexte

nds

to

φ∞

:F(bQ 1

,...

bQ `)→

F(cQ 1

,...

cQ `).

Theore

m13

(th

um

bta

ck

lem

ma)

For

any

b 1,.

..b `⊂C∗

,th

ere

exists

an

msu

ch

that

b1 m 1∈

bQ 1,.

..b

1 m `∈

bQ `⊂C∗

,dete

rmin

eth

eisom

orp

hism

type

of

bQ 1,.

..bQ `⊂C∗

over

F.

The

Thum

bta

ck

Lem

ma

implies

that

Ksa

tisfi

es

the

ho-

mogeneity

conditio

ns

and

‘excellence’.

Fcan

be

the

acfof

Qora

num

berfield

,oran

independent

syst

em

of

alg

ebra

ically

clo

sed

field

s.IfC

isre

pla

ced

by

a

sem

i-abelian

variety

,th

ese

diff

ere

nces

matt

er.

TO

WARD

SEXIS

TENT

IAL

CLO

SURE

Giv

en

V⊆

K2n

we

mig

ht

want

tofind

z 1,.

..,z

nw

ith

(z1,.

..z n

,E(z

1),

...E

(zn))∈

V.

Schanuel’s

conje

ctu

repre

vents

this

for

‘sm

all’

varieties.

We

want

tosa

yth

isis

the

only

obst

ruction.

NO

RM

AL

VARIE

TY

Let

Gn(F

)=

Fn×

(F∗ )

n.

IfM

isa

k×

nin

teger

matr

ix,

[M]:G

n(F

)→

Gn(F

)is

the

hom

om

orp

hism

takin

g〈a

,b〉t

o

〈Ma,b

M〉.

Act

additiv

ely

on

firs

tn

coord

inate

s,m

ultip

lica-

tively

on

the

last

n.

VM

isim

age

of

Vunder

M.

Vis

norm

alif

for

any

rank

km

atr

ixM

,dim

VM≥

k.

FREE

VARIE

TIE

S

Let

V(x

,y)

be

avariety

in2n

variable

s.pr x

Vis

the

pro

jec-

tion

on

x,pr y

Vis

the

pro

jection

on

y

Vconta

ined

inF

2n,exp-d

efinable

over

Cis

abso

lute

lyfree

of

additiv

edependencie

sif

for

ageneric

realization

a∈

pr x

V,

ais

additiv

ely

linearly

independent

over

acl(C).

Vconta

ined

inF

2n,exp-d

efinable

over

Cis

abso

lute

lyfree

ofm

ultip

licative

dependencie

sif

fora

generic

realization

b∈

pr y

V,

bis

multip

licatively

linearly

independent

overacl(C).

ST

RO

NG

EXPO

NENT

IAL

CLO

SURE

Let

V⊆

Gn(K

)be

free,norm

aland

irre

ducib

le.

For

every

finite

A,

there

is(z

,E(z

))∈

Vw

hic

his

generic

for

A.

This

isL

ω1,ω

-expre

ssib

le;using

uniform

CIT

(Holland,Zil-

ber)

itis

firs

tord

er.

CO

UNTABLE

CLO

SURE

Under

the

geom

etr

yim

pose

dby

δ(X

)=

td(x

,E(x

bar)−

ld(X

)

,th

eSchanuelconditio

n.

the

clo

sure

ofa

finite

set

iscounta

ble

.

OBJECT

IVE

A

Coro

llary

.T

he

models

of

Σw

ith

counta

ble

clo

sure

are

cate

goricalin

all

uncounta

ble

powers

.T

his

cla

ssis

Lω1,ω

(Q)-

axio

matizable

.

Obje

ctive

B

GENUIN

EEXPO

NENT

IAT

ION?

Schanuel’s

conje

ctu

re:

Ifx1,.

..x

nareQ-lin

early

indepen-

dent

com

ple

xnum

bers

then

x1,.

..x

n,e

x1,.

..ex

nhas

tran-

scendence

degre

eat

least

noverQ.

Zilber

showed:

Theore

m.

IfSchanuelhold

sinC

and

ifth

e(s

trong)

exis-

tentialclo

sure

axio

ms

hold

inC,

then

(C,+

,·,exp)∈

EC∗ st.

