Logic Colloquium 1984: Proceedings

LOGIC COLLOQUIUM '84 Proceedings of the Colloquium held in Manchester, U.K. July 1984

Editors

J . B. PARIS A. J. WILKIE G. M. WILMERS Department of Matherriatics University of Manchester Manchester, U. K .

AMSTERDAM

NORTH-HOLLAND NEW YORK OXFORD .TOKYO

OELSEVIER SCIENCE PUBLISHERS B.V., 1986 All rights reserved. N o part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87999 4

Published by: Elsevier Science Publishers B.V. P.O. Box 1991 1000 B Z Amsterdam The Netherlands Sole distributors for the U .S . A . and Canada: Elsevier Science Publishing Company. Inc. 52VanderbiltAvenue NewYork. N.Y. 10017 U.S.A.

Ukuy of Congregl Catalogingin-PublicationData

Logic Colloquium (1984 : Manchester, Greater Manchester) Logic CoUoquium '84. (Studies in logic and the foundations of mathematics ; v. 120) Bibliography : p 1. togic, Symbolic and mathematical--Congresses. I. Paris, J. B. 11. Wilkie, A. J. (Alec J.) 111. Wilmers, G. M. N. Series. 1984 5U.3 86-4462 QA9.AU.63

.

ISBN

0-444-8799-4

PRINTED IN THE NETHERLANDS

Dedicated to Alfred Tarski, 1901- 1983

vii

PREFACE Logic Colloquium '84, the European Summer Meeting of the Association for Symbolic Logic, was held at the University of Manchester from 15th July to 24th July, 1984. The main themes of the conference were the model theory of arithmetic, and the semantics of natural languages. The present volume constitutes the proceedings of this conference. Invited lectures at the conference were given by:

Z. Adamowicz (Warsaw), J. Barwise (Stanford), P. Clote (Boston), J. Denef (Leuven), C. Di Prisco (Caracas), J.E. Fenstad (Oslo), L. Harrington (Berkeley), H. Kamp (Stanford), L. Kirby (New York), J. Knight (Notre Dame), A. Macintyre (Yale), B. Poizat (Paris), P. Pudlhk (Prague), J. Saffe (Freiburg), P. Schmitt (Heidelberg), S. Simpson (Munich), R. Solovay (Berkeley), S. Thomas (Freiburg), C. Toffalori (Florence), L. Van den Dries (Stanford), A. Wilkie (Manchester), H. Wolter (Berlin), C. Wood (Connecticut), H.Woodin (Cal. Tech.), A. Woods (Kuala Lumpur). In addition to the invited lectures there were many contributed papers. Abstracts of most of these may be found in the report of the conference in the Journal of Symbolic Logic. Most, but not all, of the papers corresponding to the invited lectures are in this volume. In addition there are some papers in the volume which do not correspond to contributions made at the conference. The organizing committee of Logic Colloquium '84 consisted of P.H.G. Aczel, J.B. Paris, A.J. Wilkie, G.M. Wilmers, and C.E.M. Yates. The conference was supported financially by the Bertrand Russell Memorial Logic Conference Fund, the British Academy, the British Council, the British Logic Colloquium, the Logic, Methodology and Philosophy of Science Division of the International Union for the History and Philosophy of Science, the London Mathematical Society, the Royal Society, and the University of Manchester. On behalf of the organizing committee we wish to thank the above-mentioned institutions for their support. We also owe a debt of gratitude to all those people who by their generous help and sound advice contributed to the success of the con-

viii

Preface

ference. In particular we would like to thank the following secretarial staff of the mathematics department at Manchester University: Mrs. P. McMunn, Rosemarie Horton, Kath Smith, Beryl Sweeney and Stephanie Worrall. Jeff Paris Alex Wilkie George Wilmers

LOGIC COLLOQUIUM '84 J.B. Paris, A.J. Wilkie, and G.M. Wilmers (Editors) 0 Elsevier Science Publishers B. V. (North-Holland), I986

1

SOME RESULTS ON OPEN AND DIOPHANTINE INDUCTION

Zofia Adamowicz Institute of Mathematics of the Polish Academy of Sciences Warsaw, Poland

The paper contains some results on extending a %-ring by adding to it a zero of a given polynomial. The main of them is Lemma 1. These results are applied to build a model for a special fragment of diophantine induction having a bounded set of primes. The technics of the paper is based on a purely number-theoretic result which is the sublemma of Lemma 1. Fix an ul-saturated model M of Peano arithmetic, such that M E N . Let M* be the fraction field of M, M the real closure of M. ) I

Definition 1. Let x,yEM. We say that y is much bigger than x such that is the second coordinate of the t'th solution of the equation x2 - (a2 - ,)y2 = 7". The study of

@

Let cp(t) E @ form

if there are polynomials p,q

(3x1

has been inspired by the study of

... xn)(p(t,xl ...Xn)

= 0

p '.,

such that

q(t,xl

&

y

cp

... Xn)

is of the 0)

and the following is satisfied: 1)

2)

p is totally unbounded. xk) parametrize p as in Def. 2 . Assume that Bi(t,xl Then for every t E M - N there are x1 xn E M such that t Q : x1 ...a: Xk, x ~ = +ei(,xl ~ xk) and xn) 2 0 . q(t,xl

...

...

. ..

...

3)

For every t E M * there are arbitrarly large that t ax1= Q:xk , xk+, = ei(t,X1 xk) and q(t,xl Xn) '> 0 .

...

5)

... x n E M *

Fur every t E M * there are x1 are values of Z-polynomials at t, k+i ei(t,xl li

=

ei(t,xl

... xk)

such

such that

x1

... xn

... xk).

is of the form

I: gr,i(t)nr,i(xl

r= 1

... x n E M *

...

...

4)

x1

-.. xk)

gr,i are rational functions of braic functions of x, xk.

...

Example. Consider the formula cpE(t),

where t

and

~

for a fixed

~

,

a E 2.

... ~ (xk) x

~ are alge-

The polynomial po

3

Some Results on Open and Diophantine Induction

is defined as follows:

...

po(a,t,xl,x2 xl0) = 0 x4 2 - (a2 - 1,x; = 1

iff

2 2 x2 = 2x4 x 1x 8 2 2 x5 - (a2 - 1)x2 = 1 2 2 x6 = a + (x5 - a)x5 2 2 2 x7 - (X6 - l)x3 = 1

x3 x3

-

L

x1 = x x 9 5 t = 2 x1x 1 0 '

This definition has been taken from [ 2 ] , Chapter 6, where we have changed the notation ~yIylly2,x,x1,A,x2>to ~ X ~ , X ~ , X ~ , X ~ , X ~ , X ~ x7>. Take k = 3. Then t,x1,x2,x3 become independent parameters and the parametrizing functions have the form x4 = e1(t,x1,x2,x3) = el(xl) = Jl+(a2-1)x: x5 = e2(t,x ,x2,x3) =

e (x,)

=

Jl+(a2-1)x2 2

x6 = e3(t,x ,x2,x3) =

e 3 (x2)

=

2 2 a + (e2(x2)-a)e2(x2)

2

2 2 x7 = e4(t,x ,x2,x3) = e4(x2,x3) = Ji+(e3(x2)-1)x 3 x2 = e5(x1,x2) = x8 = e5(t,~1,x21x3) 2 2e1 (xl)xl x -x 3 1 x9 = e6(t,x1,x2,x3) = e6(x1,x2,x3) = e: (x,)

x3-t x3 xl0 = e 7 ( t , ~ 1 , ~ 2 1 =~ 3e7(t,xl,x3 ) = 2x = 1 2x1 Observe that ( 1 ) - ( 5 ) (1)

-

of Definition 3 are satisfied for

~

l 2x1

t

Ulo.

If N < t e x l e x 2 e x 3 then ei(t,x1,x2,x3)EM. We take x1 to be the second coordinate of the t'the solution 2 of the equation X2-(a2-l)Y = I , x2 to be a sufficiently big 2 2 solution of this equation divisible by 2e1(x1)x1, and x3 to be the second coordinate of the t'th solution of the equation 2 2 =l.Then t K x 1e x 2 e x 3 , ei(t,x1,x2,x3)E M and x 2-(B3(x2)-1)Y x > t. 1 (3) Let t a x , a x Z a x 3 be sufficiently large elements of M such

(2)

Z. ADAMOWICZ

4

that x is the second coordinate of a certain solution of 2 2 l 2 X -(a -l)Y =I, x2 is the second coordinate of another sufficiently large solution of this equation, and x3 is the second coordinate of any sufficiently large solution of 2 2= l . Then x4=e (x,) , x =e (x,) , x6=e3(x2) and x 2-(e3(x2)-1)y 1 5 2 x =e (X2,x3) are in M and x a =85 (x1x 2) , x 9=86 (x1,x2,x3), x ~ ~ = ~ ~ ( ~ ,are x ~in , xM*. ~ ) Moreover xl,t. (4) Let x 1=1, x 2=0,x 3 =t, x 4=a, x 5=1, x 6= 1 , x 7=1, x 8 ' 0 , x 9=t-1, xlO=O. Then ~ ~ + ~ = e ~ ( t , x ~ ,and x ~ ,x1 x ~ ) xl0 are as required. ( 5 ) Only e 7 dependends on t and it is of the required form.

,

...

We have the following theorem: Theorem 1. There is a model N of open induction in which the set of primes is and (Vt)v(t) bounded, every number> 1 has a prime divisor, N*:Q holds in N for every c p E Q. Especially, N satisfies induction for cp E Q. Hence follows that N p (Vt)cpE(t) for every a E Z. Moreover N F (Va)(vt)cpz(t). Before proving the theorem let us explain its content.

Work in

In this theory we can define the following relations:

IT1.

is the second coordinate of the Ro(a,t,y) *y equation X 2-(a2- l ) Y 2= 1 t Rl(a,t,y)*y = a

t'th solution of the

R3(a,t,Y) - Y = R4(t,y) *y = t!. For every relation there are polynomials Ri(a,t,y) (* (3x1 If we define Si(a,t,y) then

-

pi

and

... xn) (pi(a,t,y,xl...xn) ... xn) L 0 )

=

qi 0

such that &

qi(a,t,Yrxl

(3x1

... xn) (pi(a,t,y,xl ... xn)

qi(a,t,y,xl

... xn)

=

0

&

:0 )

I Z 1 k R .E S . . 1

1

We say that Si pretends Ri. sarily in a weaker theory.

It is

Ri

in

Izl

but not neces-

From Theorem 1 it follows, as we shall see later, that the totality of So,S1,S2,S3 in the sense that (Va,t)(3y)(Si(a,t,y)), together with open induction and the existence of prime divisors does not

Some Results on Open and Diophantine Induction

5

p r o v e t h e i n f i n i t y of t h e s e t of p r i m e s , u n l i k e t h e t o t a l i t y of

RorR1,R2nR3-

I n f a c t , w e w i l l p r o v e a s t r o n g e r theorem. Theorem 2 . For e v e r y c o u n t a b l e 2 - r i n g A s u c h t h a t AcM* t h e r e i s a countB s u c h t h a t A c B c M * and B s a t i s f i e s open i n a b l e 2-ring d u c t i o n + t h e t r u e t h e o r y of r a t i o n a l s + t v e r y number l a r g e r t h a n 1 h a s a prime d i v i s o r + s e t of p r i m e s i s bounded + ( v t ) c p ( t ) + + (va)( V t ) c p g ( t )

from

B

onto

for

Moreover, t h e r e i s a r i n g homomorphism

cp E Q .

A.

F o r t h e n o t i o n of a 2 - r i n g

see [31.

The proof of t h e theorem i s b a s e d on t h e f o l l o w i n g lemmas: Lemma 1 .

A 5 M* b e a c o u n t a b l e 2 - r i n g . L e t t E A , W E $ , aEA. Then B such t h a t A 5 B 5 M*, A is a t h e r e is a countable 2-ring

Let

homomorphic image of Lemma 2. Let A M*

and

B

be a c o u n t a b l e

t h e r e i s a countable homomorphic image of

B d q c p ( t ) C @: ( t ) ) .

2-ring,

%-ring B and

and l e t

...

Lemma 3. L e t J b e a n o n - s t a n d a r d i n i t i a l segemnt of c o u n t a b l e 2 - r i n g s u c h t h a t A $ J , A f l J 5 M. there i s a countable xo

2-ring

h a s a prime f a c t o r

homomorphic image of

u

in

... xnEM*.

x1

s u c h t h a t A 5 B 5 M* x1 xnEB*.

B

such t h a t

B B

Then

is a

M. L e t A 5 M* b e a L e t xoEA - J. Then

5 B 5 M*,

A

such t h a t

A

uEJ

and

B n J A

c_

M,

is a

B.

Lemma 4 ( W i l k i e ) Let A 5 M * b e a c o u n t a b l e 2 - r i n g , and l e t a E M b e r e a l a l g e b r a i c o v e r A. Then t h e r e i s a c o u n t a b l e 2 - r i n g B such t h a t A 5 B 5 M*, A i s a homomorphic image of B and t h e r e i s a y E B such t h a t y 5 a < y + 1 . The proof of t h e Theorem c o n s i s t s i n a n a p p r o p r i a t e i t e r a t e d u s e of t h e lemmas. W e s t a r t from a 2 - r i n g A. 5 M*. Lemma 1 serves t o add s o l u t i o n s of

p(t,xl

.. .

xn) = 0

&

q(t,xl

... x n )

1. 0 ,

Lemma

2 t o c l o s e f r a c t i o n f i e l d s of t h e o b t a i n e d L - r i n g u n d e r Skolem f u n c t i o n i n M*, Lemma 3 t o d e s t r o y p r i m e s b i g g e r t h a n J and add i n g prime f a c t o r s t o them ( n o t e t h a t s i n c e w e add prime f a c t o r s i n J , and t h e p a r t s of t h e 2 - r i n g t h a t l i e i n J c o n s i s t of i n t e g e r s , t h e prime f a c t o r s remain prime d u r i n g t h e whole c o n s t r u c t i o n ) and Lemma 4 s e r v e s t o close t h e r e s u l t i n g 2 - r i n g u n d e r i n t e g e r p a r t s of i s t h e u n i o n of a l l 2 - r i n g s a l g e b r a i c r e a l s . The r e q u i r e d model M o b t a i n e d i n t h e c o n s t r u c t i o n . I f w e a p g l y t h e lemmas ( c o n s t r u c t t h e a p p r o p r i a t e e x t e n s i o n s A. 5 A., 5 then we e a s i l y ensure t h a t ), Mo s a t i s f i e s open i n d u c t i o n ( v i a S h e p h e r d s o n ' s t h e o r e m ) , t h a t a l l

...

i t s primes l i e i n

J,

that

Mo+

Vtcp(t)

for

cpE Q

and t h a t

A,

2. ADAMOWICZ

6

is a homomorphic image of Mo (we summarize the homomorphisms occurring in the construction). To show that Mo satisfies the true theory of rationals we show that M t < M*. Proof of Lemma 1. Assume that W E 0

and

...

W(t) cs (5x1, q(t,xl,

...

,xn)(p(t,x,, ,xn) 5 0 ) .

...

...

,xn) = 0

&

Let €I1, parametrize p as in Definition 2 . If f(xl, ,xn) is a polynomial with coeficients in Q then we consider the following algebraic function

...

...,Xk,eI(T,Xl,...,Xk),...,en-k(T,X1,...Xk)

ef(T,X,...,Xk)=f(X1,

We need the following sublemma which is formulated and proved in the usual analysis over IR. Sublemma. For every d E N there is a constant c EN such that for every poly,Xnl of degree less than d and for all nomial f E Q I X l , E I R k + l satisfying ~ E Q , c < t, (tlt2)'< x1 for some

...

...

t l t 2 E N such that have either pf(t,xl,

t

=

...

ef(t,Xl, ,Xk) I > 1.

...

-

1' t2

, (parameters of f)c < x l , xy < xi+l we

,Xk)

does not depend on

X1

... Xk

or

Proof. Let s l ,

... ,sn< d. Consider the following function: s S k+l ... en-k(~,xl,... XI .. . Xkk el (T,X1, .. . ,Xk) ~~

S

S

,xk)

n

.

By ( 5 ) of Definition 3 this function is of the form 1

'

r=1grsl...sn(T)cs 1

.

where

grsl.. sn functions:

We can expand series

r=m

are fational functions and

"(X1t

5 S1".

r, S1...Snr(Xl,

- m

z

...s nr(xl'".'xk)

.. .

...

,xj-l,

...

,Xk)

x 1. + 1 ,

...

'nr

are algebraic

in a descending fraction power ,Xk).xs 1

for every j = 1 , ,k, where nr are algebraic functions, S E N - {Ol, and there is a constant c 1 such that for all ,xk> satisfying xyl<xi+l, c1 < x l , <xl,

. ..

Some Results on Open and Diophantine Induction

...

(xl, 5sl . . . ~ n , r I n d e e d we f i r s t expand ty".

r

-m

,Xk)

=

7

I:

llr(X1

r=m

- - .X j - l r X j + l ... x k , x3s

a s a f u n c t i o n of

5s1.. . s n , r

xk " i n i n f i n i -

W e obtain

r=m 'r,k Then w e expand e v e r y

qrrk

xk-l

a s a f u n c t i o n of

W e continue.

"in infinity".

...

For an a p p r o p r i a t e c o n s t a n t c , , for a l l <xl, ,xk> s a t i s f y i n g e v e r y series w e o b t a i n i s a b s o l u t e l y c o n v e r g e n t c1 < x l , < xi+l, , rlr,kt r e s p e c t i v e l y . Changing s u i t a b l y and r e p r e s e n t s n t h e o r d e r of t h e summation and g r o u p p i n g t h e t e r m s w e o b t a i n t h e r e q u i r e d e x p a n s i o n s . W e have

XI'

=

...

z a s l . . . s n < d S1"'Sn

grs r=l 1..

S u b s t i t u t i n g t h e expansions f o r expansions f o r

ef

for

.s n ( T ) 5 s 1 . . . s n r ( X 1 . ' . X k ) '

5s1...s

...

j = 1,

k:

we o b t a i n t h e following

n

-m

=

I

sl..

:

.

a sl.. sn Emk

(X1

1 s1

...sn r t l q r s l .

There i s a c o n s t a n t

c2EN

...

. . s n ( T ) ~ r ~ r j ss nl

f'

of

X. i s

(xl...xj-l,xj+l...xk)

such t h a t f o r every

t >c 2

and e v e r y

3

Z. ADAMOWICZ

8

.. .

...

polynomial f E [ X I , ,Xnl either ef t l X 1 , ,Xk) does not depend on X I . -. txk or there is a j = j(f,t) such that the expansion (3) of ef(T,x I ,Xk) for j = j(f,t), T = t contains a non-zero term with a positive exponent o X and j(f,t) ,Xk) does not depend on X . ef(t,Xl, j(t,t)+l' " ' "k' Proof of the claim. Consider the family of algebraic functions

...

...

(nr'rjsl... s n

j = I...k,

:

r' = I ,

...

r = 1

...

,ml

U

1, s1

...

s n < d,

Ill.

We choose a maximal subset of this family which is linearly independent over Q as a set of functions of X I , ,Xk.

{nl,

Let it be

j,r,sl

.sn nrlrjsl.. =

I

,npl.

...

sn,rl 5 1 there are rational such that 'bElrjsl.. . s n

Then for every b:'rjsl..

...

...

. 1n'(

a

.xj-lxj+l* . .xk)

. ..

.. .

n 1 (XI, ,Xk)+ +bFIrjs np(xlI"',xk) - brlrjsl... s n n Let c2 be so big, that any polynomial equation involving T and all the rational numbers which occured so far and which do not depend on Xk (i.e. involving the parameters of grs TI as l . . .sn, XI .sn

...

(T)' b:lrjsl..

.sn

for

j = I . ..k, i = l...p, r = l...l,

s

.sn < d,

r' = l...m) with powers at most d and with coeficients at most being satisfied for a t > c2 is satisfied identivally in T .

d

Let t > c2 and f be given. Assume that Bf(t,X l...Xk) depends on some of X1...Xk. Let j(f,t) be the largest j such that Bf(t,X l...Xk) depends on X.. Denote 1 later j(f,t) by jo. We show that there is a non-zero term with a positive exponent of X . in the expansion (a) of Bf(t,X l...Xk) for j = jo. 10 Suppose, otherwise, that all terms with positive exponents of X . 30 ,X. X. Xk. Also all terms vanish for all X I , Jo-lt lo+l with positive exponents of Xjo+l, ,Xk vanish, by the choice of

...

...

...

jo in suitable expansions tively.

(n) for j

=

jo+ 1,

On the other hand there is a negative exponent its coefficient does not vanish.

... $'

,j = k

of

X

10

respecsuch that

Some Results on Open and Diophantine Induction

It follows that the rational numbers a s l . . . s n tisfy the following system of linear equations

A jLjo

A r'=l

9

for

sl...s n


j - 0 r'=l sl...sn c2 we infer that ( A ) is satisfied for every t. Now we have to define a new constant c 3 E N . Let c 3 > c 1 be such that 1)

t if(tlt2)C3<x1, for some t l t 2 E N such that t = -t2 l as1.. sn -a' S1". ' n' ) c 3 < x 1 for and (a' sl.. . s lc3 < x l , (a; l.. . s n a"sl sn

.

. ..

.. .

s1 sn

c2

(*) A (***)

Now we choose i)

E

f. [acoJ(t,xl,

...

,x.

1-1

a EN,

aC3

1 1 1 >--j-

3s

(t,X1,

... xjTl,Xj+l, ...

and that

...

acoJ(t,xl,

I

does not depend

,Xk)

,x.

3-1

)

is not inte-

a'sl.. .sn a sl.. .sn - a" S1..'Sn c3 a' a" < t . s, sn sl.. . s ' n

if there is a solution

then there is one with

-t E N

.. .

so that

satisfies all the non identities occuring in the system

(A)

ii)

are as in ( 1 ) and

X.

f

provided that

,xk

there are

Gl,

...

-

,xn E Z

2

such that

-c3 t < x l , xc3

< z1. + 1 ~

Some Results on Open and Diophantine Induction :k+i

- -

... ,xk)

= Bi(t,xl,

- -

and

... I,:,

g(t,xl,

11

LO.

Such a

e x i s t s by ( 2 ) of D e f i n i t i o n 3 and " u n d e r s p i l l " . Then ( A ) i s s a t i s f i e d f o r t = t because i t ' s e q u a t i o n s a r e s a t i s f i e d f o r every t . I t f o l l o w s t h a t t h e system ( * ) h a s a s o l u t i o n s , ..sn' S 1 * * * S n < which s a t i s f i e s ( * * * ) .

-

W e can t a k e

asl.. . s

r a t i o n a l b e c a u s e t h e c o e f i c e n t s of

(*)

,( * * * )

a r e r a t i o n a l ( t h e y a r e of t h e form

r

1=1 g r s r grs 1=1

0

. .. s n ( t ) b r l r j o s l . .. s n

By t h e c h o i c e of

c3

-

).

-

Sl...S

-

n - a"

-

Take

...

xl,

-

,xn

n

.

a s i n (ii)

a"

n

s l . . .sn

sl.. . s

-

a' sl.. .s '

w e c a n assume

3'

-

a

i

-

W e have

EC3 < x

,xc3
1

..xk)-rn

ml) =

ml n en-k(t,~l... xk)-rml 1 €A

m.

4

Consider (2) 'Phen lg (a4.. .al.. I > 1. It follows that B is disretely ordered. It remains to show that A is a homomorphic image of kj.

.

Define h(g(al...al,

=

Clearly

x -r1 "4 ~

"I

...

1 x h. I K I= b > a .

Such an extension is called an I-eltension of M and denoted by M < I K An initial segment I of M is ( n + 1)-eztendible (in M) if there exists an I-extension K of M in which I is n-extendible. This notion is due to J. Paris. Finally, K is said to be an n-elementary eztension of M , denoted by M:K, C,-formula 4 and u belonging t o M ,

M

if for any

I= 4 ( u ) iff K I= 4 ( u ) .

Recall the

Theorem 1. (CI-1) For integers n , k

M

2 1,

I= BE,+& iff M in (M)$M

Implicit in the proof of Theorem 1 is the equivalence also with the combinatorial property: for any A, partition F: Mk + (0,.. . , a } with a E M , there exists i 5 a for which F - ' ( i ) is unbounded in M k . As well, the "boldface" version holds, which will be used later: An initial segment I C, M satisfies BE:,, if I A; (I):, iff for any coded partition F: I" -+ (0,. . . , a } there exists i 5 a such that F ( i )is unbounded in I".

'

The same proof yields the boldface version of Theorem 1: for n, k 2 1 and any initial segment I of M --+ I I= BE^,^ iff I ( I ) : , .

A;,

Ultrafilters on Definable Sets of Arithmetic

J . Paris kindly pointed about t,he above remark arid

41

its immediate

Corollary 2. (CI-1) ( n 2 1 ) . For any initial segment I of M . I is n-Ramsey if and only if I I= BE:, Concerning partial elementary end extensions and collection, we have

Thcorerii 3. (CI-3) ultrafilter on M.

(ti

2 1 ) If M satisfies BEn.I then there exists a E,-complete En-

Using this. we can obtain a cardinality free version of a result due to Kirby-Paris [KP].

+

Corollary 4. ((21-3) ( n 2 1 ) M satisfies BE,, I if and only if M admits a proper ( n 1)elementary end extension. The MacDowell-Specker Theorem on proper elementary end extensions of models of Peano arithmetic is cardinality free. so this corollary was to be expected. Theorem 5 . (CI-3) ( n 2 1 ) For countable models, M I= BE,, I if and only if there exists the following tower hf + n . c MI m and n E I- + F ( z , n ) = io

1

(The idea is that FIZ x I' is almost constant.)

U = { A C_ I: A is M-coded and 3 n E N(X, C A)}, where the sequence X,, 3 X I 3 Xz 3 . . . is defined as follows: Using countability of M. let {gn: n E N} enumerate infinitely often all .2f-coded partitions of I 2 having range bounded in I. Let X,, = I. Given X,, Let

suppose that

g , l I x X , , : I x X , - + (0... . . a } .

By the property ( * ) let, Xn+, c X, be M-coded and unbounded for which there exists i, 5 a and a n unbounded but not necessarily M-coded set 2 satisfying V r E Z3r E

IVs E I

I> .q

r and S E X,,]

4

This determines Xn and terminates the description of ultrafilter on M-coded subsets of I. Now letting

gn(z,.q) =

U.

io].

Clearly

K = {f:f is an M - coded mapping from I into M } / U

we claim that I is 1-Ramsey in K

U

is a complete

so

P.CLOTE

This latter suffices, since it is clear that 1-Ramsey and 1-extendible are synonomous. For I to be 1-Ramsry in K . consider any K-coded partition G: I -+ ( 0 , . . . , a } with a E I. So for z,y E I G ( r ) = y ++ K )=O(z. y, c) where O is a C, formula and by L6S' Lemma,

Define the M-coded partition

F:P by

+

q r i),=

and

++

{o, . . . , Q + I } M

I= e(z, y,c(i))

F ( z , i ) = a + 1 otherwise

By construction of U. there exists an unbounded M-coded subset Y of I belonging to and an unbounded but not necessarily roded subset Z of Z and io 5 Q + 1 such that VzE Z3mE IVnE

I

I n 2 m and

n E

Y

4

[

Then by completeness of so

and so

U,

-+

J

F ( r , n )= io .

Fix any z E Z and let m E I be such that V ~ IE n 2 m and n E Y

U

1

F ( r , n ) = i, .

{iEY: i l m } E U

{i E I: M K

I= e(z,i,,c(i)} E u I= qz,i,,

c)

F ( z ) = io

By hypothesis that the range of F is contained in (0.. . . , a } , it follows that io 0 and exp(;;) = y) is not "semi-analytic".)

I

--that each s e t

I

In conventional terms:

let

e(xl, ....xm) be a formula in the language of

(R. 0 and (hence) f'(d) < 0. But F(c,O) = F(c,f(c)) = f'(c) > 0 , and F(d.0) = F(d,f(d)) = f'(d) < 0 , so by the intermediate value theorem F(x,O) vanishes a t some point between c and d. What about the zeroset of a two-variable Pfaffian function f ? To simplify the discussion, assume f is defined on all of R2, does not vanish identically, and that y = f(x1.x2) satisfies

where F1 and F2 are Pfaffian functions on R2 introduced "earlier". Let us assume inductively that the decomposition conjecture holds for Pfaffian functions introduced "earlier". (The relevant functions are F1(xl,x2,0) and F2(x1,x2,0).) Let C be the

L. VAN DEN DRlES

66

"curve"

f(x1,x2) = 0. I t is not difficult t o prove the following three properties:

(1) Only finitely many vertical lines x1 = rl,...,xl = rk are tangent is a singular point of C we count x1 = r a s tangent t o C.)

There is a natural number intersects C & a t most M (2)

(3) Outside of the pieces of the form each of -whose two lirn a(x,) = *-. x -a

M

such that for

r t rl.....rk

m.

C

. (If (r.4

t h e vertical line x1 = r

vertical lines x1 = rl....,xl = rk the curve C consists of disjoint - & an analytic function on an interval I,, x2 = a(xl), & endpoints a is either in (rl,....rk.f=l, or satisfies In the latter case we say that x1 = a is a vertical asymptote t o

a

I

C.

(See picture, where

C

is decomposed in graphs of functions

x,= 5

x, --a

x, = 5

To verify the conjecture for the function (4)

Each of the functions

a

b

a a s above.)

f

now amounts t o proving:

Ipiecewise) Haffian.

(5) There are only finitely maw functions

a

as

(3). dx

2

Now (4) is easy: if x2 = a(x,) is a piece of C. then +,= (x1,x2,0). a s follows from differentiating the identity f(xl.a(xl)) = 0. Hence a LS piecewise Haffian. (We ignore h e y a technicality: one has t o split up R2 into finitely many Pfaffian cells such that -1. (x1,x2,0) is well defined and Pfaffian on the open cells.) F2

We now turn t o (5). In view of (l),(2), (3) this property will follow from: (6 ) The curve

C has only finitely m a w vertical asymDtotes. The discussion of (6 ) merits a section of its own: it leads t o the new concept of tameness, which, I hope, will be the key t o proving the conjecture.

Tarski's Problem and Pfaffian Functions

61

VI. Tameness:

use of infinite numbers To detect the points a c R for which x1 = a is a vertical asymptote t o the curve C: f(x1,x2) = 0. we take a positive infinite hyper real b c *R,and note that the equation f(xl.b) = 0, or the equation f(xl,-b) = 0. must have a root a' infinitely close t o a. So i t is enough t o show that the function f(xl,b) has only finitely many zeros in *R. But f(xl.b) is Pfaffian on *R,in an obvious sense. Provided we extend our inductive hypothesis t o such (nonstandard) Pfaffian functions - keeping however the standard meaning of finite - we may conclude that f ( 3 , b ) has only finitely many zeros. (The inductive hypothesis applies, since the one variable function f(xl,b) comes "earlier" than the two variable function f(x1.x2).J Our attempt t o verify the conjecture inductively must therefore be carried out simultaneously over R and *R. R. Here a new Let us now consider a three variable Pfaffian function f: R3 aspect comes into play, and the role of *R becomes more prominent. The technicalities (domains of definition, etc.) are more severe then in the two variable case, but, grosso modo, the verification of the conjecture for f reduces to proving the analogue of point (6) above: Let S C R3 be the surface f(x1.x2.x3) = 0. Think of the x3-axis as the vertical axis. Define As(S) t o be the set of points (a,.a2) c R2 such that the vertical line x1 = al, x2 = a 2 is an asymotote t o S. The analogue of (6) is a s follows: As(S) is contained in a Pfaffian curve, i.e., in the union of finitely many non-oDen Pfaffian cells in R ~ . To prove this, we take an infinite hyperreal b and consider the equation f(xl,x2.b) = 0 on (*R12. Since the two variable function f(xl.x2.b) comes before the three variable function f(xl,x2,x$, the inductive hypothesis applies and tells us that the "curve" f(xl,x2,b) = 0 in ('R) is, modulo finitely many lines, a union of graphs a(x,) = x2, a ranging over finitely many Pfaffian functions on intervals in *R. Let us consider such a function a, and assume for simplicity that its domain is all of *R. Keep in mind that we a r e not interested so much in the subset r(a) C (W2 but rather in the set of points of R2 that a r e infinitely close t o r(a) since it is these points t h a t are in As(!% So it suffices t o establish the following tameness property of a:

-

There a r e a1 0 on A, under transcendental operations, such a s f f fr (r c R) and f log(f). The proof of the theorem will be given in [vdDG] and is along the lines sketched above: the main point is the (inductive) verification of the tameness property (T). The theorem implies a considerable amount of "0-minimality". Expressed in logic jargon: Let the s e t S C Rm be defined by a formula

-

-

-

-

where Ql,...,Qn are guantifiers 3. V and + & 4 guantifier free formula in the language I- 0, ...,iH> 0

axil.. . a X i M 1

(the ring generated over

%(B) by the restrictions t o

J

B of the partials of

f).

Note that if % is a differential cell system on RM then % is also a differential cell system on RM. Of course, coherence is generally not inherited by % from A. For that we need further assumptions on 3 and f which we discuss now. (2.9) Definition. A cell system on, ' R M < m, is called Waff closed in a cell A' on RM*' if for each pair of cells A c d[M) and B E 3'[M + 11, such t h a t B maps onto A under the projection RM+l --. RM, the ring %(A) is f f a f f closed in d'(B1. system

Lemma. ~ e tA be cell system on RM. M < -, and f an anahtic function cell A E d[M]. Then d is coherent if & followinK conditions are satisfied: AIRM-' is Pfaff closed in 3, and there is a coherent differential cell system d' on RM+' such t h a t d ' l R M = % and f is ffaffian relative t o 3(B) for some cell B c d'[M + 11 which maps onto A under the projection R.' (2.10)

on an oDen

-

L.VAN DEN DRIES

80 (When M = 1 we call

dlRM-l

Pfaff closed in

d

by convention.)

The proof is a tedious verification. One has t o use the following f a c t which is also important for other reasons. Let A. c d[M]. A. C A. Then each function g c d < f > (Ao) can be written as -

-

where G

c d'(Bo). Bo the inverse image of A. under the projection B A. (d' B are a s in the lemma.) This f a c t implies that if the (old) systems d and d' consist of Pfaffian cells and functions, so does the (new) system d. (By virtue of condition (P3). see (1.16).)

and

From the definition of 6, it should be fairly clear that Q can be built up by starting with the system R [Xll, carrying out the operations d AN, d d[XM+l] and d d when they apply, and taking unions when an % increasing chain has been built. (Here it is understood that the operation d is only applied when the hypothesis of lemma (2.10) is satisfied for an earlier constructed system A'.) Some basic properties of Pfaffian cells and functions follow from this way of generating Q. (2.11)

-

-

-

-

(2.12) Proposition. 0 k a coherent differential system. Remark. The proof of this result is typical for many of our arguments and therefore we give it in detail.

m. BIXl]

is a coherent differential subsystem of @ I R . Take a maximal coherent differential subsystem d1 of @ i R . Of course d1 is Nash closed since C @ I R . Consider d1[X2]. I t is a coherent differential subsystem of @ I R 2 dl C and its restriction t o R equals dl. Take a maximal coherent differential subsystem d2 of @ ( R 2 with d2 IR = A]. We claim that d2 is Nash closed and that d l is Pfaff closed in d2. Nash closedness is clear from maximality. If d1 were not Pfaff closed in d2 there would be an analytic function f on a cell A c d 2 [ l l = A l [ l l such that f 6 dl(A) and f is Pfaffian relative t o d2(B), for some cell B c d2[21 projecting onto A. Then d2 would be a coherent differential system properly extending dl but still contained in 9 1 R . (See lemma (2.10)and the comments following it.) This contradicts the maximality property of d l . The claim is proved. d2[X31 is a coherent differential subsystem of S I R 3 whose restriction t o R2 equals d2. Take a maximal coherent differential subsystem d3 of @ ( R 3 with d31R2 = A2. As with d2 it follows that d3 is Nash closed and d2 is Pfaff closed in d3.

Tarski's Problem and Pfaffian Functions

81

Continuing in this fashion we obtain a sequence (an) such t h a t each dn is a Nash closed coherent differential subsystem of @IRn. with IRn = An and dn Pfaff closed in The "union" d, of the An's is then a cell system contained in I? which satisfies the closure conditions (Pl), (P2) and (P3) of (1.16). ((P$ follows from coherence of dw.) Since Q is the least such system we must have d, = I?. Now is a coherent differential system. use that A, (2.13) Remark. I t follows from the proof that an alternative definition of Q is as the least differential cell system on R' which satisfies (P,) and (P$. (I.e., condition (P,) of (1.16) could have been replaced by the requirement that the rings @(A). A c Q[ml, are closed under the operators a/axi. i = 1,...,m.) (2.14) To formulate the next result i t is convenient t o call an analytic function f on an open cell A C Rm a differentially algebraic function if the integral domain R has finite transcendence degree over R. Here R is the R-algebra generated by the

partials

a i l + ** .+im ' 1 'm a x , . . .ax,

f

of

R-algebra of analytic functions on

a/axi.

(2.15) ProDosition. algebraic.

f

(including f A

containing

itself). f

Alternatively, it is the least

and closed under the operators

Each Pfaffian function on an open Pfaffian cell

differentially

The proof is along the same lines a s the proof of the previous proposition, using the two lemmas below. The first one is well known. (2.16)

Lemma.

If

A C Rm is an open cell and

3 a set of differentially algebraic i l + . .+i

functions on

A, then

R, the R-algebra generated by the partials

a ax

1

1

l .

.ax,"

f,

with f E 3, is closed under the operators a/&,, and all its members are differentially algebraic. Moreover, each analytic function on A which is Nash over R is. differentially algebraic.

(2.17) Lemma. Let A C Rm and B C Rm+l be open cells with B projecting onto A. Let the analytic function f: A -. R be Pfaffian relative t o an R-algebra T of analytic functions on B. Suppose that T is closed under the operators Waxi. i = 1,...,m+l, that all functions in T are differentially algebraic, and that T contains t h e coordinate function Xm+l. Then f is differentially algebraic.

82

L. VAN D E N D R I E S

Proof.

For each function

G

an R-algebra morphism from

in T

T

we put

G+x) = G(x.f(x)). so t h e map

into t h e ring of analytic functions on

A.

G

-

Gf

is

Suppose now

t h a t y = f(x) satisfies t h e partial differential equations & / a x i = Fi(x,y), i = 1,...,m where Fi c T. This can also be written as a f / a x i = (Fi)f. Since R has finite transcendence degree over R i t suffices t o show that its image under t h e morphism

G

-

Gf

contains

f

and is closed under t h e operators

contains f because f = (Xm+llf. Closure under g = Gf. G E T . Then a simple computation gives:

-

(2.18) Question. analytic p

f: I

a/axi

a/axi.

The image

is seen as follows:

When is a differentially algebraic function locally Pfaffian?

R, I a n interval if f satisfies an equation p(y,y('),

suppose

(True for

Y ( ~ ) )= 0

where

is a nonzero real polynomial.)

(2.19) Several mathematicians asked m e whether substituting Pfaffian functions in a Pfaffian function gives again a Pfaffian function.

This is indeed t h e c a s e but a precise

formulation of t h e general result, let alone i t s proof, is messy, so we only indicate here a proof of t h e special c a s e of composition with a one-variable Pfaffian function. (2.20) Prouosition.

Let

f c Q(U, I

an

g c Q(A). A an open Pfaffian @(A). Proof. The statement is certainly t r u e if f belongs t o €t[XI]. Further, if d is a cell system on R, J C @ l R , such t h a t t h e statement is t r u e for all f in A . then t h e statement is true for all f in AN. Take a maximal coherent subsystem d l of P l R such t h a t t h e statement is t r u e for f in dl. Then d 1 is Nash closed. Consider d = Al[X2]. I t is easy t o verify t h a t has the following property (*):

cell, such t h a t

g(A) C I.

Then

f

o

interval.

g c

-

If F E A(B), B = (a,@[ 2 &I& 242). and g c @(A). A an open Pfaffian cell, such t h a t AA) C I. then t h e function (x.r) F(g(x),r), with domain Q. (Note: a o g , B o g a r e in f a c t (extended) functions in (aog,Bog)A. belonas

(*)

Q

since

a,B

a r e in

241) W

(-m,+-).)

Let us say t h a t a cell system

d

on

R2 is dl-good if d is a coherent

subsystem of @ I R 2 such t h a t d ( R = d1 and d satisfies property (*). It is easy to check t h a t if d is dl-good then is also dl-good. I t is more work t o show:

2

(**)

d

dl-gOOd.

4

is Pfaff closed in

J1

Tarski's Problem and Pfaffian Functions

83

To see this, let 0: I + R be analytic, and Pfaffian relative t o d(B) where B = (a.s)I c %[2], say O'(t) = $(t.e(t))(t c I), with # c d(B). We claim: (***) The statement of the theorem holds for each function f J1. (By = dl. and therefore (**).) maximality of dl, claim (***) implies To prove assertion (**I), consider a function f in dl and a function g c @(A), A c Q[m], such that g(A) C domain (f). We have t o show that f o g c @(A). We first consider the case f = 4. Then

where the function #i is defined on the cell ( a o g . s o ) A by #i(x,r) = # ( g ( x ) , r ) - e (x). By property (*) the functions #i are Pfaffian, hence + o g is Pfaffian. T h h takes care of the case f = 0. For general f the case that domain (f) Q I is trivial since then f must belong t o dl. (See (2.8), clause (2a).) So we assume that domain (f) C I. For simplicity of notation we assume even that domain (f) = I. Then, see (2.10). we have f(r) = F(r.Mr)) (r c I). for some function F c J(B). So (fog)(x) = F(g(x),(+og)(x)). Define H(x.r) = F(g(x),r) for (x,r) c (aog,80g)A. Then H is Pfaffian by property (*). Since 0 o g is also Pfaffian it follows that the function f o g , which satisfies (fog)(x) = H(x,(Oog)(x)). is Pfaffian. This finishes the proof of claim (***), and therefore we have established (**). Let d2 be a maximal cell system on R2 which is dl-good. (So is Nash closed.) I t is easy t o verify that d2[X3] has a certain property relative t o d2 which is analogous t o the property (*) which d2 has relative t o d l . A s before we express this by saying that d2[X31 is d2-good. (+-goodness is meant t o include the property of being a coherent subsystem of @ l R 3 whose restriction t o R2 equals d2.) One can then prove, a s we did for dl-good systems, that is Pfaff closed in each d2-good system. Let 4 be a maximal d2-good cell system on Continuing in this fashion we obtain a sequence (an) such that each An is a Nash closed coherent Then the subsystem of @ l R n . with IRn = dn and dn Pfaff closed in of the An's is necessarily equal t o Q, by definition of Q. This "union" d shows that @ I R = J1, and proves closure of Q under composition with functions in @1R.

+

+

3.

~

93.

Zerosets of Pfaffian Functions.

Comuleteness of Cell Systems.

(3.1) Definition. A cell system d on RM is called comdete if d is coherent and the zeroset Z(f) of each nonzero f c d(A), A an open cell in d[m]. 1 5 m 5 M. is

L. V A N DEN DRIES

84

contained in t h e union of finitely many cells in belong t o (Note:

Rm of dimension

< m

that

A.

for

m = 1 this means t h a t each nonzero function

has only finitely many zeros.

In particular, BIXl]

f E A(I), I

an interval,

is complete.)

(3.2) One can show t h a t if a cell system d on RM is complete then there is for each function f E $(A) ( A an open cell in AIm]. 1 5 m 5 M) a decomposition B of Rm consisting of cells in

d

such t h a t

A

is a union of cells in

constant sign (-1. 0, or 1) on each cell of

3

contained in

d

and such that

In particular, t h e Decomposition Conjecture s t a t e s t h a t t h e cell system complete.

In view of t h e way we generate

Q

f

has

A. Q

is

i t is therefore of vital importance t o know

under what conditions completeness is preserved by t h e various operations on cell systems. The algebraic operations a r e well behaved in this respect a s t h e following two propositions show.

The first one is essentially trivial.

(3.3) ProDosition. If A is a complete cell system on Nash closed then drat and d N a r e complete. (3.4)

ProDosition.

d[XM+l]

If

RM, M < -, and

is a complete Nash closed cell system on

d

AIRM-'

is

RM, M < -, then

is complete.

(3.5) The proof of (3.4) uses Tarski's elimination theory and t h e following real analytic version of a topological lemma in tojasiewicz [t.p. 1071.

-

Lemma.

Let C be a connected analytic manifold and cO,...,cd: C R analytic functions, not all zero, such t h a t t h e number of comDlex zeros of t h e polynomial cd(x) Yd + cd-l(x) Yd-' +...+ c 0(x) is constant, i.e.. does not depend on x E C. Then R such t h a t for each x E C t h e real there a r e analytic functions y1

is coherent. We have t o show that each nonzero function g c % (J) has only finitely many zeros in i t s interval of definition J. To simplify notations let us assume J = I. Then we can write g(x) = G(x,f(x)), for some function G E d'(B). (See (2.10).) Since d' is complete the zeroset Z(G) of G has a particularly simple form, and in f a c t we may reduce t o the case that Z(G) = r I ( h l ) V...V rI(hk). where hl 0 on I: then LE is a Hardy system on R.) Let ?f be a Hardy system on R. If ?f is not Nash closed we can extend 21 t o the larger Hardy system ?fN on R which is Nash closed. So let us assume that # is already Nash closed. Then BIX,] is a Hardy system on R2, hence #' = #[X21N

-

-

L. V A N DEN DRlES

86 is a Hardy system on for some

B

E

-

R2, with

is an analytic function

f: I

If % is not Pfaff closed in #' then there an interval. which is Pfaffian relative t o #'(B)

# ' ( R = #. R. I

%'I21 projecting onto I. such t h a t f d %(I). Then # is a Hardy

R, by (3.8), and is strictly larger than

system on

Now t h e adjunction process just

#.

in t h e role of

described can be started over again, with #

These considerations lead t o t h e following conclusion.

#.

For each Hardy system

on

R there & a smallest Nash closed Hardy system BP on R such t h a t EP and B ' T P f a f f closed in XP[X2jN. (Note: if # C @ ( R , then RP C Q I R . )

%

2 #

The Hardy system (BIXll)p is already quite large: i t contains the functions exp, sin x (-$r < x < &), i t contains each indefinite integral of each of i t s functions, and is closed under composition of functions. (The last statement can be proved along t h e same lines a s proposition (2.201.)