(C,+

,·,exp)

has

the

counta

ble

clo

sure

pro

pert

y.

VERIF

YIN

GEXPO

NENT

IAL

CO

MPLET

ENESS

We

want:

For

any

free

norm

al

Vgiv

en

by

p(z

1,.

..z n

,w1,.

..w

n)=

0,

with

p∈Q[z

1,.

..z n

,w1,.

..w

n],

and

any

finite

Ath

ere

isa

solu

tion

satisf

yin

g

(z1,.

..z n

,E(z

1),

...E

(zn))∈

V.

and

z 1,.

..z n

,E(z

1),

...E

(zn)

isgeneric

for

A.

VERIF

IED

EXPO

NENT

IAL

CO

MPLET

ENESS

Mark

er

has

pro

ved.

Ass

um

eSchanuel.

Ifp(x

,y)∈Q[x

,y]and

depends

on

both

xand

yth

en

ithas

infinitely

many

alg

ebra

ically

independent

solu

tions.

This

verifiesth

en-v

ariable

conje

ctu

refo

rn

=1

with

stro

ng

rest

rictions

on

the

coeffi

cie

nts

.

The

pro

ofis

ath

ree

orfo

urpage

arg

um

entusing

Hadam

ard

facto

rization.

MO

DEL

THEO

RET

ICCO

NT

EXT

Any

κ-c

ate

goricalse

nte

nce

of

Lω1,ω

can

be

repla

ced

(for

cate

goricity

purp

ose

s)by

considering

the

ato

mic

models

of

afirs

tord

er

theory

.(E

C(T

,Ato

mic)-

cla

ss)

Shela

hdefined

anotion

ofexcellence;Zilber’s

isth

e‘rank

one’case

.

Theore

m14

(Shela

h1983)

IfK

isan

excellent

EC(T

,Ato

mic)-

cla

ssth

en

ifit

cate

goricalin

one

uncounta

ble

card

inal,

it

iscate

goricalin

all

uncounta

ble

card

inals.

Theore

m15

(Shela

h1983)

If2ℵ n

<2ℵ n

+1and

an

EC(T

,Ato

mic)-

cla

ssK

iscate

gorical

inallℵ n

for

all

n<

ω,

then

itis

excellent.

An

exam

ple

with

Hart

show

sth

ein

finitely

many

inst

ances

ofcate

goricity

are

necess

ary

.

The

cate

goricity

arg

um

ents

were

‘Morley-s

tyle

’.Less

mann

has

giv

en

‘Bald

win

-Lachla

n’st

yle

pro

ofs

-sh

ow

ing

models

prim

eover

quasim

inim

alse

ts.

First

Ord

erto

infinitary

Str

ongly

min

imalis

tofirs

tord

er

as

Quasim

inim

alexcellent

isto

Lω1,ω

.

But

the

analo

gy

slip

sw

ith

considera

tion

of

Lω1,ω

(Q).

UNIV

ERSAL

CO

VERS

When

isth

eexact

sequence:

0→

Z→

V→

A→

0.

(1)

cate

goricalw

here

Vis

aQ

vecto

rsp

ace

and

Ais

ase

mi-

abelian

variety

.

Zilber

showed

‘the

thum

bta

ck

lem

ma’is

suffi

cie

nt.

(and

true

–w

hen

A=

(C,·)

.

CO

NVERSELY

Apply

ing

Shela

h’s

theore

m,Zilber

showed:

if

0→

Z→

V→

A→

0.

(2)

iscate

goricalup

toℵ ω

then

the

arith

metic

state

ments

of

the

‘thum

bta

ck

lem

ma’are

true

for

A.

MESSAG

E

The

analy

sis

ofnum

berth

eore

tic

pro

ble

ms

using

infinitary

logic

pro

vid

es

excitin

gopport

unitie

sfo

rcontinuin

gth

eal-

most

100

yearin

tera

ction

betw

een

modelth

eory

and

num

-

ber

theory

. GO

FO

RT

HAND

MULT

IPLY

htt

p:/

/w

ww

2.m

ath

.uic

.edu/

jbald

win

/m

odel.htm

l