(3.10) We now proceed t o generalize these results t o Hardy systems on RM where M > 1. Here a new phenomenon, t h e existence of vertical asymDtotes t o t h e zeroset, may cause trouble. Let M > 1 and let A C RM be an open cell, say A = (a,fl),A. (Look at A as t h e union of vertical intervals parametrized by t h e points of R A ; K: RM '-'R IS ' t h e usual projection.) Z a subset of A. (E.g..

-

Z = Z(f) for some analytic function

Definition.

( 1 ) We say that a point

if there is a point

RA. (q,p) = def'((l - t) q + t p l O t h a t (x,ux) E Z for all x (N.B.. i t may be t h a t (2)

q c

A ~ ( z . A ) = def.(p c

(If expects

Z

f

on

A.)

lies on

p c KA

a vertical asymptote

2

q f p , such t h a t

< t < 1) C %A, and an analytic function

u: (q.p)

c (q,p), and either

1i m

a =

-m

R A J ~lies

or

fl

=

1 i m u(x) = a(p), or

+z.rp

x+ P

-

A,

R. such

u(x) = ~ l p .)

on a vertical asymptote t o z in A). f on A, and f f 0, one naturally

is t h e zeroset of an analytic function

dim As(Z.A) < dim Z.)

(3.17) Lemma. Let # be a Hardy system on RM, 1 < M < -, such that ?f(RM-' is Pfaff closed in B . Let A C RM be an open cell in #. and f: A -R an analytic function such t h a t Y = f(x) satisfies a y / a x i = Fi(x,y), i = 1,...,M. where t h e Fi a r e analytic functions on an open s e t B C R'+', with TA(f) c 9. Suppose t h e following three conditions a r e satisfied: A X (0)C B As(Z(f1.A)

Then

Z(f) C D1 V..V Dk for suitable cells

For each cell

=

-

-

(1) (2) (3)

and each function x Fi(x,O): A R belongs t o #(A). D C A in #, of dimension < M, we have (f ID) c #(D).

6.

Dj C A

in

%

of dimension

< M.

Tarski's Problem and Pfaffian Functions (Weierstrass preparation, real analytic continuation, and a refinement of the Key Lemma in the Introduction a r e the principal ingredients.) (3.12) The proof of this important lemma is long, we do not give i t here.

(3.13) The main consequence of lemma (3.11) is a generalization of proposition (3.8) to

Hardy systems on RM: Theorem. Let X be a Hardy system on RM, 1 < M < -, such that #IRM-' is Pfaff closed in 8 . Let f be an analytic function on a cell A E #[MI such that f is Pfaffian relative to X'(B) where X ' is a Hardy system on RM'l with #' IRM = X and B c #'[M + 11 projects onto A. Then X is a Hardy system on RM if and only if the following condition on asymptotes is satisfied: For each function h c #(Ao). where A. C A, A. c #[MI, the set As(Z(fh).AO),with f h = (f IAo) - h, is contained in the union of finitely many cells in RM-l which belong t o X and have dimension < M - 1.

Proof. (Sketch) The idea

is the same a s in the proof of proposition (3.8). Lemma (2.10) and completeness of # enable us t o reduce t o the case where lemma (3.11) can be applied. See also parts V, VI of the Introduction.

(3.14) Comment. The theorem is useful since it reduces an M-variable problem (on zerosets of M-variable functions) t o an (M-1)-variable problem, i.e., the verification of the "condition on asymptotes." For example, if M = 2 we have t o verify that certain plane curves have only finitely many vertical asymptotes. For M = 3 we must consider a certain collection of surfaces S C R3 = R2 X R, and check that for each surface S there is a curve C C R2 of a certain type such that all vertical asymptotes t o S intersect R2 in C. I have been able t o verify this condition on asymptotes in quite a few cases, by means of the notion of tameness. (See parts VI, VIII of the Introduction.) To work with tameness it is essential t o develop the theory of Hardy systems and F'faffian functions also over *R (without changing the notion of 'finite'). This development will be carried out in [vdD 61. (3.15) Let us finish this paper by pointing out an algebraic consequence of completeness.

If & is a complete rationally closed cell system on R then it is easy t o see that each nonzero function f c &(I), I an interval, is of the form

-

where u c %(I) has no zeros and rl, ...,rk are the distinct zeros of f. I t follows that &(I) is a principal ideal domain and that r (X, - r) &(I) is a bijection of I onto the set of maximal ideals.

-

87

s m a ma NVA '1

88

Tarski's Problem and Pfaffian Functions

89

References

IBI

M. Boshernitzan, New "Orders of Infinity", Journal d' Analyse Mathematique,

41 (1982). 130-167. [Cl

G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, Automata Theory and Formal Language, 2nd G. I Conference, Kaiserslautern. pp. 134-183, Berlin, Springer-Verlag. 1975.

[DI

B. I. Dahn, The Limit Behaviour of Exponential Terms, t o appear in Fund. Math.

[vdDl]

L. van den Dries, Bounding the Rate of Growth of Solutions of Algebraic Differential Equations and Exponential Equations in Hardy Fields, Report.

23 pp., Stanford University, Jan. 1982 (unpublished). L. van den Dries, Analytic Hardy Fields and Exponential Curves in the Real Plane, Amer. J. Math. 106 (1984). 149-167. L. van den Dries, Remarks on Tarski's problem concerning (R,+;,exp),

Logic

Colloquium 1982, pp. 97-121. Ed. by G. Lolli. G. Longo and A. Marcja, North-Holland, 1984. [vdD4]

L. van den Dries, Tarski's elimination theory for real closed fields, preprint.

IvdD51

L. van den Dries, Definable sets in 0-minimal structures, in preparation.

[vdD6]

L. van den Dries, Elimination Theory for a class of transcendental equations, in preparation. E. A. Gorin. Asymptotic Properties of Polynomials and Algebraic Functions of Several Variables, Russian Math. Surveys 16.

[Hardt]

R. Hardt, Semi-Algebraic Local-Triviality in Semi-Algebraic Mappings, Amer. J. Math., 102 (19801,291-302. G. H. Hardy, Orders of Infinity, Cambridge, 1910.

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90

[Khll

A. G. Khovanskii. On a class of systems of transcendental equations. Soviet Math. Dokl.. 22 11980). 762-763. A. G. Khovanskii, Fewnomials and Pfaff Manifolds. Proc. Int. Congress of Mathematicians, Warsaw 1983.

IKh31

A. G. Khovanskii. Real Analytic Varieties with t h e Finiteness Property and Complex Abelian Integrals, Functional analysis and i t s applications (Translated from Russian). 18 (1984), 119-127.

[K-P-S]

J. Knight. A. Pillay, C. Steinhorn, Definable Sets in Ordered Structures I!. preprint. S. tojasiewicz. Ensembles Semi-Analytiques, mimeographed notes, IHES. 1965.

A. Pillay and C. Steinhorn, Definable Sets in Ordered Structures I, preprint. ,

I

J. J. Risler. Complexite et Geometrie Reelle (d'apres A. Khovansky), Seminaire

Bourbaki. no. 637, Nov. 1984. [Rol

M. Rosenlicht, The Rank of a Hardy Field, Trans. AMS 280 (1983). 659-671.

IS1

M. Singer, Asymptotic behavior of solutions of differential equations and Hardy fields, preliminary report, SUNY a t Stony Brook. 1976 (unpublished).

[TI

A. Tarski, A Decision method for Elementary Algebra and Geometry, 2nd ed. revised, Berkeley, and Los Angeles, 1951.

IWI

H. Wolter. this volume.

L O G K COLLOQUIUM '84

91

J. 6. Paris, A . J . Wilkie, and C.M. Wiliiiers (Editors1 0 Blsei,ier Science Piiblisliers 6. V. lNorfh-Holland/, 1986

SITUATION SCHEMATA AND SYSTEMS OF L O G I C RELATED TO SITUATION SEMANTICS J e n s Erik Fenstad U n i v e r s i t y o f Oslo

Logic and l i n g u i s t i c s have o v e r t h e a g e s l i v e d i n a s o m e t i m e s uneasy r e l a t i o n s h i p . From t h e s t r o n g m e d e v i a l b r e w o f m e t a p h y s i c s , l o g i c a n d grammar c h a r a c t e r i s t i c o f t h e m o d i s t a e ( e g . t h e G r a m m a t i c a Spec u l a t i v a o g Thomas o f E r f u r t , 1 3 1 5 ) w e h a v e g o n e t o t h e modern' extreme o f t h e America1 s t r u c t u r a l i s t school (L. B l o o m f i e l d , 2. H a r r i s , 1 9 3 0 - 5 0 ) , where " t h e o r y " s i g n i f i e d a scheme f o r d i s c o v e r y p r o c e d u r e s rather t h a n a n a t t e m p t a t e x p l a n a t i o n and u n d e r s t a n d i n g . Such w e r e t h e e x t r e m e s : t h e r e w e r e a l w a y s w e l l r e a s o n e d p o s i t i o n s i n t h e m i d d l e g r o u n d , e g . 0. J e s p e r s e n ' s P h i l o s o p h y of Grammar from 1 9 2 4 . But i t s e e m s f a i r t o s a y t h a t N. Chomsky's S y n t a c t i c S t r u c t u r e s f r o m 1 9 5 6 marked a t h e o r e t i c a l r e n e w a l o f l i n g u i s t i c s c i e n c e . From a l o g i c i a n p o i n t o f v i e w w e s h o u l d n o t e t h a t w i t h Chomsky o n e bond w a s f o r g e d between t h e t w o s c i e n c e s c e n t e r i n g around formal l a n g u a g e t h e o r y , s y n t a c t i c p a r s i n g a n d o t h e r aspects of "computational linguistics". Meaning w a s a l w a y s a r e c o g n i z e d " b o x " i n t h e Chomskian schema, b u t i t w a s o n l y w i t h t h e work of R . Montague f r o m 1 9 6 7 t h a t a t e c h n i c a l l y a d e q u a t e meaning component w a s j o i n e d t o t h e s y n t a c t i c p a r t , and a r i c h e r p i c t u r e emerged. Of n e c e s s i t y t h e r e w a s i n a n i n i t i a l p h a s e s o m e t i m e s c o n f u s i o n a n d m i s u n d e r s t a n d i n g - n o t a l w a y s of t h e q u i t e k i n d , a n d n o t a l w a y s o f s u b s t a n c e . Sometimes matters o f n o t a t i o n t o o k p r e c e n d e n c e ; s o m e t i m e s g r a n d e m p i r i c a l c l a i m s w a s r e a d o f f a r b i t r a r y n o t a t i o n a l schemata. Today w e r e c o g n i z e t h e f u n d a m e n t a l c o n t r i b u t i o n s of N . Choinsky a n d R . Montague, b u t we a p p r o a c h t h e c u r r e n t problems i n t h e s p i r i t o f a c u m m u l a t i v e s c i e n c e , e x p l o i t i n g t h e i n s i g h t s a n d r e s u l t s of t h e research community, which means both t o a c c e p t a n d t o r e j e c t . I t i s d o u b t f u l whether h i g h e r o r d e r i n t e n s i o n a l l o g i c a s d e v e l o p e d by R . Montague i s t h e r i g h t way t o s t r u c t u r e t h e " w o r l d " : w h a t i s beyond d o u b t i s t h a t Montague s e t a n e x a m p l e f o r w h a t i t means f o r a t h e o r y o f grammar t o j o i n l i n g u i s t i c form a n d m e a n i n g . Remark. N . Chomsky a n d R . Montague a r e t h e " p u b l i c f i g u r e s " o f t h e t h e o r e t i c a l r e n e w a l . A s a l o g i c i a n I would l i k e t o r e c a l l a "missed opportunity". Around 1 9 5 0 Y. B a r - H i l l e l r e s u r r e c t e d t h e c a t e g o r i a l grammar o f S . L e s n i e w s k i ( 1 9 2 9 ) a n d K. A j d u k i e w i c z ( 1 9 3 6 ) . Few h a v e r e a d L e s n i e w s k i ' s paper " G r u n d z u g e e i n e s n e u e n s y s t e m s d e s G r u n d l a g e n d e r M a t h e m a t i k " : h i s i d e a s were made more a c c e s s i b l e t h r o u g h A j d u k i e w i c z ' s p a p e r " D i e S y n t a k i s c h e K o n n e x i t a t " The 1 9 5 0 ' s w a s t h e t i m e o f g r e a t hopes f o r m a c h i n e t r a n s l a t i o n a n d B a r - H i l l e l i n t e n d e d

.

92

J.E. FENSTAD

c a t e g o r i a l grammar as t h e p r o p e r t h e o r e t i c a l frame-work enterprise.

for this

B u t t h e e n t e r p r i s e f a i l e d , c h i e f l y b e c a u s e t h e r e was no a d e q u a t e t h e o r y of meaning. But something d i d e x i s t a t t h e t i m e . Hans Reichenbach had i n c l u d e d a c h a p t e r on " c o n v e r s a t i o n a l l a n g u a g e s " i n h i s t e x t book from 1947, Elements of Symbolic Logic, u s i n g as h i s t e c h n i c a l tools h i g h e r o r d e r l o g i c e x t e n d e d by c e r t a i n " p r a g m a t i c " operators.

The t w o p a r t s remained s e p a r a t e , no one s a w or was a t a l l i n t e r e s t e d i n how t o c o n n e c t the c a t e g o r i a l a n a l y s i s of Ajdukiewicz w i t h t h e s e m a n t i c a l a n a l y s i s of Reichenbach, which, p e r h a p s , i s a b i t n o t e worthy s i n c e one o f t h e e x p l i c i t s o u r c e s f o r L e s n i e w s k i ' s c a t e g o r i e s was R u s s e l ' s t h e o r y o f t y p e s , t h e v e r y l o g i c a l formalism Reichenbach b u i l t on. However, i n d e p e n d e n t l y of B a r - H i l l e l and Reichenbach, H a s k e l l 9 . Curry r e a d a p a p e r i n t h e l i n g u i s t i c s e m i n a r conducted by Z . H a r r i s , Some l o g i c a l a s p e c t s o f g r a m m a t i c a l s t r u c t u r e . The p a p e r was w r i t t e n i n 1948, b u t w a s f i r s t p u b l i s h e d i n 1961. I n it Curry g i v e s a n anal y s i s o f t h e t r a d i t i o n a l p a r t s o f s p e e c h i n t e r m s o f combinatory l o g i c , t h u s combining both a s y n t a c t i c and a s e m a n t i c a n a l y s i s . B u t no one t o o k n o t i c e a t t h e t i m e . Acknowledgement. I n t h i s t a l k I r e p o r t on some j o i n t work w i t h P.K. Halvorsen, T. Langholm and J . van Benthem. T h i s w i l l be p u b l i s h e d i n E q u a t i o n s , Schemata and S i t u a t i o n s ( t o a p p e a r ) : h e n c e f o r t h r e f e r r e d t o a s ESS. From l i n g u i s t i c form t o meaninq The meaning o r i n f o r m a t i o n a l c o n t e n t o f an u t t e r a n c e i s d e t e r m i n e d by a number of c o n t e x t u a l f a c t o r s a s w e l l a s t h e l i n g u i s t i c form i n a s t r i c t s e n s e ( i . e . t h e phonology, morphology and s y n t a x ) , and t h e i n t e r p r e t a t i o n m u s t s a t i s f y a l l t h e c o n s t r a i n t s imposed by a l l t h e r e l e v a n t a s p e c t s o f t h e l a r g e r " u t t e r a n c e s i t u a t i o n " . There seems t o b e no s o l i d e m p i r i c a l e v i d e n c e t h a t one component h a s primacy o v e r t h e o t h e r s i n a r r i v i n g a t t h e meaning c o n t e n t , t h u s t h e r e i s no nece s s i t y i n t h e o r t h o d o x p o i n t o f view which r e q u i r e s o n e t o c h a n n e l t h e f u l l i n f o r m a t i o n a l c o n t e n t of t h e u t t e r a n c e and i t s c o n t e x t through a t r a d i t i o n a l s y n t a c t i c s t r u c t u r e i n o r d e r t o a r r i v e a t t h e meaning o f t h e u t t e r a n c e . On t h e c o n t r a r y , e x p e r i e n c e h a s shown t h a t one had t o resort t o a number o f c o m p l i c a t e d d e v i c e s i n o r d e r t o c o n s e r v e s t r i c t composit i o n a l i t y i n t h e p a s s a g e from a s t a n d a r d s y n t a c t i c s t r u c t u r e t o a r e l a t i o n a l s e m a n t i c form. C o m b i n a t o r i a l i n g e n i u i t y c o u l d overcome m o s t d i f f i c u l t i e s , and t h e l i t t e r a t u r e on Montague grammar i s f u l l o f p r e t t y examples. But i n g e n i u i t y can be a n a r t i f i c e . And, more s e r i o u s l y , d i f f i c u l t i e s remained, e . g . w i t h a n a p h o r i c r e f e r e n c e . Awareness o f t h i s l e d u s i n ESS t o choose a d i f f e r e n t p e r s p e c t i v e . W e would l i k e t o r e p r e s e n t t h e c o n s t r a i n t s which a r e imposed o n t h e i n t e r p r e t a t i o n o f an u t t e r a n c e by i t s c o n t e x t u a l and l i n g u i s t i c c o n s t i t u e n t s t h r o u g h a c u m u l a t i v e system o f c o n s t r a i n t e q u a t i o n s ,

Situation Schemata and Systems of Logic

93

reducing thereby the problem of systematically determining the meaning content to the problem of finding a consistent solution to the constraint equations. In our actual system in ESS we are more modest. We have chosen a format which we call situation schemata as a theoretical notion convenient f o r summing up information from linguistic form and certain other aspects of the utterance situation. In ESS we present a grammar for a fragment of English in the style of lexical functional grammar ( L F G ) , see Kaplan and Bresnan ( 1 9 8 2 ) , i.e. we have a simple context-free phrase structure augmented by constraint equations which are introduced in the main by the syntactic rules and in the lexicon, but could also possibly come from other features of the utterance situation. The first step taward the semantic interpretation is to seek a consistent solution to the constraint equations which, if it exists, can be represented in an array or tabular form. Such a representation, which is analogous to an f-structure in LFGtheory, is what we call a situation schema. There is some similarity between the notion of a situation schema and the intuitive notion of "logical form", which should not be confused with the notion of a well-formed expression in a standard logical formalism. A simple declarative sentence has a main "semantic predicate" relating a number of "actors" playing various "r6les". Both the predicate and the r8le actors play can be modified (by adjectives, adverbs, prepositional phrases, relative clauses). This suggests the following basic format for situation schemata. Let @ be a simple declarative sentence. Situation Schema of

@

:

This means that SIT.@, which we use to abbreviate the situation schema associated with @, is a function with argument list REL, ARG n, LOC (and possibly others as we will see in the ARG 1 , . next section). The value of SIT.@ on the arguments is given by the predicate and r6les of the sentence These values can either be simple, e.g. SIT.$ (ARG k) can be the name of an actor, or it can be a complex entity, i.e. SIT.@ (ARG k) can be some complex NP. In the latter case we get a new function or subordinate "box" as value determined by the structure of the NP (the determiner, the noun, the optional relative clause). The value of SIT.@ on LOC will be derived from the tense marker of

. .,

@.

@.

Remark. We have presented the notion of situation schema as a refine-ment of the LFG notion of f-structure: recall also the "syntaxe structurale" of Tesnidre ( 1 9 5 9 ) . We could also have started from the "discourse representation systems" of Kamp (1981 1, which gives a level of representation comparable to the "logical form" of situation schemata. Note, hmever, that we do not claim any psychological reality for our representional level. It would be inte-

J.E. FENSTAD

94

resting, even pleasing, but not necessary sis. The notion of situation schema could various semantic theories, in particular, partially designed to fit - the format of Barwise and Perry (1983).

for our theoretical analyalso be abstracted out of the format fits - and was situation semantics, see

We shall recall some basic notions from situation semantics, see Barwise and Perry (1983), Barwise (these proceedings). Situation semantics is grounded in a set of primitives: S

situations

A

locations individuals

D R

relations

For our immediate purposes we do not worry about the ontological status of the primitives; as mathematicians we assume that they come with some structure. A minimal requirement is that each relation in R is provided with an a-rity, i.e. a specification of the number of arguments slots or r6les of that relation. For the moment we impose no structure on the set of situations and individuals. The set A is or represents connected regions of space-time. Thus A may be endwed with a rich geometric structure, and should be if we were to give an analysis of seeing that which would correctly classify verb phrases describing spatio-temporal processes. Here we are much more modest and assume that A comes endowed with two structural relations: < temporally precedes

temporally overlaps,

0

to account for a simple-minded analysis of past and present tenses. Primitives combine to form facts which are either located or cated. Let r be an n-ary relation, R a location and al, individuals. The format of a basic located fact is

e-

... fan

at at

I R

...,an;l ...,an;O

:

r,al,

:

r,al,

where the first expresses that at location R the relation r an; the second expresses that it holds of the individuals a l , does not hold. The basic format of unlocated facts is

...,

. . . ,an;1 . .,an;O

r,al, r,al,.

In addition to basic facts we have "atomic" assertions concerning the location structure: I

L

0

1'

R

1'

I

wholly temporally precedes 1' and I' temporally overlap.

Situation Schemata and Systems of Logic

95

s determines a set of facts, but is not in the settheoretic sense a set of facts. This distinction is not so important for our present discussion, but will be of crucial importance in the semantic analysis of attitude verbs (see Barwise and Perry (1983). p. 223) The "primitive" relation connecting situations, locations, individuals and relations is

A situation

.

in s in s

: :

...,an:l ...,an:O

at I( : r,al, at 1 : r,al,

the first expressing that in the situation s at location I( the relation r holds of a l , an. We have a corresponding reading of the second and of the unlocated versions.

...,

The meaning of a (simple declarative) sentence @ is a relation between an utterance situation u and a described situation 8 . We shall use the situation scheme SIT.@ to spell out the connection between u and s, and we write the basic meaning relation as

u [sIT.$]s. We refer the reader to Barwise and Perry (1983). Chapter 6, for a full discussion and motivation for the chosen relational format of the meaning relation. In the next section we shall give a formal definition of situation schema associated to a fragment of English and use this to give a semantic interpretation of the fragment in a system of situation semantics.

Situation Schemata Take a simple sentence such as A girl handed the baby a toy

This sentence can be generated by a simple context free grammar. Indeed, any proposed grammatical analysis of such a specimen will be almost identical to the context free analysis: S

+

NP VP Det

+

N V

+

+ +

+

NP VP Det N V NP NP a, the girl, baby, toy handed

This gives the following syntax tree for the sentence:

J.E. FENSTAD

96

handed

the baby

a

toy

We want to convert this tree into a format better adapted for further semantic processing. From the point of view of "logical form" we have a main predicate handed and three r6les a girl, the baby, a toy. The standard contex free analysis gives us a tree where one of the r8les is separated from the predicate and the other actors. In this case it is not difficult to supplement the contex free structure with "constraint equations" leading to a functional form of the type "a girl" "the baby" "a toy" "past tense locationBvl

ARG 2

1

LOC

We shall not enter into details of how to convert the syntax tree to schematic form: this is discussed in detail in part 1 of ESS. As remarked above we follaw the pattern of LFG theory, and the reader is also urged to consult Kaplan and Bresnan ( 1 9 8 2 ) . The schematic form above is only a first step toward a full unravelling of r6les actors play. In ESS we have proposed a formal definition of situation schema adapted to a simple fragment of English. The definition is given as a set of rewriting rules, using standard notational conventions including the Kleene *-notation: SIT.SCHEMA SIT REL" ARGi LOC POL IND SPEC COND CONDlOC

+ + +

+ + +

+

+ + +

...

(SIT)RELn ARGl ARGn LOC POL <situation indeterminate, IND (SPEC COND (SIT.SCHEMA)*) IND CONDloC I0111

{l<entity>) (SIT) REL ARGl POL RELloc ARGl ARG2

Here entity stands for a proper name, quantifier for determiners and RELloc is either < or o such as a, the, some, most, (see our discussion of the basic ideas of situation semantics above) The value of POL is either 1 or 0, compare the basic format of facts in situation semantics.

...,

.

91

Situation Schemata and Systems of Logic

In ESS we have an algorithm which the context free structure and the constraint equations converts a grammatically correct sentence of the fragment to a situation scheme, and every correctly generated situation schema is the schema derived from a sentence of the fragment. This algorithm has been implemented as an extension of the implementation of the LFG algorithm. With our simple declarative sentence we arrive at the following situation schema: REL hand ARG 1

IND

IND 1

SPEC

a

I ' ,

girl

CoND

ARG 2

IND SPEC COND

IND 2 the

POL ARG 3

I IND

I IND

I

IND 3

ARG 1

LOC

1

IND4

IND3

I

The complete situation schema is a function of arguments REL, ARGl, ARG2, ARG3, LOC. The value on REL is a "constant", the 3-ary relation constant hand. The values on the other arguments are composite functional structures. Let 9 be our sample sentence, and let SIT.9 denote the above situation schema. Let u be an utterance situation and s a described situation. We have to explain the basic meaning relation U[SIT.$]S. This we cannot do in full details in this talk, see ESS part 2 for the complete story. The reader will see that we have a fairly straight-forward recursion on the occurrences of SIT.SCHEMA within SIT.9. There are a number of problems with quantifier scopes, anaphorie reference and definite descriptions which we evade in

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98

a g a i n we r e f e r t h e r e a d e r t o E S S . c o n c e n t r a t e o n a few s i m p l e p o i n t s . this brief exposition,

H e r e we

i s a f u n c t i o n (or, r a t h e r , h a s a f u n c t i o n a s v a l u e ) which a n c h ors the d e s c r i b e d f a c t r e l a t i v e t o the d i s c o u r s e l o c a t i o n as g i v e n by u . T e n s e c a n be i n t e r p r e t e d i n many ways: t a k i n g a s i m p l e m i n d e d r e f e r e n t i a l and " u n l o c a t e d ' ' p o i n t o f view w e s a y t h a t ( t h e p a r t i a l f u n c t i o n ) g a n c h o r s t h e l o c a t i o n o f S I T . $ , S I T . $ LOC, i n t h e d i s course s i t u a t i o n u i f

LOC

w h e r e 1, i s t h e d i s c o u r s e l o c a t i o n d e te r m i n e d by the r e l a t i o n of temporally precede.

u

and


0 ,

b.

comes before choosing the Cn-type of

is not one of the first b

b

Cn+l-

comes before choosing the Cn+l-type of

b < c, then choosing the C -type of Choosing the C -type of

if it is consistent with the

5 n.

tasks on the list.

should do.

Worker n

then Worker n

-

If c.

Choos-

Witnessing a C -type

is the first to suggest 1 replaces the constant

by a different one, and the other workers make further suggestions. The constant that is finally adopted is one suggested by Woker

(D(N))'

is a job for Worker At stage 0 ,

Worker s

Worker n

n does nothing new.

-

- pns-ls-l.

Worker n

0 < j < s,

pn k = p n

0.

Putting a number e

into

0. 0 does nothing but determine pn o. For many stages s, That is, pn - pns-1 for all s'i 1 and

-

never does more than one thing new at a given stage s,

pn and the new work is always displayed in pnss.

that

It may be that for some i such s s-1 for k < j, pn k = p n j-l for j c k < s . and

pnss # pnss-l. This happens when Worker n corrects a mistake or does some fol- '-li for all low-through on a witnessing task. Or, it may be that p ni-'n s-1 i < s, and pn # pn s-l. This happens when n attempts some new task in

J.F. KNIGHT

108

forming

n' s . s-1 At stage s > 0, Worker n checks for j 5 i in deciding whether n' j s-1 s-1 0. Suppose that to make pn - pn i. There is never any mistake in pn s-1 s-1 there is no mistake in pn for j < i. If pn # P, i-l, then n' i must have remained unchanged (as t varied) since state i. In checking s-1 i i, Worker n considers the support of p '-li to be the support of pn pn If n discovers a mistake, then pn - pns-li-l for (i k < s , and the corrected version of

pn

'-' (if available at stage

s)

appears in

The

pnSs.

upper part of the support for pn (the part produced by Workers n + 1 and n) i is the same as for but the lower part of the support (the part produced by pn i, co Worker n - 1, if n > 0 ) has grown. Worker n will look at P,-~ s , not just m in making the correction. pn-l i, If there are no mistakes discovered at stage s , Worker n may do some follow-through on a witnessing task.

does at

Whatever follow-through Worker n

'-'

stage s will make and will make pn the same as Pn s-1 - n' s-2' s-1 s-l except for a change in witnessing constant. In choosing the new witnesspn ing constant, Worker n Worker n

counts on Worker n

If

r6,x)

+

pn-l

m

and does not look up at all.

1 to wait while this is going on.

Cn-type, then Worker n may initiate the task of wit-

is a

r@,x)

nessing

looks down at

in forming

pn s

by choosing a special witnessing constant

and trying to find a type that amalgamates r(?i,x)

n > 0, Worker n waits for Worker n

Then, if

pnss-l.

chosen constant b

will alert Worker n

trying to witness r ( 8 , x ) .

Worker n

-

-

-

-

1

to respond.

1 will replace the constant

b

The is

by a new

that is consistent with

n's

choice

in waiting for Worker n

-

to

of type.

Then Worker n

respond.

Finally, Worker 0 suggests a constant that everyone can accept.

1 joins Worker n

b

from

1 to the fact that Worker n

and will choose a type for b'

constant b',

with the type C(b,v)

2

If r = Rn j' then the witness-

Each constant carries the history of apparently correct work. and

Worker n

first attempts to witness

'

-f

r(c,x)

will be (n,j,z,(n,s)). ing constant used in p n s with their Gzdel numbers.) Suppose that at stage

s,

at stage

(Here tuples are identified t > s,

Worker n

discovers a

The mistake will actually appear in pnt-lr for some 1, where is just like pnSs except for the n' r (The fact that the work on the task, the choice of amalgam type, appears

mistake in this attempt. r

such that

constant.

s

5

r

5

t

-

t-l instead of t-l results from the follow-through procedure.) If in n' r n' s Worker n has a new amalgam type to try at stage t, then this appears in p n t' + with the witnessing constant (n,j,c,(n,t)). If stage s

r

is a

C -type for some r > n, and Worker n

to respond to Worker n

T ( f , x ) , then n

+

1's

first attempts at

use of a witnessing constant a

will use the constant b = a^(n,s)

in

pnss.

for

If Worker n

finds

109

Effective Construction of Models a mistake and attempts to do its part of this task again at stage t,

then the

constant used in p will be aA(n,t). n t Suppose that Worker n is doing the follow-through on the witnessing of r(z,x),

where

r

Worker n

is either a

C -type for r > n.

Cn-type or a

Suppose that in

either initiated the witnessing or made an initial response to

pnss, something that Worker n

+

1 did on the task.

Suppose that no mistake has been

and follow-through is still called for.

found up to stage t

Let b

constant appearing in pnt-lt-l, and suppose that Worker n

be the

is choosing the new

t

constant to use, with the same type, in pn t. There are two cases. First, suppose that there is a "stable" response from Worker n this means is that and

b

OD

pn-l

has a constant d

are both descended from the constant introduced by

there have been responses by

Worker k

-

1.

What

with a history indicating that it

for all k < n.

n at stage

s,

and

In this case, Worker n

in pntt. If no mistakes are found later, this will become The follow-through process is complete. Second, suppose there is no stable

uses the constant d 03

pn t. response from n

-

m

1 (pn-l does not have any witnessing constant descended from the one introduced by Worker n at stage s). Then in p tt, Worker n

uses the constant b' = bA(n,t).

At the next stage, pntt will be erased, and a

If Worker n

m

will not appear in p n t' finds nothing to correct at stage s, and n does not

new constant will be used in p t+lt+l,

so

b'

anticipate any further response from Worker n

'.

-

1, then n may attempt some-

The method for choosing which task to attempt, if any, thing new in forming pn s will be discussed below. The rules will assure that if Worker n attempts to do the task of choosing the Cn-type of

b,

then b
s .

n > 0, then there is at most one constant appearing in p and not in n s Worker and the constant tells what task was attempted in forming pnss. 0 may introduce more than one new constant at a time, in trying to control the If

jump of the open diagram. If Worker n is to attempt some new task in forming p

then the fol-

s,

lowing conditions must be met.

(1) Worker n

+

1 must appear to have done, in pn+l

s-1

all tasks of Type (a) that come before the task that Worker n (2)

Worker n must have done, in pnsj

for j 5 s

-

j

for j 5 s - 1, is now considering

1 (in an apparently

correct fashion), all possible tasks that come before the one being considered. (3)

If n > 0, then Worker n

-

1 must have done, in P,-~

everything that could be done before Worker n (4)

m

for j

5

s,

does more.

The task being considered must be among the first s

must be some one of the first s tasks that still needs work.

tasks, so there

J.F. KNIGHT

110

(5)

I t must seem t o

Worker n

t h a t t h e t a s k being considered can a c t u a l l y s-1 f o r j 2 s - 1, and n p n+l sj

be done so as t o m a i n t a i n c o n s i s t e n c y w i t h

must b e a b l e t o see f o r s u r e t h a t t h e proposed If

-

Worker n

1 c o u l d o n l y do f i n i t e l y many t a s k s .

that for a l l

pn-l

t > to,

m

m

= pn-l

c a n do, i n c l u d i n g f o l l o w - t h r o u g h

When

p

Worker n - 1 r e s p o n d s . Worker r

T h e r e would be some Workern-1

( f a r above

n

s-1'

'n-1 then such

to

Workern-1

must l o o k

Worker n - 1 i s

Worker

n

repeats certain

+

i n i t i a t e s t h e w i t n e s s i n g of

n)

S .

h a s done a l l t h a t i t

c a n t e l l when

worker n

g o i n g t o be f o r c e d i n t h i s way t o l o o k a t Suppose t h a t

'-'

on w i t n e s s i n g t a s k s , t h e n

f o r new c o n s t a n t s a p p e a r i n g a b o v e . things u n t i l

m

is consistent with n s s-ly n e v e r d i d a n y t h i n g new beyond what i s i n P,

Worker n

p

r(c,x).

Worker r - 1 w i l l p r o b a b l y hand down some i n c o r r e c t r e q u e s t s f o r h e l p i n w i t n e s s i n g , t h e n a c o r r e c t r e q u e s t , a n d t h e n some i m a g i n a r y r e q u e s t s . requests don't contribute t o Could t h i s p r e v e n t

Worker n

m

pr-l

U

tEu from e v e r

The i m a g i n a r y

b u t t h e y may b e s e e n i n

t,

m

U

tEu

'r-2

g e t t i n g t o work on c e r t a i n t a s k s ?

pose t h a t t h e r e is some f i r s t Type ( a ) t a s k , c h o o s i n g t h e

Cn-type

of

t'

Supt h a t is

b,

n o t d o n e b e c a u s e o l d w i t n e s s i n g t a s k s ( e a r l i e r on t h e master l i s t ) k e e p i n t e r Suppose t h a t t h e most complex of t h e o l d w i t n e s s i n g t a s k s i n v o l v e s a

fering.

Worker r

C -type.

tie o l d t a s k s .

w i l l e v e n t u a l l y s t o p i n t r o d u c i n g new w i t n e s s i n g c o n s t a n t s f o r n < m < r.

Let

+

Worker m

If

1 p r o d u c e s some

p dlj

(that

Worker m

w i l l l o o k a t when d e c i d i n g what t o d o n e x t ) w i t h no new w i t n e s s i n g conk s t a n t s f o r t h e o l d t a s k s , t h e n Worker m w i l l n e c e s s a r i l y p r o d u c e some m ' k 1 w i l l l o o k a t ) w i t h no new w i t n e s s i n g c o n s t a n t s f o r t h e o l d ( t h a t Worker m

-

tasks.

T h i s means t h a t e v e n t u a l l y

of

b.

r

= Rn+li

;low d o e s and

the f i r s t

c

Worker n

t h e s e matches

c($,y),

and

u s e s a l i m i t e d p o r t i o n of Rn+'-index

i.

If

n

where

P";

namely,

f i n d s t h a t o n e of

then

Rn-l

n

by

pn-l

with the

Rn-index

I f o n e of

i.

knows t h a t t h e t y p e s are c o n s i s t e n t .

Other-

j' assumes t h a t t h e y are n o t c o n s i s t e n t .

n

How d o e s

Worker n

poss-l.

poss-l

=

a s s o c i a t e d by extends

z,

is some

k

x(z,z), Po

up t o

attempt a t s t a g e

s

t h e t a s k of c h o o s i n g a

s-1

Worker 0 h a s g u e s s e s a t p1 s-1 L e t t h e g u e s s a t p1 be where

F i r s t , consider

and knows

let

t y p e s associate:

s

t h e s e f u l l y matches

b?

r(2)

C -type

R n . up t o s , t h e n n assumes t h a t t h e t y p e s are c o n s i s t e n t . 3 S i m i l a r l y , i f r = R n . a n d 1 = Rn-' then a t stage s , n

looks a t the f i r s t

for

n

s,

3 types associated with the

s

Otherwise, n o t .

wise,

c h e c k t h e c o n s i s t e n c y of

At stage

= Rn.?

w i l l be a b l e t o choose t h e

Worker n

n = 0.

where

z(t,?)

w i t h t h e index

s.

such t h a t

r(t),

.

s-1

for

I-(;)

C -type

j < s = R

1

i, and

Worker 0 l o o k s a t t h e f i r s t s t y p e s j s e a r c h i n g f o r some A(G,;,z) such t h a t A 0 l o o k s a t t h e f i r s t s R - i n d i c e s t o see i f t h e r e

Then 0 0 R k matches

= Ro

i,

A,

UP

to

S .

I f these searches are successful,

Effective Construction of Models Worker 0

then

t h i s , however, s i s t e n t with

h a s i n mind a s s i g n i n g t h e

j

Worker 0

sistent,

pn s-l

knows

r(z)

= R~+'

m

Pn-1 s ency of

r

Therefore, i f

is con-

U

c

a p p e a r s t o b e con-

u n t i l a f t e r t h i s t a s k a p p e a r s t o b e done.

Worker m

pn-l

Pn s-l

Let

i'

r(z) U C(x,y)

of u C, Rn+l -index (i,j'),

B e f o r e doing poss

is c o n s i s t e n t , t h e n a n a p p r o p r i a t e t y p e

C

n

= C(a,c),

where

and of

P,+~ s-1 'n+1 s-1

C(x,y) = Rn

C(x,y) U @(z,$,z,z).

looks a t t h e f i r s t

s

s-1

for

where

checks t h e c o n s i s t -

I n checking t h e consistency

t y p e s a s s o c i a t e d by

Pn

with t h e

Rn. up t o S . Then i,j' nJ s Rn-indices a s s o c i a t e d by Q with t h e pair ++ -+-+ ++ -+ RnR = A(x,y,z) s u c h t h a t A(x,y,z) u @ ( x , y , z , u )

looks a t t h e f i r s t hoping t o f i n d some

Pn

j < s

I-(:),

be

and l e t

j'

Worker n

++

hoping t o f i n d some

i,

++

++ + n-1 @(x,y,z,u) = R k.

where

Worker n

has guesses at

L e t t h e guess a t

S.

++

++

Worker n

U

t > s)

n > 0.

and

= @(z,z,b,&,

r

:AzAb.

may make e r r o r s , b u t t h e r e i s no lower worker t o worry a b o u t .

Next, c o n s i d e r and

to

c o n s i d e r s t h e t a s k t o b e p o s s i b l e and r e f r a i n s from going on

t o t h e next t a s k ( a t stages Worker 0

r

knows t h a t i f

w i l l e v e n t u a l l y t u r n up.

k

k

Worker 0 makes a f i n a l check t h a t t h i s proposed s-1 p1 f o r i 5 s.

Worker 0 and i n d e x

Ro-index

111

t h a t matches

is c o n s i s t e n t . Note t h a t mind a s s i g n i n g t h e t h i s proposed

p

c o n s i s t e n t and

to

2!

2zAb.

for

j

is consistent unless it

If

j < S.

'n s-1 w i l l e v e n t u a l l y t u r n up.

A

Worker n

has i n

makes a f i n a l check t h a t

is s t a b l e ) , then

(i.e.,

a p p e a r s t o b e c o n s i s t e n t , Worker n

U C

Worker n

is consistent with 1s " c o r r e c t "

b e c o n s i s t e n t , and t h e d e s i r e d

r

@

I f t h e s e a r c h e s are s u c c e s s f u l , t h e n

Rn-index

n s

u

w i l l not believe t h a t

Worker n

actually is consistent.

C

r

U C

U

0 should

is

Therefore, i f

considers t h e t a s k t o be possi bl e

and d o e s n o t go - on t o t h e n e x t t a s k ( a t l a t e r s t a g e s ) u n t i l a f t e r t h i s t a s k h a s been done.

s-1

s-1 '-' s-1

Pn+1

s

'n

s-1

f o r j < s - 1 and Pn+l j - Pn+1 s-1 i s c o n s i s t e n t , t h e n once Worker n i s c o r r e c t a b o u t

s-1

If

s-1

and i t s c o n s i s t e n c y w i t h

t a s k i s done c o r r e c t l y o r n o t a t a l l . that

Worker n

-

p

s-l,

--

t h e r e w i l l b e no m i s t a k e s

the

T h i s i s i m p o r t a n t b e c a u s e i t means

1 c a n n o t do t o o many t a s k s u n t i l

Worker

n

has finished

t h i s one. How d o e s r(:,u)?

Worker n

p +n+; s-1 = A ( c , d , a ) , where ~

and l e t

'n s-1 at t h e f i r s t s one t h a t , up t o

i n i t i a t e a t stage s

s-1

Suppose t h a t

using t h i s type f o r

p

S.

Rn-index

Worker n

t o use i n

S.

C(Z,d),

A(x,z,y) = R n Pn

a p p e a r s t o complete

t h a t matches t h i s one up t o mind t h e

+++

+

t y p e s a s s o c i a t e d by s,

the taskofwitnessing a

appears t o be j' with t h e

+++

A t stage

Rn+l-index

A(x,z,y) U I'($,u),

looks f o r a n

s,

Rn-index

C(x,z) = R Worker n i.

then k < s

I f t h i s s e a r c h is s u c c e s s f u l , then

Pn s .

C -type

+ +

where

n+l

i,

looks

I f there is n

considers f o r a type n

has i n

The c h o i c e of c o n s t a n t was d e s c r i b e d above.

112

J.F. KNIGHT

Worker n

is consistent with pnss If t h e search is unsuccessful, then

makes a f i n a l c h e c k t h a t t h e proposed

s-1

for

co

and w i t h

j < s

pnV1 S . assumes t h a t t h e t a s k d o e s n ' t need d o i n g and g o e s on.

Worker n

Worker r ,

Suppose t h a t a w i t n e s s i n g t a s k w a s i n i t i a t e d by

r > n,

and

++

appears t o be and

L:(:,y)

C(a,b),

where Let

= Rn+li.

Worker n index

i s t r y i n g t o respond i n

Worker n

i.

least up t o

++

pnss-l

= A(a,c),

Cn-types

s

I f t h e r e i s one, t h e n

S .

t y p e t h a t matches

up t o

0

that

n

k

in

m

and

L:

If

S .

k

A

U

t h a t extends

looks f o r an

Rn-index

is consistent, then

Worker 0

with t h e

+

R1-index

s,

a t least up t o

+++

..

J

0

A

Worker n-1

(D(IJO)'?

into

e

+

r ( x ) = R1

hopping t o f i n d a t y p e D(N)

would c a u s e

Worker 0

If

S .

up t o k

t h i s with

for

'-'

g o e s on t o

Suppose t h a t

++

and l e t

i'

s

= C(a,b), Po s-1 t y p e s a s s o c i a t e d by Po

+++

A(x,y,z)

S .

to

IpeD(OZ)(e)

++

finds

A(z,b,c),

c,

aAbAc i n

having

t o c o n v e r g e , w i t h a com-

w i t h a c o n v e r g e n t computa-

R - i n d i c e s t o see i f t h e r e is some

I f t h e searches are successful, then

j
0 and v(p) i s n o t d i v i s i b l e by 2 i n t h e value group. This i s e a s i l y checked. Indeed, using v a r i a n t s of t h e above has a unique s t r u c t u r e of valued f i e l d s u b j e c t e q u i v a l e n c e s , one sees t h a t Q, only t o t h e c o n d i t i o n t h a t v(p) i s n o t i n f i n i t e l y d i v i s i b l e . A more e l a b o r a t e argument u s i n g H e n s e l i z a t i o n s and uniqueness of Henselian s t r u c t u r e ( s e e [Nagata 19621) w i l l show t h a t Q has a unique s t r u c t u r e of valued f i e l d i f w e demand P

128

A. MACINTYRE

only t h a t v(p) is n o t i n f i n i t e l y d i v i s i b l e t h a t t h e r e s i d u e class f i e l d i s n o t a l g e b r a i c a l l y c l o s e d . But, perhaps u n f o r t u n a t e l y , i t i s p o s s i b l e t o f i n d a s t r u c t u r e o f v a l u e d f i e l d o n Q w i t h v ( p ) > 0, v ( p ) i n f i n i t e l y d i v i s i b l e , and P t h e r e s i d u e c l a s s f i e l d a l g e b r a i c a l l y c l o s e d . I d o n ' t know i f t h i s h a s b e e n Anyway, t h e example i s s o u n i n t e r e s t i n g t h a t I omit d e t a i l s . pointed out before. S i m i l a r c o n s i d e r a t i o n s apply t o

Fp((t)),

p r o v i d e d one u s e s

i n the definitions.

t

But i n f a c t t h e v a l u a t i o n i s a l g e b r a i c a l l y d e f i n a b l e w i t h o u t u s i n g t , a n d t h i s i s an i n s t a n c e o f a n i n t e r e s t i n g g e n e r a l theorem. C . U. J e n s e n , ( u n p u b l i s h e d ? ) c o m p l e t i n g e a r l i e r work of Ax a n d F r e y , showed t h a t i f K i s a n o t a l g e b r a i c a l l y c l o s e d v a l u e d f i e l d s a t i s f y i n g H e n s e l ' s Lemma t h e n t h e v a l u a t i o n i s a l g e b r a i c a l l y definable

.

From t h i s w e c a n deduce t h a t f o r a l l l o c a l l y compact f i e l d s e x c e p t C one h a s an I t i s a worthwhile e x e r c i s e t o a l g e b r a i c a l l y d e f i n a b l e b a s i s f o r t h e topology. show t h a t f o r C one c a n n o t d e f i n e a b a s i s . 3.2. The p r e c e d i n g i m p l i e s t h a t w e can i n t e r p r e t , f o r e a c h l o c a l l y compact f i e l d e x c e p t R and C, P r e s b u r g e r a r i t h m e t i c , i . e . t h e t h e o r y o f t h e o r d e r e d g r o u p Z. I t i s q u i t e e a s y t o see t h a t P r e s b u r g e r i s n o t i n t e r p r e t a b l e i n C , b u t The p r o o f of t h e l a t t e r r a t h e r less o b v i o u s t h a t i t c a n n o t be i n t e r p r e t e d i n R. seem t o need c y l i n d r i c d e c o m p o s i t i o n f o r R ( s e e [ C o l l i n s 19751).

*

3.3. Analopy 2. T h i s c o n c e r n s t h e t o p o l o g y on t h e m u l t i p l i c a t i v e group K , where K i s r e s p e c t i v e l y R o r 2QP. K = R . The group o f s q u a r e s (K ) i s o f i n d e x 2 i n K*, a n d t h e two c o s e t s

*

~

d i s c o n n e c t K*. The c o s e t s (which are e v i d e n t l y open i n K) are r e p r e s e n t e d by 1 and -1. Each c o s e t h a s i n K a s i n g l e boundary p o i n t 0. K = Q

n 3 2.

F i x any

(K*)"

i s of f i n i t e i n d e x i n

open i n K) are r e p r e s e n t e d by e l e m e n t s o f boundary p o i n t 0.

Z.

The c o s e t s (which are

K*.

Each c o s e t h a s i n

3.4. Analogy 3. I n b o t h c a s e s (K*)" i s e f f e c t i v e l y open i n e x p l a i n e d by t h e f o l l o w i n g two remarks. 3.4.1.

K = R.

3.4.2.

K = Q P and

x c (K*)n

If

.

x C (K*)"

and

IX-Y~

< 1x1

then

3.5.1.

3.5.2. K = R.

in

If

(K*)"

The n o t i o n i s

.

I

f C K[x] then

* n y € (K )

a single

Take t h e u s u a l n o r m a l i z a t i o n of Ix-yI P-(1+2v(n)). Ix then y

3.5. Analogy 4. Formal Completeness Schemata. v e r s i o n which f o l l o w s from 3.4.

of

K*.

K

and a r b i t r a r i l y c l o s e t o

I f i r s t g i v e an u n c m v e n t i o n a l

a

f

takes v a l u e s i n two c o s e t s

f ( a ) = 0.

The u s u a l s c h e m a t a a r e :

Sign Change Scheme: [a,b].

If

f(x)

changes s i g n o n

[a,b]

K=Q H e n s e l Scheme: I f f C Z p [ x ] , f monic, and f P' modulo p , then f has a unique r o o t a congruent t o

then

f

has a root

has a simple r o o t modulo

p.

Remarks (1). [ C h e r l i n 19761 h a s a s u g g e s t i v e d i s c u s s i o n o f t h e above a n a l o g y . ( 2 ) There a r e many u s e f u l v a r i a n t s o f t h e H e n s e l Scheme. See f o r example [Ribenboim 19681.

p

Twenty Years of p-adic Model Theory 3.6. and

129

Analogy 5. F i n i t e E x t e n s i o n s . The o n l y a l g e b r a i c e x t e n s i o n s of C, and t h e a b s o l u t e G a l o i s group GR is 2 / ( 2 ) .

By K r a s n e r ' s Lemma ([Lang 19641) one c a n show t h a t f o r e a c h f i n i t e l y many e x t e n s i o n s o f d e g r e e

n.

n

Not a l l e x t e n s i o n s o f

e a c h normal e x t e n s i o n i s s o l v a b l e ( [ S e r r e 19651).

R

are

R

QP has O n l y a r e normal, b u t

Qp

The p r o f i n i t e group

is

G QP

t o p o l o g i c a l l y g e n e r a t e d by f i n i t e l y many e l e m e n t s , and g e n e r a t o r s and r e l a t i o n s a r e known ( [ S e r r e 19651). This knowledge i s h i g h l y r e l e v a n t t o a l o g i c i a n aiming t o e l i m i n a t e q u a n t i f i e r s i n some n a t u r a l language. The key problem of t h i s e l i m i n a t i o n w i l l b e : a.

When does

+

+ ... +

alx

anxn

have a r o o t ?

This can b e s u g g e s t i v e l y r e p h r a s e d a s : Which of t h e f i n i t e l y many e x t e n s i o n s o f dimension a.

+

a x 1

+

... + a

5 n!

p r o v i d e s t h e r o o t s of

xn?

One e x p e c t s t o need a u x i l i a r y p r e d i c a t e s c o r r e s p o n d i n g t o each of t h e f i n i t e l y many t y p e s of e x t e n s i o n of dimension 5 n! For example, f o r

R,

of t h e s q u a r e s i n

R

*.

t h e e x t e n s i o n s of dimension

5

the

nth

powers a r e r e l e v a n t f o r

$4.

METHODS FOR UNDERSTANDING

Q

P'

and o f c o u r s e

correspond t o t h e c o s e t s

2

The remark a b o u t s o l v a b i l i t y of

should suggest t h a t QP 3.1 confirms t h i s . G

Th(R) :

4.1. Q u a n t i f i e r e l i m i n a t i o n . T h i s was t h e o r i g i n a l method of T a r s k i . I t was l a t e r e c l i p s e d by t h e methods of model-completeness and s a t u r a t e d models, b u t from t h e 1 9 7 0 ' s on, w i t h t h e emphasis on complexity o f computation, i t was a g a i n prominent. The n a t u r a l language f o r t h e e l i m i n a t i o n i s t h a t of o r d e r e d r i n g s . one c a n n o t e l i m i n a t e q u a n t i f i e r s , a s t h e example e l i m i n a t i o n problem e a s i l y r e d u c e s t o t h a t f o r

+

( ~ Y ) [ P ( Y , x ,= 0 where

p

j

2

= x)

Without o r d e r

shows.

The g e n e r a l

-+

Ant\

and t h e

(3y)(y

j ( ~ , x )> 01

q

5

are i n

Z[y,;].

Another i n e v i t a b l e problem i s t o c a l c u l a t e ,

+

a s t h e f u n c t i o n of x, t h e number of z y o s o f p ( y , z ) i n R. Except f o r t h e x s u c h t h a t a l l y c o e f f i c i e n t s o f p ( y , x ) v a n i s h , t h i s number i s bounded by t h e f o r m a l y-degree o f p. Once t h e e x a c t number o f r o o t s is known, t h e r o o t s can b e d i s t i n g u i s h e d b y t h e i r p o s i t i o n i n t h e o r d e r d', if p i s l a r g e enough t h e n any f o f d e g r e e d i n n v a r i a b l e s h a s a n o n z e r o s o l u t i o n . Of c o u r s e n i s i r r e l e v a n t ( p u t some v a r i a b l e s e q u a l t o z e r o ) , s o i n

Twenty Years of p-adic Model Theory f a c t w e g e t a f u n c t i o n F(d) s o t h a t i f n t d2 and degree d i n n v a r i a b l e s has a nonzero s o l u t i o n i n

139

p 2 F(d)

then any

f

of

QP.

The n a t u r a l i t y of t h e method has been confirmed by a s e r i e s of examples showing t h a t F(d) cannot be chosen t o be 0. One should c o n s u l t t h e r e c e n t [LewisMontgomery 19831 f o r r e f e r e n c e s , and f o r some new i n f o r m a t i o n of a q u a n t i t a t i v e k i n d . The method of Ax-Kochen does n o t guarantee a p r i m i t i v e r e c u r s i v e F, r e c u r s i v e F. For what improvements l o g i c can give though i t does give an C0-

i s n o t Ci f o r any i. See Lewiss e e S e c t i o n 10. I t i s now known t h a t Q P Montgomery f o r an i n t r i g u i n g c o n j e c t u r e about t h e counterexamples f o r f i x e d p. The s t r u c t u r e o f t h e s e counterexamples i s c e r t a i n l y a n a t u r a l t o p i c i n t h e advanced d e f i n a b i l i t y theory of Q

P'

5.6. Defects i n t h e analogy. There has been almost no p r o g r e s s on understanding t h e elementary theory of F p ( ( t ) ) . F u r t h e r , one has t h e ominous r e s u l t t h a t F p ( ( t ) ) w i t h c r o s s - s e c t i o n i s undecidable ( d e s p i t e t h e f a c t t h a t c r o s s - s e c t i o n i s allowed i n t h e Ax-Kochen a n a l o g y ) . Apparently Ax f i r s t proved t h i s , and i t was r e d i s covered by Jacob. The d e t a i l s i n s p i r e d C h e r l i n ( [ C h e r l i n 19841) t o show t h a t f o r valued f i e l d s of c h a r a c t e r i s t i c p t h e r e i s no p o s s i b i l i t y of Ax-Kochen isomorphism theorems p r e s e r v i n g c r o s s - s e c t i o n and r e s i d u e f i e l d . I n f a c t , almost always the l a s t "and" can be r e p l a c e d by a n "or".

Ax's main o b s e r v a t i o n i s t h a t i n t h e f i e l d of r e s i d u e

0,

i . e . t h e series

F ( ( t ) ) one can d e f i n e t h e elements P Zantn where a-l = 0 . (The underlying computa-

t i o n i s well-known i n l o c a l - c l a s s - f i e l d

theory.)

Recall t h a t

t

i s not definable.

5.7.

Algebraic e x t e n s i o n s of Q Since every f i n i t e e x t e n s i o n of Q i s of t h e P' P Q (a) where a i s a l g e b r a i c over Q (an easy consequence of K r a s n e r ' s P is i n t e r p r e t a b l e i n Q and s o d e c i d a b l e . Lemma), every f i n i t e e x t e n s i o n of Q P P However, t h e s e methods a r e a l i t t l e too crude f o r i n f o r m a t i v e d e f i n a b i l i t y r e s u l t s . form

The i s s u e w a s n o t immediately considered. However i n 1 9 7 4 Kochen published an e l e g a n t paper o u t l i n i n g t h e m o d i f i c a t i o n s n e c e s s a r y f o r t h e analogues of t h e o r i g i n a l Ax-Kochen isomorphism theorems. For unramified e x t e n s i o n s no modifications a r e needed, b u t i n t h e g e n e r a l c a s e one has t o d i s t i n g u i s h an a a s above. I n p a r t i c u l a r , t o c a p t u r e t h e theory of Q (a) one needs more than j u s t t h e r e s i d u e P f i e l d and t h e degree o f r a m i f i c a t i o n . There a r e no s u r p r i s e s , and adding crosss e c t i o n makes no e x t r a complication. Note t h a t Hypothesis B a p p l i e s . I do n o t know i f anyone wrote o u t t h e Ax-Kochen analogy i n t h i s s e t t i n g , b u t t h e r e a r e no d i f f i c u l t i e s i n f i n d i n g a n a t u r a l formulation. Because of l a c k of space I leave the exercise t o the reader. (The v e r s i o n i n Kochen i s n o t q u i t e what I have i n mind.) Recently B e l a i r , van den Dries and I noted t h a t even i f K = Q (a), K P need n o t be of t h e form L(p) where L G Q P' I n f i n i t e Extensions. As pointed o u t i n 5.4 t h e Ax-Kochen approach t o t h e mixed c h a r a c t e r i s t i c c a s e (valued f i e l d of c h a r a c t e r i s t i c 0, r e s i d u e f i e l d c h a r a c t e r i s t i c p) works b e s t f o r f i n i t e r e s i d u e f i e l d . The reason i s t h a t i n o t h e r cases t h e r e w i l l be no obvious way t o keep t h e r e s i d u e f i e l d f i x e d from t h e beginning of t h e c o n s t r u c t i o n of an isomorphism, w i t h consequent complications i n t h e study of non-immediate e x t e n s i o n s . Recall t h a t t h e r e i s no problem under Hypothesis 0 (cf. 5.4).

Ax and Kochen found a b r i l l i a n t t r i c k f o r reducing t h e mixed c a s e under Hypothesis B t o Hypothesis 0. This i n v o l v e s an e l a b o r a t i o n of t h e o b s e r v a t i o n t h a t s a t u r a t e d p-adically c l o s e d f i e l d s a r e ( g e n e r a l i z e d ) power s e r i e s f i e l d s over Q P'

A. MACINTYRE

140

[Kochen 19741 e x p l o i t e d t h e o b s e r v a t i o n t o prove d e c i d a b i l i t y of t h e maximal unramified e x t e n s i o n of Q Here w e have a Henselian f i e l d K where t h e r e s i d u e P c l a s s f i e l d is a l g e b r a i c a l l y c l o s e d , t h e v a l u e group i s 2 , and v ( p ) = 1. The i d e a is t o look a t a s a t u r a t e d e x t e n s i o n L w i t h v a l u e group r ( n e c e s s a r i l y s a t u r a t e d ) . 2 is a convex subgroup of r , and s o r/2 has a n a t u r a l o r d e r . The map

.

gives a new v a l u a t i o n

v1 : L* -+ r/Z.

L* i s Henselian w i t h r e s p e c t t o

vl,

and

r e a d i l y seen t o be s a t u r a t e d .

r/Z i s d i v i s i b l e and s a t u r a t e d .

v1

has c h a r a c t e r i s t i c

r i n g of

v.

F

The main p o i n t i s t h a t t h e r e s i d u e f i e l d F f o r can be i d e n t i f i e d thus. Let V be t h e v a l u a t i o n

Consider t h e p r o j e c t i v e system of r i n g s

jective l i m i t acteristic

0.

R.

0.

F

R

V/pn

(n 2 l ) ,

and i t s pro-

can be i d e n t i f i e d w i t h

V/npnV, which i s a domain of charn i s n a t u r a l l y isomorphic t o t h e f i e l d of f r a c t i o n s of R.

It follows then from t h e Hypothesis 0 a n a l y s i s t h a t L w i l l be i d e n t i f i e d , a s a f i e l d w i t h v a l u a t i o n vl, once F i s i d e n t i f i e d a s a f i e l d . Since v(p) = 1, t h e argument of shows t h a t v (L,v) w e need only i d e n t i f y F.

i s algebraically definable, so to identify

We may suppose F 5 L. The r e s t r i c t i o n of v t o F t a k e s v a l u e s i n 2 , and F i s complete under v. The r e s i d u e f i e l d of v on F i s ( c a n o n i c a l l y ) t h e r e s i d u e f i e l d of v on L , and s o i s s a t u r a t e d . As Kochen p o i n t s o u t , F i s then uniquely determined a s a Witt-Teichmuller c o n s t r u c t i o n o v e r the r e s i d u e f i e l d .

I t follows e a s i l y t h a t t h e theory of K is axiomatized by t h e theory of i t s r e s i d u e f i e l d , p l u s t h e i n f o r m a t i o n t h a t t h e v a l u e group i s a 2-group and v(p) = 1. The d e c i d a b i l i t y of t h e maximal unramified e x t e n s i o n of Q follows. (Note t h a t P Kochen s l i p s up by i d e n t i f y i n g t h e above f i e l d w i t h t h e maximal cyclotomic extens i o n of Q .) P By t h e above technique one may handle a l l Hypothesis B f i e l d s , a s f a r a s complete axiomatizations a r e concerned. This i n c l u d e s a l l unramified e x t e n s i o n s of Q P' and t h e r e i s no t r o u b l e i n h a n d l i n g f i n i t e r a m i f i c a t i o n . But, as f a r a s I know, t h e r e has been no s y s t e m a t i c s t u d y of model-completeness, l e t alone q u a n t i f i e r elimination. There a r e two i n t e r e s t i n g i n f i n i t e e x t e n s i o n s of

Q which e l u d e t h e above methods. P The f i r s t i s t h e maximal a b e l i a n e x t e n s i o n , w i t h a l g e b r a i c a l l y c l o s e d r e s i d u e f i e l d

and value group

{m/pk : m,k C 2 ) .

The o t h e r i s t h e t o t a l l y ramified e x t e n s i o n g o t

p j th r o o t s of u n i t y , f o r a l l and value group as above. gy a d j o i n i n g t h e

j.

This has r e s i d u e f i e l d

F P

A c l u e t o t h e i r a n a l y s i s comes from some work of van den Dries (unpublished, e a r l y 1980's). H e looked a t t h e g e n e r a l mixed c h a r a c t e r i s t i c Henselian c a s e , w i t h no r e s t r i c t i o n s on v a l u e group r. I n s t e a d o f t h e convex subgroup Z of I' one now t h e convex subgroup generated by v ( p ) . The r o l e of r/Z i s taken considers rP' by T/rp. With t h i s m o d i f i c a t i o n , v1 i s c o n s t r u c t e d as b e f o r e . The r e s i d u e f i e l d

admits t h e d e s c r i p t i o n given b e f o r e , v i a R = & V/pnV. By a t y p i c a l l y c l e v e r s h o r t p r o o f , van den Dries (unpublished) obtained t h e following completeness theorem:

Twenty Years of p-adic Model Theory The theory of t h e mixed c h a r a c t e r i s t i c Henselian f i e l d c h a r a c t e r i s t i c p, v a l u a t i o n r i n g V and v a l u e group t h e o r i e s of (T,v(p))

and

V/pnV,

K,

r,

141 w i t h r e s i d u e f i e l d of a s determined by t h e

n ? 1.

I n o t i c e d i n w r i t i n g t h i s paper t h e f a c t t h a t f o r n > 1 V/pnV i s a W i t t cons t r u c t i o n ([Jacobson 19641) over V/pV, so i n van den Dries' r e s u l t one needs only (I',v(p)) and V/pV.

These n e a t o b s e r v a t i o n s have n o t y e t produced new a p p l i c a t i o n s . Even f o r t h e i n f i n i t e e x t e n s i o n s l i s t e d above V/pV i s from a l o g i c a l p o i n t of view q u i t e complex ( a l o c a l r i n g of c h a r a c t e r i s t i c p with l o t s of n i l p o t e n t e l e m e n t s ) .

5.8. Cohen's Elimination. Cohen's work was published i n 1969, b u t a v a i l a b l e i n 1966-7. The most obvious advances over Ax-Kochen-Ersov a r e (i)

a p r i m i t i v e r e c u r s i v e d e c i s i o n procedure f o r

(ii)

Qp;

p r i m i t i v e r e c u r s i v e bounds i n t h e Ax-Kochen analogy.

However, t h e following elements of h i s proof have proved of g r e a t e r i n t e r e s t and utility : (1) H i s e l i m i n a t i o n t a k e s p l a c e i n s i d e a f i x e d Henselian f i e l d , using only Hensel's Lemma, and i n p a r t i c u l a r , no g l o b a l Ostrowski-Kaplansky theory i s needed;

(2)

H e g i v e s a procedure f o r " i s o l a t i n g " t h e r o o t s of polynomials, and simul-

taneously reducing c o n d i t i o n s t h e c o e f f i c i e n t s of f .

+

(the

kth

r o o t of

f)

t o s i m p l e c o n d i t i o n s on

Cohen's proof i s r a t h e r demanding. The r e a d e r i s l u r e d i n t o i t by a memorable r e a l analogue ( a q u a n t i f i e r e l i m i n a t i o n f o r R u s i n g only t h e Sign Change Property and a s i m i l a r i s o l a t i o n / e l i m i n a t i o n technology)

.

I t i s n o t a t all easy t o say e x a c t l y what Cohen proved. Anyone w i t h a s p i r a t i o n s t o r e s e a r c h i n this a r e a m u s t m a k e a p a t i e n t s t u d y of t h e paper (and w i l l be rewarded). Roughly, e f f e c t i v e f u n c t i o n s ( l i k e kth r o o t ) can be e l i m i n a t e d i n favour of a s t o c k of f u n c t i o n s on the v a l u e group and on t h e v a r i o u s r e s i d u e r i n g s , t o g e t h e r w i t h c r o s s - s e c t i o n and a c o n s t a n t f o r an element of v a l u a t i o n 1. Cohen works under t h e assumption of c h a r a c t e r i s t i c 0 f i e l d and d i s c r e t e v a l u a t i o n i n t o a 2-group.

How t h e n can he o b t a i n i n f o r m a t i o n about

Fp((t))?

H i s g e n e r a l method produces,

f o r a b a t c h of f ( x ) of degree 4 n, what he c a l l s an e f f e c t i v e graph f o r f . That i s , v a r i o u s f u n c t i o n s a s s o c i a t e d w i t h t h e geometry of f a r e e l i m i n a b l e i n terms of f u n c t i o n s a t t h e l e v e l of t h e v a l u e group o r t h e r e s i d u e r i n g s . For given n, t h e r e d u c t i o n w i l l be v a l i d even i n c h a r a c t e r i s t i c p, provided p i s b i g g e r than a c e r t a i n p r i m i t i v e r e c u r s i v e f u n c t i o n of n. ( T y p i c a l l y t h e r e d u c t i o n s f a i l because of s i n g u l a r i t i e s of f t h e r e s i d u e f i e l d . ) Then one shows e a s i l y t h a t i f

+

i s given

Z/pn

and

Fp[t]/tn

agree on

+

for

n

l a r g e enough, thus y i e l d i n g

t h e Ax-Kochen analogy. F ( ( t ) ) w i l l reach t h a t P i f we a c q u i r e s y s t e m a t i c d e s i n g u l a r i z a t i o n techniques i n c h a r a c t e r i s t i c p .

This approach s t r o n g l y s u g g e s t s t h a t o u r understanding of of

Qp

Observations on t h i s can be found i n [Roquette 19771, [McKenna 19801, and v a r i o u s t a l k s by Denef. I n p a r t i c u l a r one should be looking f o r r e s t r i c t e d d e c i s i o n / e l i m i n a t i o n methods, s a y f o r f as above w i t h some l i m i t a t i o n s on i t s complexity.

142

A. MACINTYRE

5.9. The analogue of H i l b e r t ' s 17th Problem. I n 4.6.8 I mentioned Robinson's approach t o H i l b e r t ' s 17th Problem. T h i s s u b t r a c t s nothing from t h e p a r t of t h e o r i g i n a l proof connecting order and sums of squares, b u t t r i v i a l i z e s t h e s p e c i a l i z a t i o n argument. [Kochen 19691 e s t a b l i s h e d a very s a t i s f y i n g p-adic analogue. The o r d e r - t h e o r e t i c " p o s i t i v e d e f i n i t e " is replaced by the v a l u a t i o n - t h e o r e t i c " i n t e g r a l d e f i n i t e " -e s s e n t i a l l y x z 0 i s replaced by v(x) 2 0. (Note t h a t t h i s i s not e n t i r e l y reasonable -- by our e a r l i e r discussion v(x) 5 0 corresponds t o 1x1 5 1.) The main d i f f i c u l t y i s t o f i n d t h e analogue of the sums of s q u a r e s / o r d e r l i n k . Kochen found such an analogue. One notable d i f f e r e n c e i n the two cases is t h a t on R one can bound, i n terms of t h e number of v a r i a b l e s t h e complexity of t h e sums of squares r e p r e s e n t a t i o n . No such bound i s known i n Kochen's analogue. 56.

TEN YEARS LATER:

As i n d i c a t e d above, some important work of consolidation was done by Z i e g l e r and Weispfenning. Beyond t h a t , the scene quietened very r a p i d l y . The next important advance came from a r e c o n s i d e r a t i o n of some p e r i p h e r a l work I had done on t h e R/Qp analogy i n 1966. Robinson 1959 had proved t h a t t h e theory of p a i r s ( L , K ) , of real-closed f i e l d s with K a proper dense s u b f i e l d of L , i s complete. H i s proof i s not a t t r a c t i v e . He proved model-completeness i n a language which c e r t a i n p r e d i c a t e s f o r a l g e b r a i c dependence over K. I noticed i n 1966 completeness theorem has a memorable proof by s a t u r a t e d models. About t h e same time Cohen sketched a c o n s t r u c t i v e e l i m i n a t i o n theorem f o r t h i s problem.

I went on ( i n 1966) t o e s t a b l i s h Analogy 11. The theory of p a i r s (L,K) where K , L a r e real-closed (resp. p-adically closed) and K i s a proper dense s u b f i e l d of L i s complete. I n both cases t h e proof uses t h e theory of c l o s u r e s and some lemmas about dense transcendence bases. I n 1974 I had many d i s c u s s i o n s with P e t e r Winkler i n connection with h i s t h e s i s begun under Robinson and f i n i s h e d under my d i r e c t i o n . We catalogued t h e f a m i l i a r t h e o r i e s i n terms of a l g e b r a i c boundedness and Vaughtian p a i r s ( s e e Macintyre 19751). Algebraic boundedness f o r R comes from t h e q u a n t i f i e r - e l i mination, and the l a c k of Vaughtian p a i r s f o r Th(R) comes from T a r s k i ' s observ a t i o n t h a t i n f i n i t e d e f i n a b l e sets have i n t e r i o r , t o g e t h e r with t h e lemma t h a t an i n t e r v a l i n an ordered f i e l d has t h e c a r d i n a l i t y of t h e f i e l d . The l a t t e r i s used i n Analogy 11. 'Ibis s e t m e thinking f i r s t about a l g e b r a i c boundedness and and then about an a b s t r a c t (topology-free) formulation Vaughtian p a i r s f o r Q P' of t h e theorem about dense embeddings. Straightaway I c a m up a g a i n s t t h e problem whether an i n f i n i t e d e f i n a b l e s u b s e t of Q has i n t e r i o r . P

The answer i s "0, i f one allows cross-section a s p r i m i t i v e , and yes i f one does n o t . The former i s easy ([Macintyre 1976]), b u t t h e l a t t e r i s n o t . One has t o understand the d e f i n a b l e sets, and t h a t tends t o mean t h a t one must e l i m i n a t e q u a n t i f i e r s i n terms of some n a t u r a l p r i m i t i v e s . I solved t h e problem by focussing on elements which had occurred i n Ax-Kochen, but were used t h e r e only a s a book-keeping device ( t o reduce t o the case of pure value-group). The r e l e v a n t p r i m i t i v e s a r e t h e s e t s Analogy 12. (a) The language and P2.

field

R

pn

of

nth

powers.

has q u a n t i f i e r - e l i m i n a t i o n i n terms of t h e f i e l d

Twenty Years of padic Model Theory (b)

all

The Pn.

field

Q

P

143

has q u a n t i f i e r - e l i m i n a t i o n i n terms of t h e f i e l d language and

I t can be shown t h a t f o r

Q

P From 1 2 t h e r e follow quickly:

all

Pn

a r e needed.

Analogy 1 2 .

(K = R

or

Q ).

I n f i n i t e d e f i n a b l e sets have i n t e r i o r s .

Analogy 13. pairs.

(K

= R

or

Qp).

Th(K)

P

i s a l g e b r a i c a l l y bounded and has no Vaughtian

.

Analogy 1 4 . (K = R o r Q See [ P i l l a y 19831). P n o t have the independence property.

Th(K)

i s unstable b u t does

My proof of q u a n t i f i e r - e l i m i n a t i o n i n [Macintyre 19761 used q u i t e a l o t of t h e

Ax-Kochen paraphernalia. Much l a t e r [Weispfenning 19841 gave a p r i m i t i v e recurs i v e q u a n t i f i e r - e l i m i n a t i o n i n my formalism.

I n a very u s e f u l t e x t published i n 1984, P r e s t e l and Roquette extended the method As w e l l a s t h e Pn one must d i s t i n g u i s h a generator t o f i n i t e extensions of Q P a as i n 5.7.

.

I do not know i f t h e case of i n f i n i t e (even unramified) extensions has been checked out.

Tung r a i s e d i n 1984 t h e p o s s i b i l i t y t h a t a i s i r r e l e v a n t f o r quantifier-elimination. IYasumuto 19851 r e f u t e d t h i s f o r a ramified extension using i d e a s of [ E . Robinson 1985Bl. However, van den Dries

57.

DEVELOPMENTS I N THE

Pn

FORMALISM:

7.1. Model-theoretically t h e next s t e p should have been t o i d e n t i f y t h e universal i n the P -formalism. I n f a c t t h i s has been done only r e c e n t l y , by theory of

B

[Robinson 19831 and [ B e l a i r 19851, whose work I d i s c u s s l a t e r . For real-closed f i e l d s w e have q u a n t i f i e r - e l i m i n a t i o n using e i t h e r < 0

by

D 2 ( f ( xl , . . . , x n ) ) .

Dn9

I n [Robinson 19831 t h e f o l l o w i n g a n a l o g y w a s o b t a i n e d : Analogy 19. using

A,V

(K

=

or

R

and t h e

Q

P

1.

Any d e f i n a b l e open s u b s e t o f

Kn

can be defined

Dn.

7.5. S p e c t r a . The work o f B e l a i r a n d Robinson, though p e r f e c t l y n a t u r a l i n t h e c o n t e x t o f t h e c l a s s i c a l model t h e o r y o f f i e l d s , w a s a c t u a l l y a f o u n d a t i o n a l component of t h e c o n s t r u c t i o n o f t h e p-adic s p e c t r u m of a commutative r i n g . Here one w a s p u r s u i n g t h e a n a l o g y w i t h t h e i n t e n s i v e l y s t u d i e d real s p e c t r u m o f a commut a t i v e r i n g ( s e e Coste-Roy 19821. T h e r e i s a w e a l t h o f material o n t h e t o p i c , f o r example i n [Robinson 19831 i n which one w i l l f i n d many s o p h i s t i c a t e d i d e a s from c a t e g o r i c a l l o g i c . There had b e e n r e g r e t t a b l y l i t t l e t r a d e between c a t e g o r i c a l l o g i c a n d t h e more c o n v e n t i o n a l a p p l i e d model t h e o r y , u n t i l R o b i n s o n ' s work. H e u s e s t h e work o f Cohen and m e , b u t g i v e s back f i n e i n f o r m a t i o n o n c o h e r e n t a x i o m a t i z a t i o n s as w e l l as p r e s e r v a t i o n theorems f o r v a r i o u s classes o f l o c a l r i n g s w i t h p-adic s t r u c t u r e . Both h e and B e l a i r r e l a t e s p a t i a l l y of t h e p-adic s p e c t r u m t o r i g i d i t y o f p-adic c l o s u r e . Lack o f s p a c e p r o h i b i t s f u r t h e r d i s c u s s i o n of t h i s e x c i t i n g work ( i n d i s p e n s i b l e f o r any f u t u r e p-adic a l g e b r a i c g e o m e t r y ) . I recommend t h a t t h e r e a d e r m a k e a s e r i o u s s t u d y o f the t h e s e s [ B e l a i r 19851 and [Robinson 19831, and t h e p a p e r s [Robinson 19858, B ] a n d I B e l a i r 1985Al.

7.6.

More o n d e f i n a b l e s e t s i n

Kn.

Analogy 18 was r e a l l y v e r y weak, u n l e s s w e

c o u l d b e t t e r v i s u a l i z e t h e d e f i n a b l e sets i n Kn. T h i s h a s b e e n done i n [van den Dries-Scowcroft 19851. I w i l l c i t e j u s t two more a n a l o g i e s : Suppose S Km h a s nonempty i n t e r i o r , and Analogy 20. (K = R o r 9,). n S = U Si, e a c h Si d e f i n a b l e . Then some Si h a s nonempty i n t e r i o r . i=l

Think o f t h i s as a "formal Baire theorem". Analogy 21.

(K = R

e q u a l t o a union

n

or

U Si,

i=l

1. L e t S b e a d e f i n a b l e s u b s e t o f Km. P where e a c h Si i s d e f i n a b l e , and f o r e a c h

Then

Q

i

S

is

either

Si

i s open o r h a s no i n t e r i o r and i s homeomorphic by a b i a n a l y t i c p r o j e c t i o n a l o n g

c e r t a i n c o o r d i n a t e a x e s t o a n open s u b s e t o f some

Kr

where

r < m.

7.7. Dimension. I f w e m a k e a s l i g h t r e f o r m u l a t i o n o f Analogy 21 w e r e a c h a p l a u s i b l e d e f i n i t i o n o f real and p-adic d i m e n s i o n . Namely, w i t h K f i x e d , d e f i n e n t h e d i m e n s i o n o f S as t h e l a r g e s t d s u c h t h a t S i s a u n i o n U Si, e a c h i=1 Si d e f i n a b l e , and f o r some i Si i s homeomorphic, b y a b i a n a l y t i c p r o j e c t i o n a l o n g c e r t a i n c o o r d i n a t e a x e s , t o a n open s u b s e t o f d i m e n s i o n (S) = --).

Kd.

(For

S =

+

put

This w a s one o f t h e p r o p o s a l s o f [ v a n d e n Dries-Scowcroft 19851. I t l e a d s i n t h e p-adic case t o a n a t t r a c t i v e d i m e n s i o n t h e o r y ( a t l e a s t p a r t i a l l y a n t i c i p a t e d by [Robinson 1 9 8 3 1 ) .

146

A. MACINTYRE

Writing

dim

f o r dimension one then has t h e following remarkable analogy:

Analogy 2 2 . (i)

dim

(K = R

or

Km = m;

Q ).

P

(ii) I f X and Y a r e d e f i n a b l e s u b s e t s of Y; and dim(X) 5 dim(Y) i f X (iii) I_f X Ken i s d e f i n a b l e and X(d) = { a : d i m ( L ) = d ) i s d e f i n a b l e ; (iv)

X-,

=

{6

Km

dim(X

(a,g)

:

U

Y ) = max(dim(X), dim(Y)),

€ X}

t h e n f o r every

d € N

Definable maps do n o t i n c r e a s e dimension;

(v) I f X i s d e f i n a b l e , the Z a r i s k i c l o s u r e X.

dim(X)

i s equal t o the algebraic-geometric dimension of

( i i i ) i s of course remeniscent o f s t r a t i f i c a t i o n formulas i n s t a b i l i t y t h e o r y , and I would expect i t t o be worthwhile t o attempt an axiomatic dimension theory subsumThere i s a l r e a d y some unpublished work on groups i n g t h e c a s e s of C, R and Q P' (by van den Dries and Scowcroft) which confirms t h i s . I t appears a l s o t h a t i t w i l l be necessary t o have a f i n e r dimension theory f o r t h e c r o s s - s e c t i o n formalism. Scowcroft has work i n progress on t h i s .

$8. DENEF'S APPLICATION: 8.1.

Z i s t h e p r o j e c t i v e l i m i t of t h e f i n i t e r i n g s Z/pn, one knows P i f and only i f f has a s o l u f h a s a zero i n Z f € Zp[;l,...,xn] P t i o n i n each Z/p

Since

that for

.

Before passing t o t h e f i n e r a n a l y s i s of t h e above, one should observe t h a t t h e x ] unsolvable i n remark above a l r e a d y i m p l i e s t h a t t h e set of f € Z[xl,

...,

2 i s r e c u r s i v e l y enumerable. For i f f i s unsolvable, t h i s w i l l be revealed by P enumerating t h e f i n i t e r i n g s Z/pn and v e r i f y i n g u n s o l v a b i l i t y of f i n one of those.

I n 1963 Nerode managed t o show t h a t t h e s e t of

(a)

if

f

(b)

t h e s e t of a l g e b r a i c elements of

...,

x 1 solvable i n 2 P Nerode's method depended on two

f € Z[xl,

i s r e c u r s i v e l y enumerable, and thereby r e c u r s i v e . observations :

has a s o l u t i o n , i t has a s o l u t i o n a l g e b r a i c o v e r Z

P

Q;

is a recursive ring.

An a l t e r n a t i v e would have been t o prove ( t h e t r u e r e s u l t ) t h a t t h e r e i s a r e c u r s i v e i f and only i f f i s s o l v a b l e f u n c t i o n p ( f ) such t h a t f i s s o l v a b l e i n Z

P i n Z / p n f o r n = p ( f ) . A r e s u l t of t h i s kind, f o r v a r i e t i e s s i n g l e f was given i n [Birch-McCann 19671.

V

r a t h e r than a

I t i s Hensel's Lemma ( s u i t a b l y g e n e r a l i z e d ) which provides such a bound. For 2 p # 2, and v ( a ) 2 0 , x -a i s s o l v a b l e i n Z i f and only i f P v( a ) +1 i s a s q u a r e modulo p

example, f o r

a

L i f e would be much s i m p l e r i f t o t e s t s o l v a b i l i t y of f one needed only t o test s o l v a b i l i t y mod p. Even t h e above example shows t h a t t h i s d o e s n ' t work. Hensel's Lemma adds s u i t a b l e hypotheses of n o n s i n g u l a r i t y mod p.

Twenty Years of padic Model Theory

147

This s u g g e s t s c l a s s i f y i n g

2 s o l u t i o n s i n terms of t h e i r s i n g u l a r i t i e s mod pn, P The c l a s s i f i c a t i o n i s c o d i f i e d by v a r i o u s Poincare' s e r i e s , as follows. f r C Zp[xl xm]. Let Nn be t h e number of s o l u t i o n s of

each n. Fix f l fl =

,...,

,..., ... = f

= 0

mod pn,

from p-adic s o l u t i o n s .

and l e t

Nn

be t h e number of such s o l u t i o n s coming

Define t h e formal s e r i e s

m

and

P(T) =

-

Nn

E

n=O

T" N

Borevic and S h a f a r e v i c (page 63) c o n j e c t u r e d t h a t P(T) i s a r a t i o n a l f u n c t i o n of T, and t h i s was proved by Igusa and Meuser u s i n g r e s o l u t i o n of s i n g u l a r i t i e s . There a r e v a r i o u s r e l a t e d s e r i e s i n s e v e r a l v a r i a b l e s . the number of s o l u t i o n s P(T,U) =

N

n.J

pn

which l i f t t o s o l u t i o n s

For example, l e t mod pn+'.

Nn .j Consider then

be

TnUj.

[Denef 19841 showed t h i s i s r a t i o n a l , and r e l a t e d t h e r a t i o n a l i t y t o t h e l i n e a r growth of t h e f o l l o w i n g f u n c t i o n y. y ( n ) i s d e f i n e d a s t h e l e a s t y 2 n such t h a t f o r any s o l u t i o n

mod py

t h e r e i s a p-adic s o l u t i o n congruent t o

7

mod p". See a l s o IGreenberg 19661. Notice t h a t y i s d e f i n a b l e , n o t i n the pure language of f i e l d - t h e o r y , b u t i n a n a t u r a l many-sorted language having i n p a r t i c u l a r A s Weispfenning showed, t h e r e i s a good q u a n t i f i e r a s o r t f o r t h e value-group. e l i m i n a t i o n f o r t h i s formalism, and Denef observed t h a t one could deduce from t h i s alone t h a t t h e r e i s a f i n i t e p a r t i t i o n of N i n t o congruence c l a s s e s such t h a t on each c l a s s , and f o r s u f f i c i e n t l y l a r g e arguments, y i s l i n e a r . Denef's main achievement was t o show t h a t b o t h ? and P a r e r a t i o n a l , without using r e s o l u t i o n of s i n g u l a r i t i e s , r e l y i n g only on q u a n t i f i e r - e l i m i n a t i o n and some r e f i n e d work on d e f i n a b i l i t y . By s t a n d a r d formal manipulations t h e r a t i o n a l i t y i n e i t h e r case can be reduced t o : Theorem (Denef). subset.

h

Let

bounded on Then

S.

Let

S

b e a d e f i n a b l e s u b s e t of

b e a d e f i n a b l e f u n c t i o n from Let

e C N , e >_ 1.

Qm

Suppose t h a t

P

Qm contained i n a compact P' t o Q p such t h a t Ih(x) is

I

v(h(x)) C e Z U

I-)

for

x C S.

( S C R , s >. 0) f l h ( x ) Isle * ldxl S i s a r a t i o n a l f u n c t i o n of p-'.

The a n a l y s i s i s r e l a t i v e t o normalized Haar measure on r e q u i r e s proof t h a t

Ih(x)

Isle

i s measurable.

I t i s e a s i l y s e e n t h a t one need only c o n s i d e r

Also, w i t h o u t l o s s of g e n e r a l i t y

h

S

Q

P'

Note t h a t i t even

given by a c o n d i t i o n

Pn(f(x)).

i s simply a r a t i o n a l f u n c t i o n .

Denef's main i d e a i s a process of s e p a r a t i o n of v a r i a b l e s , which depends on a of a r a t h e r d e t a i l e d a n a l y s i s , of Cohen's type, of t h e graph, v i s a v i s t h e Pn, g e n e r i c polynomial g. This e n a b l e s him t o express t h e above i n t e g r a l a s the product of a s i m i l a r one i n dimension m-1 and a "Poincare' s e r i e s of Presburger

148

A. MACINTYRE m -2 kiAi(S)

type"

J(s) = (kl,

z

...,km) C

i n P r e s b u r g e r , and t h e techniques of Meuser,

Pi'l

L

where

i s a s u b s e t of

L

Zm

definable

a r e l i n e a r polynomials w i t h i n t e g e r c o e f f i c i e n t s .

Ai

-S

i s a r a t i o n a l f u n c t i o n of

J

p

.

By

Denef's paper i s n o t easy t o summarize. Aside from t h e main theorem, t h e r e a r e important c o n t r i b u t i o n s t o e f f e c t i v e Skolemization and e f f e c t i v e graphing. A t t h e end t h e r e a r e i n t r i g u i n g remarks about t h e p o s s i b i l i t i e s f o r analogous r e s u l t s w i t h cross-section. Apparently r a t i o n a l i t y can be o b t a i n e d , b u t l e s s i n f o r m a t i o n about the m u l t i p l i c i t y of p o l e s . Methodologically i t i s worth p o i n t i n g o u t t h a t Denef's proof needs n o t only quant i f i e r - e l i m i n a t i o n (over-view of d e f i n a b l e sets) b u t a l s o t h e e x i s t e n c e of d e f i n a b l e Skolem f u n c t i o n s . This has some b e a r i n g on t h e uniformity i n p of Denef's result. $9.

ALL

p

SIMULTANEOUSLY:

The o r d e r may be confusing.

The proof of t h e Ax-Kochen analogy involved

Since t h e r e s i d u e f i e l d s a r e uniformly i n t e r p r e t a b l e i n t h e and II F D D valued f i e l d s , one c e r t a i n l y cannot understand t h e theory of a l l ( o r almost a l l ) Q On t h e o t h e r hand i t without understanding t h e theory of a l l ( o r almost a l l ) F P w a s f a i r l y c l e a r from t h e Hypothesis 0 case t h a t one would understand t h e s e g l o b a l The l a t t e r t h e o r i e s of Qp once one understood t h e g l o b a l t h e o r i e s f o r t h e F P' were mastered hy [Ax 19681, and t h e consequences f o r d e c i d a b i l i t y were s t r e s s e d .

i l Fp((t)),

.

Less a t t e n t i o n was paid a t t h a t time t o d e f i n a b i l i t y . I n f a c t Ax showed q u a n t i f i e r e l i m i n a t i o n f o r t h e above g l o b a l t h e o r i e s , i n a language w i t h e x t r a p r e d i c a t e s S o l i n t e r p r e t e d by S0ln(Xl

)...,

Xn)

+

* (3y)(y"

XIY

n- 1

+

... + xn

= 0).

The e l i m i n a t i o n i s uniform f o r a l l t h e o r i e s of p s e u d o f i n i t e f i e l d s . Macintyre-van den Dries 19811. We want to make uniform

(in

p)

t h e e l i m i n a t i o n theory of $6.

1985A1) t h a t t h e r e i s a bound, independent of

p,

c o s e t r e p r e s e n t a t i v e s of

Pi.

Qg.

F i r s t note ( [ B e l a i r

f o r t h e i n d e x of

This s u g g e s t s adding t o t h e language of f i e l d t h e o r y , f o r each

See [Cherlin-

n,

(Pi)

in

constants f o r

To combine t h i s w i t h t h e e l i m i n a t i o n theorem of t h e

previous paragraph, one can appeal t o t h e comprehensive s t u d y made by [Delon 19811, and o b t a i n a uniform e l i m i n a t i o n based on t h e t h e above c o n s t a n t s , and p r e Pn, which h o l d of xl, x i f f t h e xi are i n the valuation ring d i c a t e s Sol, and

Soln(xl,

...,x

...,

h o l d s modulo t h e v a l u a t i o n r i n g .

)

The e x a c t r e s u l t i s s p e l l e d

o u t i n [ B e l a i r 198583. The i m p l i c a t i o n s of t h i s have n o t been worked o u t s y s t e m a t i c a l l y . sequence i s t h a t any s e t {p : Q p I= +], where @ i s f i r s t - o r d e r , form

{p : Fp I=

$1,

where

'p

is first-order.

One amusing coni s a l s o of t h e

I49

Twenty Years of padic Model Theory

A d e s i r a b l e a p p l i c a t i o n would b e t o u n i f o r m i t y of D e n e f ' s r e s u l t . This i s prima f a c i e b l o c k e d b e c a u s e t h e r e i s "0 u n i f o r m d e f i n a b i l i t y o f Skolem f u n c t i o n s f o r t h e S e e [ B e l a i r 198583. I s t i l l b e l i e v e , however, t h a t f u r t h e r e f f o r t s i n t h i s Qp. d i r e c t i o n are w o r t h w h i l e .

$10.

COMPLEXITY THEORY:

10.1. From t h e e a r l y 1 9 7 0 ' s s i g n i f i c a n t p r o g r e s s w a s made i n u n d e r s t a n d i n g t h e c o m p l e x i t y ( u s u a l l y i n terms o f c o m p u t a t i o n t i m e , b u t sometimes i n terms o f s p a c e ) o f t h e c l a s s i c a l d e c i d a b l e t h e o r i e s . I t i s n o t a b l e t h a t some o f t h e new a l g o r i t h m s i n v o l v e d r e f i n e m e n t s of b o t h q u a n t i f i e r - e l i m i n a t i o n and e f f e c t i v e S k o l e m i z a t i o n . L e t us f o r example c o n s i d e r t h e i d e a s i n t h e work o f C o l l i n s and Monk-Solovay Th(R).

A prenex sentence

(1)

polynomials

f(xl,

(Q,x,)

...,xn)

... (Qnxn) ...,x n l .

C Z[xl,

A(xl,.

. ., x

)

has various c o n s t i t u e n t

+

The t r u t h o f

how the z e r o p o s i t i v i t y and n e g a t i v i t y s e t s o f t h e many " r e g i o n s o f e q u i v a l e n c e " .

f

on

is t o b e d e t e r m i n e d by

decompose

into finitely

Rk

( 2 ) One h a s t o compute a f i n i t e s e t o f a l g e b r a i c n e e d s s o t h a t t h e t r u t h o f e q u i v a l e n t t o t h e t r u t h of r e l a t i v i z e d t o t h e a l g e b r a i c sample set.

+

+

is

One s h o u l d c a l c u l a t e r a p i d l y t h e t r u t h - v a l u e o f t h e r e l a t i v i z a t i o n .

(3)

( 2 ) and ( 3 ) are o b v i o u s l y r e l a t e d t o e f f e c t i v e e l i m i n a t i o n and S k o l e m i z a t i o n . Roughly, t h e q u a n t i t a t i v e r e s u l t o b t a i n e d i s t h a t t h e r e are c o n s t a n t s that with

0

as above a n d

m = l e n g t h of

A,

C1,

C2

such

t h e sample s e t can b e chosen of s i z e

.

C1( Z e n ) 2C2" ( s e e below) 2 , and Q c a n b e t r e a t e d i n t i m e 2 The s i z e of a r e a l a l g e b r a i c number i s t h e l e a s t l e n g t h o f a n i n t e g r a l p o l y n o m i a l o f which i t is a root. 10.2.

The a n a l o g u e o f t h e above o u g h t t o b e done f o r

.

Since interwets P cQp P r e s b u r g e r a r i t h m e t i c , and t h e l a t t e r c a n n o t b e d e c i d e d i n t i m e 22 ([FerranteRackoff 1 9 7 9 ] ) , i t i s r e a s o n a b l e t o c o n j e c t u r e t h a t Q and P r e s b u r g e r h a v e t h e P same t i m e c o m p l e x i t y f o r a d e c i s i o n p r o c e d u r e . This h a s c e r t a i n l y n o t b e e n has an elementary recure s t a b l i s h e d , and i n f a c t no p r o o f i s a v a i l a b l e t h a t Q s i v e decision-procedure. P Q

1 0 . 3 . Oddly enough, a t t h e l e v e l o f t h e Ax-Kochen a n a l o g y , e l e m e n t a r y r e c u r s i v e bounds h a v e b e e n o b t a i n e d . By methods a n a l o g o u s t o t h o s e o f C o l l i n s , Monk and S o l o v a y , [Brown 19781 p r o v e d : If

+

i s a s e n t e n c e o f t h e language of valued f i e l d s ,

m = length

0,

and

211m P 2 2

2

Z2

and F p ( ( t ) ) a g r e e a b o u t 0. I t i s n o t e w o r t h y t o o t h a t methods of t h i s then Q P k i n d are used i n t h e i m p o r t a n t p a p e r s [Kiehne 1979A, B] on c o n s t r u c t i v e modelcompleteness

.

A. MACINTYRE

150

10.4. Concluding Remarks. Very l i t t l e o f a s y s t e m a t i c n a t u r e h a s b e e n p u b l i s h e d on e f f e c t i v e d e f i n a b i l i t y t h e o r y f o r Q ( o r e v e n R!). There i s a w e a l t h of P t o p i c s -- Newton's method, f a s t f a c t o r i z a t i o n o f p o l y n o m i a l s , d i o p h a n t i n e a p p r o x i m a t i o n and e f f i c i e n t s a m p l e s e t s . I t seem t o m e l i k e l y t h a t t h i s w i l l n o t be t h e l a s t s u r v e y needed on t h e l o g i c o f t h e p - a d i c s . I hope t h e n e x t o n e w i l l r e p o r t s h i f t s o f e m p h a s i s t o and f r o from d e f i n a b i l i t y t o d e c i d a b i l i t y . REFERENCES : [ A r t i n 19671. E. A r t i n , A l g e b r a i c Numbers and A l g e b r a i c F u n c t i o n s , Gordon a n d Breach, 196 7. [ A x 19681.

J. A x , The e l e m e n t a r y t h e o r y of f i n i t e f i e l d s , Annals o f Math. 8 8 ( 1 9 6 8 ) , 239-271.

[Ax-Kochen 1965Al. J. Ax and S . Kochen, D i o p h a n t i n e problems o v e r l o c a l f i e l d s I , Amr. J o u r . o f Math. 87 ( 1 9 6 5 ) , 605-630. [Ax-Kochen 1965Bl. , D i o p h a n t i n e p r o b l e m s o v e r l o c a l f i e l d s 11, h e r . J o u r . o f Math. 87 ( 1 9 6 5 ) , 631-648. [Ax-Kochen 19661. , D i o p h a n t i n e p r o b l e m s o v e r l o c a l f i e l d s 111, Annals o f Math. 8 3 ( 1 9 6 6 ) , 437-456. [ B e l a i r 1985Al. L. B e l a i r , T o p i c s i n t h e model t h e o r y o f p - a d i c f i e l d s , Ph.D. T h e s i s , Yale, May 1985.

, Model

IBelair 1985Bl.

t h e o r y o f p - a d i c f i e l d s , P r e p r i n t 1985.

[Birch-McCann 19671. B . J. B i r c h and K. McCann, A c r i t e r i o n f o r t h e p - a d i c s o l u b i l i t y o f d i o p h a n t i n e e q u a t i o n s , Quant. J. Math. 18 ( 1 9 6 7 ) , 59-63. [ B o r e v i c h - S h a f a r e v i c h 19661. Academic P r e s s 1966. [Brown 19781. 1978.

Z.

I . B o r e v i c h and I . R. S h a f a r e v i c h , Number Theory

S . Brown, Bounds on T r a n s f e r P r i n c i p l e s

..., Memoirs

o f A.M.S.

204,

[ B r u m f i e l 19791. G. B r u m f i e l , P a r t i a l l y O r d e r e d Rings and S e m i - a l g e b r a i c Geometry, Cambridge 1979. [Cassels 19661. J. W . S . Cassels, D i o p h a n t i n e e q u a t i o n s w i t h s p e c i a l r e f e r e n c e t o e l l i p t i c c u r v e s , J. London Math. SOC. 4 1 ( 1 9 6 6 ) , 193-291. [Cassels 19671. P r e s s 1967. [Cassels 19781.

and A. F r o h l i c h , A l g e b r a i c Number Theory, Academic

,

R a t i o n a l Q u a d r a t i c Forms, Academic P r e s s 1978.

[ C h e r l i n 19761. G . C h e r l i n , Model T h e o r e t i c A l g e b r a , S e l e c t i c T o p i c s , L e c t u r e Notes i n Mathematics 521, S p r i n g e r 1976. [ C h e r l i n 19841. N o r t h H o l l a n d 1984.

, P a p e r i n L o g i c Colloquium 1982 ( e d .

G. L o l l i e t . a l . ) ,

, A. M a c i n t y r e , L. v a n d e n [ C h e r l i n , M a c i n t y r e , van den Dries 19811. Dries, The e l e m e n t a r y t h e o r y o f r e g u l a r l y c l o s e d f i e l d s , P r e p r i n t 1 9 8 1 , t o a p p e a r i n J. r e i n e angew. Math.

Twenty Years of padic Model Theory [Cohen 19691. P. J. Cohen, D e c i s i o n p r o c e d u r e s f o r r e a l and p-adic on Pure and Applied Math. XXII ( 1 9 6 9 ) , 131-151.

151 f i e l d s , Cow.

[ C o l l i n s 19751. G. E . C o l l i n s , Q u a n t i f i e r e l i m i n a t i o n f o r r e a l c l o s e d f i e l d s by c y l i n d r i c a l a l g e b r a i c d e c o m p o s i t i o n , pages 134-183 i n L e c t u r e Notes i n Cow p u t e r S c i e n c e 33, ( e d . H. Brakhage), S p r i n g e r 1975. [Coste-Roy 19821. M. Coste and M. F. Roy, La t o p o l o g i e du s p e c t r e &el, i n Ordered F i e l d s and R e a l A l g e b r a i c Geometry, (D. W. Dubois and T. Recio, e d s . ) , Contemporary Mathematics Vol. 8 , A . M . S . 1982. [Delon 19811. F. Delon, Quelques p r o p r i C t & d e s c o r p s v a l u e s e n t h g o r i e des modGles, T h e s i s , Univ. P a r i s V I I , 1981. [Denef 19841. J. Denef, The r a t i o n a l i t y o f t h e P o i n c a r e s e r i e s a s s o c i a t e d t o t h e p-adic p o i n t s o n a v a r i e t y , I n v e n t i o n e s Math. 77 ( 1 9 8 4 ) , 1-23. I E r s o v 19651. Ju. L. Ersov, On e l e m e n t a r y t h e o r i e s of l o c a l f i e l d s , Algebra i Logika 4 (1965), 5-30. IFerrante-Rackoff 19791. J. F e r r a n t e and C. Rackoff, The Computational Complexity of L o g i c a l T h e o r i e s , L e c t u r e Notes i n Mathematics 718, S p r i n g e r 1979. IGreenberg 19661. M. Greenberg, R a t i o n a l p o i n t s i n h e n s e l i a n d i s c r e t e v a l u a t i o n r i n g s , Publ. Math. IHES 31 (1966), 59-64.

[ H a s s e 19801.

H. Hasse, Number Theory, S p r i n g e r 1980.

IHonnander 19691. 1969. I J a c o b s o n 19641. 1964.

L. Hormander, L i n e a r P a r t i a l D i f f e r e n t i a l O p e r a t o r s , S p r i n g e r N. Jacobson, L e c t u r e s i n A b s t r a c t Algebra, Vol. 111, Van Nostrand,

[Kaplansky 19421. I . Kaplansky, Maximal f i e l d s w i t h v a l u a t i o n s , Duke Math. J . 9 (1942), 313-321. [Kaplansky 19451. 1 2 (1945), 2 4 3 2 4 8 . [Kiehne 1979Al. 9-36.

, Maximal

f i e l d s w i t h v a l u a t i o n s 11, Duke Math. J

U. Kiehne, Bounded p r o d u c t s , J . s e i n e angew. Math. 305 ( 1 9 7 9 ) ,

IKiehne 1979Bl. , Zeros o f polynomials o v e r v a l u e d f i e l d s , J . r e i n e angew. Math. 305 (1979). 37-59. [Kochen 19611. S. Kochen, U l t r a p r o d u c t s i n t h e t h e o r y o f models, Annals of Math. 74 (1961), 221-261. IKochen 19691. , I n t e g e r v a l u e d r a t i o n a l f u n c t i o n s o v e r t h e p-adic numb e r s . A p-adic analogue o f t h e t h e o r y o f r e a l f i e l d s , i n A.M.S. P r o c . Symp. P u r e Math. X I 1 (1969), 57-73. [Kochen 19751. K i e l 1974. 1975.

, Model t h e o r y of l o c a l f i e l d s , i n Logic Conference, (Ed. G . M u l l e r e t . a l . ) , L e c t u r e Notes i n Mathematics 499, S p r i n g e r

IKnight, P i l l a y , S t e i n h o r n 19851. Notre Dame 1985.

J . K n i g h t , A. P i l l a y , C. S t e i n h o r n , p r e p r i n t ,

152

A. MACINTYRE

[Lang 19521. S . Lang, On q u a s i a l g e b r a i c c l o s u r e , Annals o f Math. 55 ( 1 9 5 2 ) , 37 3- 390.

, Algebraic

[Lang 19641.

Numbers, Addison-Wesley 1964.

[Lewis-Montgomery 19831. D. J . Lewis and H . L . Montgomery, On z e r o s o f p-adic forms, Michigan Math. J o u r n a l 30 ( 1 9 8 3 ) , 83-87. [Macintyre 19681. A. M a c i n t y r e , C l a s s i f y i n g p a i r s of r e a l c l o s e d f i e l d s , Ph.D. T h e s i s , S t a n f o r d 1968. [Macintyre 19751. Dense embeddings I : A theorem o f Robinson i n a g e n e r a l s e t t i n g , i n L e c t u r e Notes i n Mathematics 498, S p r i n g e r 1975. [ M a c i n t y r e 19761. 41, NO. 3, 1976, 605-610.

,

On d e f i n a b l e s u b s e t s of p - a d i c f i e l d s , J . S . L .

[ M a c i n t y r e 19771. , Model-completeness, Logic ( e d . J. B a r w i s e ) , North H o l l a n d 1977. [Macintyre 1986?].

i n Handbook of M a t h e m a t i c a l

L e c t u r e s o n Real E x p o n e n t i a t i o n , book i n p r e p a r a t i o n .

, K . McKenna, L . van den [ M a c i n t y r e , McKenna, van den D r i e s 19831. Dries, E l i m i n a t i o n o f q u a n t i f i e r s i n a l g e b r a i c s t r u c t u r e s , Advances i n Mathem a t i c s 47 ( 1 9 8 3 ) , 74-87. [Manin 19711. J u . I . Manin, Cyclotomic f i e l d s and modular c u r v e s , R u s s i a n Mathem a t i c a l S u r v e y s 26 ( 1 9 7 1 ) , 7-78. IMcKenna 19801. K . McKenna, Some d i o p h a n t i n e N u l l s t e l l e n s z t z e , pages 228-247 i n S p r i n g e r L e c t u r e Notes i n Mathematics 834. S p r i n g e r 1980. [Monk 19751. L. Monk, E l e m e n t a r y - r e c u r s i v e B e r k e l e y 1975.

[Nagata 19621.

M. Nagata 19621.

d e c i s i o n p r o c e d u r e s , Ph.D. T h e s i s ,

M. Nagata, L o c a l R i n g s , Wiley I n t e r s c i e n c e , 1962.

[Nerode 19631. A. Nerode, A d e c i s i o n method f o r p-adic i n t e g r a l z e r o s o f diophant i n e e q u a t i o n s , B u l l . A.M.S. 69 (19631, 513-517. I P i l l a y 19831. A. P i l l a y , An I n t r o d u c t i o n t o S t a b i l i t y Theory, Oxford U n i v e r s i t y P r e s s 1983. [ P o i z a t 19831. B . P o i z a t , Une t h e o r i e d e G a l o i s i m a g i n a i r e , J. S . L . 48 (19831, 1151-1170. [ P r e s t e l - R o q u e t t e 19841. A. P r e s t e l and P . R o q u e t t e , Formally P-adic F i e l d s , L e c t u r e Notes i n Mathematics 1050, S p r i n g e r 1984. [Ribenboim 19681. P . Ribenboim, T h e o r i e d e s V a l u a t i o n s , d e 1 ’ U n i v e r s i t e de M o n t r e a l , 1968.

2e. e d i t i o n , L e s P r e s s e s

[A. Robinson 19591. A . Robinson, S o l u t i o n of a problem of T a r s k i , Fund. Math. 47 (1959). 179-204. [A.

Robinson 19651.

, Model

Theory, N o r t h H o l l a n d , 1965.

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Thesis,

Twenty Years of padic Model Theory

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153

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fields, Preprint,

[ J . Robinson 19651. J. Robinson, The d e c i s i o n problem f o r f i e l d s , i n Theory of Models (ed. Addison e t . a l . ) , North Holland 1965, 299-311.

[Roquette 19771. P. Roquette, A c r i t e r i o n f o r r a t i o n a l p l a c e s over l o c a l f i e l d s , J. r e i n e angew. Math 292 (1977), 90-108. [Scowcroft, van den Dries 19851. P . Scowcroft and L. van den D r i e s , On t h e strlict u r e o f semi-algebraic s e t s over p-adic f i e l d s , p r e p r i n t 1985 ( a v a i l a b l e from Stanford). [Serre 19653. J-P. S e r r e , Cohomologie Galoisienne, Lecture Notes i n Mathematics 5 , Springer 1965.

,

I S e r r e 19681.

Corps Locaux, Hermann 1968.

T . Skolem, Diophantische Gleichungen, Springer 1938.

[Skolem 19381. [ S t e i n i t z 19301.

E . S t e i n i t z , Algebraische Theorie d e r Korper, B e r l i n 1930.

[Tarski 19511. A. T a r s k i , A Decision Method f o r Elementary Algebra and Geometry, 2nd ed., r e v i s e d , Berkeley and Los Angeles 1951. [Tate 19673.

J. Tate, Global c l a s s f i e l d theory, i n [Cassels 19671.

[Van den Dries 19821. L. van den Dries, Some a p p l i c a t i o n s of a model-theoretic f a c t t o (semi-) a l g e b r a i c geometry, Indagat. Math. 44 (1982), 397-401. [Van den Dries 198483. , Remarks on T a r s k i ' s problem concerning (R, -, exp), pages 97-121 i n Logic Colloquium 1982, (ed. G. L o l l i e t . a l . ) , North Holland 1984.

+,

[Van den Eries 1984Bl. , Algebraic t h e o r i e s with d e f i n a b l e Skolem f u n c t i o n s , J.S.L. 49 (1984), 625-629. [Weil 19671.

A. W e i l , Basic Number Theory, Springer 1967.

[Weispfenning 19761. V. Weispfenning, On t h e elementary theory of Hensel f i e l d s , Ann. Math. Logic 10 (1976), 59-93. [Weispfenning 19841. , Q u a n t i f i e r e l i m i n a t i o n and d e c i s i o n procedures f o r valued f i e l d s , pages 419-472 i n Lecture Notes i n Mathematics 1103 (eds. G . Muller and M. R i c h t e r ) , S p r i n g e r 1984. [Whitney 19571. H. Whitney, Elementary s t r u c t u r e of real a l g e b r a i c v a r i e t i e s , Annals of Math. 66 (1957), 545-556. [Yasumoto 19851.

M. Yasumoto, Personal communication.

[ Z i e g l e r 19721. M. Z i e g l e r , D i e elementare Theorie d e r henselschen KGrper, D i s s e r t a t i o n , Kzln 1972.

LOGIC COLLOQUIUM '84 J.B. Paris, A.J. Wilkie, and G.M. Wilmers (Editorsj 0 Elsevier Science Publishers B. V. (North-Holland), 1986

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Bruno P o i z a t U n i v e r s i t e P i e r r e & Marie Curie Paris

L e premier j e t de c e t t e conference, dont quelques p r i v i l e g i e s , h

P a r i s e t h Wittenberg, ont eu l a primeur, e t a i t i n t i t u l g "Malaise dans l e s modbles recursivement s a t u r e s " . J ' y e x p l i q u a i s combien j e t r o u v a i s malsaines t o u t e s c e s n o t i o n s de r e c u r s i v i t k , d 6 c i d a b i l i t 6 , e t a u t r e s f a r i b o l e s , qu'on v o i t encore t r o p souvent t r a i n e r dans des exposes de Theorie des Modhles, e t j ' a v o u a i s mon d e s a r r o i devant des travaux r e c e n t s h propos de modkles r e s p l e n d i s s a n t s une notion B l a q u e l l e j ' a t t a c h a i s une s i g n i f i c a t i o n s t r u c t u r e l l e - de t h e o r i e s t r 6 s s t a b l e s , q u i donnaient l ' i m p r e s s i o n que d e s arguments r e c u r s i v i s t e s a l l a i e n t i n t e r v e n i r de faqon e s s e n t i e l l e dans des thkorbmes de s t r u c t u r e .

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Pour m e g u 6 r i r de mon malaise, c o m e j e ne pouvais Q v i t e r l e s modkles rkcursivement s a t u r e s , il m e f a l l a i t l e s n e u t r a l i s e r , e t j e r i s q u a i l a conjecture suivante : Conjecture de P o i z a t : S i T e s t une t h e o r i e com l k t e t o t a l e m e n t transcendante, dans un l a E a q e L f i n i , t o u t mod81: r e s b l e n d i s s a n t de T e s t s a t u r 6 , t o u t modele recursivement s a t u r e de T e s t om6gasature. J u l i a Knight a eu l a bonte de m'envoyer un contre-exemple h c e t t e c o n j e c t u r e avant l a tenue de c e t t e reunion. Craignant une rechute grave, j ' a i t r i t u r g dans t o u s l e s s e n s ce contre-exemple, e t j ' a i e n f i n compris que mon malaise n e v e n a i t que d'un aveuglement passager dfi h l ' i d e o l o g i e ambiante : contrairement h ce qu'on p o u r r a i t penser, l a notion de modble r e s p l e n d i s s a n t n ' a a s un c a r a c t h r e exclusivement s t r u c t u r e l , e l l e ne peut s e rkduire quelque chose de modele-thgoriquement s a i n : on peut montrer c e l a au moyen d ' u n argument mathkmatique A l a f o i s simple e t d k c i s i f .

A l'intention de ceux qui pensent que ces bonnes nouvelles de ma santd mentale n ' o n t pas un i n t e r & general, j e p r 6 c i s e que ce malaise nous donnera l ' o c c a s i o n de r e n c o n t r e r quelques beaux thborbmes de Theorie des Modkles. Ces r e s u l t a t s ne s o n t pas de moi, e t l e s Qminents savants dont j ' e x p o s e les travaux ne p o r t e n t aucune r e s p o n s a b i l i t e dans l e s p o s i t i o n s s e c t a i r e s q u i s o n t exprimees i c i .

C ' e s t a i n s i que j ' a i trouve l a voie de l a guerison :

1

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LA RECURSIVITE EN THEORIE DES MODELES

On rencontre encore a u j o u r d ' h u i des gens q u i vous p r e s e n t e n t l a d e c i d a b i l i t e de l a t h e o r i e de j e ne s a i s q u e l l e s t r u c t u r e , p a r exemp l e un corps, un groupe c o m e quelque chose de digne d ' i n t e r b t . C e t t e manie a son o r i g i n e dans l a p e r v e r s i t g de nos phres, yui, rompant avec une t r a d i t i o n m i l l g n a i r e , ont i n t r o d u i t e n mathematique

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des o b j e t s indktenninks, t o u t c e l a pour s ' o f f r i r l e p l a i s i r d'un doute ontologique h l e u r s u j e t : pour &re bien s b r de l ' e x i s t e n c e de quelque chose, il f a l l a i t en donner un algorithme de c o n s t r u c t i o n . I1 k t a i t nature1 h l'kpoque de chercher une " t h k o r i e des ensembles" q u i f c t non seulement v r a i e e t complete, mais a u s s i dkcidable, l e c a r a c t k r e algorithmique de l a logique sous-jacente &ant une garant i e de l a s o l i d i t k de l a c o n s t r u c t i o n . Mais on ne v o i t pas quel enjeu s e cache d e r r i e r e l a d k c i d a b i l i t k d'un corps de series formelles, e t on s e r a i t b i e n en peine de c i t e r un exemple oh un t e l r b s u l t a t , en s o i , s e r a i t u t i l e b quelque chose pour l a Thkorie des Modkles, en dehors du domaine s p k c i f i q u e de 1 ' 8 t u d e des modkles de 1'Arithmktique.

Ce placage a r t i f i c i e l de n o t i o n s r k c u r s i v i s t e s s u r l a Thkorie des Modkles n ' a p p o r t e r i e n non p l u s h l ' k t u d e de l a r k c u r s i v i t g , ou de l a complexit6 des algorithmes, q u i n ' y i n t e r v i e n t que p a r des techniques r o u t i n i h r e s . On montre gknkralement l ' i n d k c i d a b i l i t k de l a t h k o r i e d'une s t r u c t u r e e n y i n t e r p r k t a n t quelque chose c o m e 1'Arithmktique ; e t pourquoi donc, quand on a i n t e r p r k t k 1'Arithmktique du 2 5 O ordre dans M, e n t i r e r l e c o r o l l a i r e triomphant que M a une t h k o r i e indkcidable ? Quant aux arguments de d k c i d a b i l i t k , i l s reposent uniform6ment s u r une g e n k r a l i s a t i o n de l a These de Church, b s a v o i r que t o u t e f o n c t i o n de N dans N e s t rkcursive p r i m i t i v e . En f a i t , quand un t h k o r i c i e n des modkles proclame qu'une t h k o r i e e s t

decidable, il a en t 6 t e t o u t a u t r e chose ; il veut d i r e que c e t t e t h k o r i e e s t simple, q u ' i l s a i t l a m a i t r i s e r , q u ' i l s a i t en d k c r i r e l e s types, peut-etre mbme en c l a s s e r l e s modbles. La d k c i d a b i l i t k d'une t h e o r i e mesure, d'une c e r t a i n e faqon, l a complexitk d'un systkm e d'axiornes ; on s a i t q u ' e l l e e s t s e n s i b l e h t o u t e s o r t e de manipulat i o n s a r t i f i c i e i l e s s u r l a p r e s e n t a t i o n du langage ; e l l e ne concerne pas l e t h k o r i c i e n des modeles, dont l e souci e s t de d k c r i r e une c l a s s e de s t r u c t u r e s . P l u t 6 t que de f a i r e appel b des n o t i o n s deplackes, il d e v r a i t S t r e capable d'exprimer ses r k s u l t a t s dans un langage autonome e t adkquat.. Cela d i t , il n ' y a pas 18. grand mystere, n i de quoi provoquer un malaise : t o u t au p l u s une l k g e r e i r r i t a t i o n . Notre t r a n q u i l i t 6 d ' e s p r i t n ' a u r a k t k que lkgkrement perturbke p a r l ' a p p a r i t i o n des "mod&l e s recursivement s a t u r k s " , une s o r t e de hochet pour r k c u r s i v i s t e s ; mais l e malaise s ' i n s t a l l e quand a p p a r a i t l e Thkor&me de Ressayre qui montre, dans l e c a s dknombrable, l'kquivalence de l a s a t u r a t i o n rkcursive avec ce qu'on appelle, depuis Barwise e t S c h l i p f , l a resplendance, une notion qui semble s i n a t u r e l l e du p o i n t de vue modblethkorique ! E t l e malaise s ' e s t aggravk rkcemment, lorsque s o n t parus des travaux

de Buechler, m i g h t , P i l l a y , e t d ' a u t r e s , p o r t a n t s u r l e s modkles r e s p l e n d i s s a n t s de t h k o r i e s s u p e r s t a b l e s , ou m b m e totalement t r a n s cendantes (donc les p l u s BloignBes de l ' A r i t h m k t i q u e ) , oh il semblait que des techniques de r 6 c u r s i v i t e B t a i e n t indispensables pour mettre en Bvidence d e s p r o p r i e t k s s t r u c t u r e l l e s de c e s modkles. 2

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QUELQUES D E F I N I T I O N S

Avant d ' e n t r e r dans l e v i f du s u j e t , j e r a p p e l l e les principaux themes de l a s a t u r a t i o n rdcursive e t de l a resplendance. On considere une t h k o r i e compl&te T, dans un langage f i n i L. Un modble

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a

M de T est d i t r6cursivement s a t u r e s i pour t o u t u p l e d'Pl6ments.a). de M, e t t o u t ensemble p, r d c u r s i f e t c o n s i s t a n t . de formules f (. x .-,, il e x i s t e un element de-M qui l e s s a t i s f a i t t o u t e s .

a

Comme L e t s o n t f i n i s , l a n o t i o n d'"ensemb1e r e c u r s i f de formules" ne pose pas de problbme, s o i t qu'on d e f i n i s s e l e s ensembles r e c u r s i f s d i r e c t e m e n t comme ensembles de mots dans un a l p h a b e t f i n i , s o i t qu'on l e s d d f i n i s s e comme ensembles de nombres e n t i e r s , e t qu'on r e p r k s e n t e les formules au moyen d ' u n codage non e x o t i q u e . C e s e r a i t une n o t i o n beaucoup moins s i g n i f i a n t e s i l e langage L e t a i t dPnombrable, c a r e l l e s e r a i t s e n s i b l e h l a faqon d o n t c e langage s e r a i t p r 6 s e n t d . D B s l e s premiers mots on v o i t q u ' i l y a un v e r dans l e f r u i t i en

e f f e t , dans l a d e f i n i t i o n c i - d e s s u s , on p e u t remplacer " r d c u r s i f " p a r "rdcursivement dnumdrable" i on l e v o i t p a r une m6thode c l a s s i q u e de pl6onasme, l e pleonasme & a n t i n d i s s o c i a b l e de l a l o g i q u e du premier o r d r e : en e f f e t , s i p e s t un ensemble recursivement dnumdrable, l ' a p p a r t e n a n c e de f h p s e t r a d u i t p a r l a s a t i s f a c t i o n d ' u n e formule a r i t h d t i q u e (E u) a ( f , u ) , oh a e s t 5 q u a n t i f i c a t i o n s bornees i considdrons l ' e n s e m b l e q d e s formules g, obtenues en m e t t a n t u doub l e s n k g a t i o n s devant une formule f , l e couple ( f , u ) s a t i s f a i s a n t a i pour un 61dment de M, s a t i s f a i r e p ou q, c ' e s t p a r e i l , e t q e s t rkcurs i f , puisque l ' a p p a r t e n a n c e de g q s'exprime au moyen d ' u n e formule h q u a n t i f i c a t i o n s borndes en f o n c t i o n de l a complexit6 de g. A l ' i n v e r s e , on v e r r a i t p a r l a mdme methode, en d i l a t a n t a r t i f i c i e l l e m e n t l a t a i l l e d e s formules, qu'on p e u t remplacer " r d c u r s i f " p a r P ou N P i en cons6quence, l e s rnodgles rgcursivement s a t u r d s ne f o n t p a s d e s 7 i s t i n c t i o n s q u i s o n t p o u r t a n t fondamentales dans l ' d t u d e d e s proc6dds a l g o r i t h m i q u e s , e t il e s t d i f f i c i l e de l e s j u s t i f i e r en p r d t e n d a n t q u ' i l s pourraient avoir des applications importantes h 1'Informatique ( T h e o r i q u e ) , c o m e c ' e s t l a mode a u j o u r d ' h u i . 11 f a u t a u s s i remarquer que les modBles r6cursivement s a t u r d s forment une c l a s s e pseudo-6ldmentaire i vous pouvez l e montrer de d i f f e r e n t e s facons, s u i v a n t v o t r e degr6 de s o p h i s t i c a t i o n en " s o f t model t h e o r y " , mais l e mieux e s t de procdder de l a manibre l a p l u s d i r e c t e i pour c e l a , on s e rememore un v i e w (1958) thdorbme de Craig-Vaught, q u i a f f i r m e qu'une t h g o r i e rdcursivement a x i o m a t i s a b l e , dans un langage f i n i , a une expansion f i n i m e n t a x i o m a t i s a b l e . On a j o u t e a l o r s au langage L de T c e l u i de 1 ' A r i t h m d t i q u e e t de l a combinatoire d e s part i e s f i n i e s de M, e t a u s s i un p r d d i c a t de s a t i s f a c t i o n S a t ( f , u ) d e s t i nd h n o t e r l a s a t i s f a c t i o n d ' u n e formule f du langage L p a r un u p l e u d'dldments de M, p l u s une f o n c t i o n t ( n , u ) . Nous pouvons rassembler t o u s l e s symboles r a j o u t d s h L en un s e u l , R , e t c o n s i d d r e r 1'8nonc6 r s ( R ) q u i a s s u r e un minimum d ' a r i t h m d t i q u e , q u i d i t que t o u t uple d'P1dments de M e s t r e p d s e n t d , que l e p r d d i c a t de s a t i s f a c t i o n a l e s p r o p r i d t d s qu'on pense, e t que pour t o u t u p l e u, e t t o u s e n t i e r s n e t m, s ' i l e x i s t e un x s a t i s f a i s a n t l e s m premibres formules du n o ensemble pn d d f i n i p a r une formule a r i t h m d t i q u e h q u a n t i f i c a t i o n s bornees, une f o i s les parambtres s u b s t i t u d s p a r u, a l o r s t ( n , u ) e s t un t e l x . Je d i s que M e s t rdcursivement s a t u r d s i e t seulement s i on p e u t y i n t e r p r e t e r R de maniere h s a t i s f a i r e r s ( R ) . S i M e s t recursivement s a t u r e , i n t e r p r e t e r R de faqon s t a n d a r d i e t s i M se transforme en Un modble de r s ( R ) , l ' i n t e r p r 6 t a t i o n de 1'Arithmdtique e t de l a Comb i n a t o i r e s e r a p e u t - d t r e non-standard, i n t r o d u i s a n t d e s n o t i o n s ( e n t i e r s , u p l e s , formules) s a n s s i g n i f i c a t i o n r d e l l e , mais c e que nous avons m i s dans r s ( R ) l'empdche de m e n t i r h propos d e s n o t i o n s s t a n d a r d s dont nous avons b e s o i n ( e t en p a r t i c u l i e r l'empdche de

B. POIZAT

158

d e c l a r e r faussement qu'un e n t i e r s t a n d a r d s a t i s f a i t une formule a r i t h metique s t a n d a r d 2 q u a n t i f i c a t i o n s born6es). Passons maintenant 2 la deuxihme n o t i o n ; M e s t d i t r e s p l e n d i s s a n t s i pour t o u t Qnonc6 f ( a , R ) , dans un langage f a i s a n t i n t e r v e n i r , o u t r e d'616ments de M e t un nouveau symbole r e l a t i o n n e l R, e t L, un uple q u i s o i t c o n s i s t a n t avec l a t h e o r i e de M ( j e vewc d i r e l e t y p e de t r a n s f o r m e r M en u n modble a l o r s on p e u t i n t e r p r e t e r R de manibre de f ( a , R ) . Par exgmple un modkle r e s p l e n d i s s a n t e s t omega-fortementhornogbne, c a r s i a e t 5 o n t m&ne type, il e s t c o n s i s t a n t de supposer q u ' i l e x i s t e un automorphisme de M ( n o t i o n exprimable p a r u n QnoncB, puisque l e langage e s t f i n i ) q u i l e s dchange.

a

a),

Nous voyons que 1'6noncd r s ( R ) c i - a v a n t e s t t o u j o u r s c o n s i s t a n t avec M, ne s e r a i t - c e que p a r c e que M a une e x t e n s i o n omega-saturke, s i bien qu'un modble r e s p l e n d i s s a n t e s t r6cursivement s a t u r e ; on p e u t mgme p r e c i s e r que s i f ( S , R ) e s t c o n s i s t a n t avec M, M se t r a n s f o r m e en un rnodble (M,R) recursivement s a t u r e de c e t enonce : s i on p u t m e t t r e " r e s p l e n d i s s a n t " au l i e u de "recursivement s a t u r 6 " dans c e t t e phrase, on d i t que M e s t chroniquernent r e s p l e n d i s s a n t ; on ne s a i t r i e n de b i e n genkral s u r l a s p l e n d e u r chronique. Le Theorbme de Ressayre a f f i r m e l a r e c i p r o q u e dans l e c a s denombrab l e : s i M e s t denombrable ( l e langage 6 t a n t f i n i ) e t r6cursivement s a t u r 4 T i l e s t r e s p l e n d i s s a n t ( e t m&ne chroniquement 1 ) ; on l e moGtre en f a i s a n t B l ' i n t g r i e u r de M une c o n s t r u c t i o n de Henkin pour f ( a , R ) : comme l a n o t i o n d e consdquence e s t r6cursivement Bnum&rable, e t M recursivement s a t u r 6 , on p e u t t o u j o u r s i n t e r p r e t e r l e s temoins p a r d e s Blements de M. S i T e s t omega-catdgorique, s e s modkles r e s p l e n d i s s a n t s forment une c l a s s e pseudo-bl6mentaire : l a c o n s i s t a n c e de f ( a , R ) "e-d6pendant que du typg de-a, q u i ' e s t i s o l e , i n t r o d u i r e un symbole R(x,y) e t d e c l a r e r que f ( a , R ( a , y ) ) e s t v r a i quand a a l e type convenable. Cela n ' e s t pas v r a i en g e n e r a l : s i T e s t l a t h e o r i e d ' u n e r e l a t i o n d'Bquivalence E q u i , pour chaque e n t i e r n, a exactement une c l a s s e avec n elements, les modkles r e s p l e n d i s s a n t s de T s o n t s a t u r 6 s ( i l e s t c o n s i s t a n t de supposer que chaque c l a s s e i n f i n i e e s t en b i j e c t i o n avec l e modble, e t q u ' i l e x i s t e un ensemble A form6 d'6ldments deux-A-deux non Bquiv a l e n t s e t en b i j e c t i o n avec l e modkle), e t l a n o t i o n de resplendance ne s e conserve pas p a r u l t r a p u i s s a n c e .

On v o i t f a c i l e m e n t , en i t e r a n t omega f o i s l e Lemme de c o n s i s t a n c e d i s j o i n t e , que t o u t modkle a une e x t e n s i o n 6 l e m e n t a i r e r e s p l e n d i s s a n t e de mihe c a r d i n a l ; il e s t 6galement f a c i l e de v o i r qu'un modble s a t u r e ( i . e . kappa-satur6 de c a r d i n a l kappa) e s t r e s p l e n d i s s a n t ( e t kappa-resplendissant : v o i r c i - a p r k s ) . Par c o n t r e , il e s t souvent d e l i c a t de determiner s i un modble donne e s t r e s p l e n d i s s a n t ou non. 11 y a un lemme de d i l a t a t i o n de Schmerl q u i e s t extremement u t i l e pou; c e l a : & M e s t un modkle recursivernent s a t u r e dknombrable, il a en t o u t c a r d i n a l lambda un modgle r e s p l e n d i s s a n t de c e c a r d i n a l qu$ r e a l i s e l e s m E m e s types que M ; comme il s ' a g i t de s t r u c t u r e s omdgahomogenes, c e s deux modbles s o n t Glkmentairement e q u i v a l e n t s dans Lew.

Ce lemme e s t l e r e s u l t a t d ' u n e corrunande passee B Schmerl p a r Buechler, e t il e s t p u b l i e dans (BUECHLER 1984) ; il repose s u r un maniement assez d e l i c a t d'indiscernables. 3

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commenFons p a r g k n k r a l i s e r h l a f o i s l a n o t i o n d e r e s p l e n d a n c e e t c e l l e d e k a p p a - s a t u r a t i o n ; nous d i r o n s que M e s t kappa-resplend i s s a n t s i t o u t e t h k o r i e : d a n s un l a n g a g e f a i s a n t i n t e r v e n i r s t r i c t e ment moins de kappa nouveaux symboles, e t u t i l i s a n t s t r i c t e m e n t moins d e kappa d l e m e n t s d e M, e s t r e a l i s a b l e s u r M. J ' i n t r o d u i s kgalement, s i l e l a n g a g e d e M e s t f i n i , une n o t i o n i n t e r m k d i a i r e e n t r e l a r e s p l e n d a n c e e t l a omkga-resplendance ( q u e c e r t a i n s a p p e l l e n t r e l a t i o n - u n i v e r s a l i t k ) ; k t a n t donnk un r e e l r, j e d i s que M e s t ( r k c u r s i v e m e n t e n r ) - r e s p l e n d i s s a n t s i on a F a p o u r l e s t h k o r i e s , d a n s un langage f i n i , q u i s o n t r e c u r s i v e s e n r ; on v o i t a i s k m e n t , e n a j o u t a n t un peu d ' A r i t h m k t i q u e , que rkcursivement-resplendissant = r e s p l e n d i s s a n t . NOUS

une hypothbse d e s t a b i l i t k , l a s p l e n d e u r a t e n d a n c e h i m p l i q u e r c a r r k m e n t l a s a t u r a t i o n ; l a chose semble a v o i r Q t B remarquke pour l a p r e m i e r e f o i s p a r B u e c h l e r dans son P M .

AVeC

Montrons que, s i T e s t omkqa-stable, un modble M o m k g a - r e s p l e n d i s s a n t d e T e s t s a t u r z S o i t A une p a r t i e de M, de c a r d i n a l s t r i c t e m e n t i n f k rieur c e l u i de M, e t s o i t p un 1 - t y p e s u r A ; s o i t p ' un f i l s de-p s u r M, i s o l e des t y p e s de m e m e r a n g de Morley p a r une formule f(x,a)- ; on s a i t que p ' e s t l ' u n i q u e f i l s n o n - d e v i a n t de s a r e s t r i c t i o n q h a ; on a j o u t e au l a n g a g e un p r k d i c a t u n a i r e I , un symbole de f o n c t i o n s, on d k c l a r e que s e s t une b i j e c t i o n e n t r e M e t I, e t que I e s t une c o p i e d e l a s u i t e de Morley d e q, c ' e s t - h - d i r e que t o u t n-uple e x t r a i t d e I e s t form6 de r k a l i s a t i o n s de q i n d k p e n d a n t e s ; c o m e c ' e s t c o n s i s t a n t , Ga e x i s t e ; e t on s a i t que t o u s l e s klkments de I, s a u f un nombre f i n i s i A e s t f i n i , s a u f c a r d ( A ) s i A e S t i n f i n i , r k a l i s e n t s u r A l ' u n i q u e f i l s non-dkviant d e q, c ' e s t - & - d i r e p.

7

Une d e m o n s t r a t i n semblable montre que s i T e s t s t a b l e u n Todele d e T g u i e g t c a r d ( T )' - r e s p l e n d i s s a n t , ou m e m e s e u l e m e n t c a r d ( T ) - s a t u r k e t omkga - r e s p l e n d i s s a n t y s t s a t u r k ; de m e m e , p o u r une t h k o r i e s u p e r s t a b l e , un modble omega - r e s p l e n d i s s a n t e s t s a t u r k .

ce t y p e

de p r o p r i k t e s c a r a c t k r i s e l a s t a b i l i t 6 : p o u r une t h k o r i e i n s t a b l e , aucune c o n d i t i o n d e r e s p l e n d a n c e n e p e u t i m p l i q u e r l a s a t u r a t i o n . En e f f e t , si T e s t i n s t a b l e , on p e u t f a b r i q u e r d e maniere tr&s s o u p l e d e s modeles q u i s o n t n o n - s a t u r k s non p a s p a r c e q u ' i l s o m e t t e n t d e s t y p e s , mais p a r c e q u ' i l s e n r k a l i s e n t ! P l u s p r e c i s k m e n t , s i on p a r t d ' u n modele quelconque Mo, p u i s q u ' o n r k a l i s e d a n s M1 t o u s l e s lambda-types s u r Mo, p u i s d a n s M 2 t o u s les lambda-types s u r M1, e t q u ' o n r e p b t e lambda f o i s , l e modble M A o b t e n u h l a f i n n e p e u t p a s e t r e lambda+-saturd ( p o u r l e s d k t a i l s , l e mieux sera de c o n s u l t e r (POIZAT 198?) quand c e t e x c e l l e n t ouvrage a u r a t r o u v k un k d i t e u r ;+ en a t t e n d a n t , v o i r (POIZAT 1983)) ; i$ n e sera p a s non p l u s lambda homoghne s i p a r exemple Mo e s t lambda - s a t u r 6 . Comme d a n s c e t t e const r u c t i o n l a s e u l e c h o s e q u ' o n demande aux M, est de r k a l i s e r des t y p e s , on p e u t m e t t r e e n sandwich une a u t r e c o n s t r u c t i o n qui g a r a n t i t que M~ s o i t k a p p a - r e s p l e n d i s s a n t , p o u r un kappa r a i s o n a b l e p a r r a p p o r t lambda ; e t e n s u i t e , s i l e c a r d i n a l de M X vous semble t r o p g r o s , l e f a i r e m a i g r i r p a r un th6orerne de Mwenheim.

pour une t h e o r i e s t a b l e n o n - s u p e r s t a b l e , l a m e m e c o n s t r u c t i o n p e u t se f a i r e pour lambda = omega ( e l l e f o n c t i o n n e d a n s l e c a s stable s i lambda e s t s t r i c t e m e n t i n f k r i e u r h k a p p a ( T ) ) , s i b i e n q u ' e n t o u t card i n a l s u p k r i e u r ou k g a l h 2- me t h k o r i e n o n - s u p e r s t a b l e a d e s mod&l e s o m k g a - r e s p l e n d i s s a n t s q u i n e s o n t p a s omkga+-saturks, n i m&ne omkga+-homogbnes c o n s i s t a n t e avec c e l l e de M !

.

B. POIZAT

160

S i on y e f f a c e l e s omega e t l e s omega', ce dernier resultat a Ct6 montre d a n s (KNIGHT 19821, p a r un argument oh i n t e r v i e n n e n t d e s d e f i n i t i o n s r e c u r s i v e s de t y p e s , q u i a 6 t 6 p o u r beaucoup d a n s mon malaise : J u l i a K n i g h t montre que s i pour un lambda non-dknombrable t o u s l e s modeles r e s p l e n d i s s a n t s de T s o n t homogknes, a l o r s c e l a se p r o d u i t pour t o u t lambda non-denombrable : T e s t a l o r s n d c e s s a i r e m e n t s u p e r s t a b l e . I1 y a des t h e o r i e s s u p e r s t a b l e s p o u r l e s q u e l l e s c ' e s t l e cas ( p a r exemple t o u t e s les t h d o r i e s om6ga-stables, comme nous a l l o n s l e v o i r ) , d ' a u t r e s non : j ' a i m e r a i s q u ' o n m e d o n n s t une c a r a c t e r i s a t i o n modgle-thkoriquement s i g n i f i a n t e du c a s oh Fa se p r o d u i t , m a i s j e ne s a i s que p e n s e r : une t e l l e c a r a c t e r i s a t i o n n ' e x i s t e p e u t Gtre p a s !

NOTE

4

-

TH EO R I ES OMEGA-STABLES

(DANS UN LANGAGE FINI)

Dans une t h e o r i e T omega-stable, chaque t y p e ( c o m p l e t ) p e s t d e t e r m i n e p a r l a f o r m u l e f ( x , a ) q u i l ' i s o l e d e s t y p e s de & m e r a n g de Morley, e t l e t h e o r i c i e n d e s modgles s a i t d ' e x p e r i e n c e q u ' o n p e u t a l o r s , h p a r t i r d e c o n s i d e r a t i o n s s u r d e s f o r m u l e s , m o n t r e r d e s theorkmes q u i demandent d e s c o n d i t i o n s beaucoup p l u s g l o b a l e s dans l e c a s oh T e s t seulement stable. C e l a e x p l i q u e l a c o n j e c t u r e de l ' i n t r o d u c t i o n ; e l l e s e r a i t aisCment f a l s i f i a b l e s i l e l a n g a g e e t a i t i n f i n i , c a r l a resplendance n e f a i t i n t e r v e n i r l e l a n g a g e que f r a g m e n t f i n i p a r f r a g m e n t fini. S i nous c h e r c h o n s h p r o u v e r l a c o n j e c t u r e e n a d a p t a n t l a d e m o n s t r a t i o n de l a s e c t i o n 3 , nous pouvons, p a r un seul-enonce, imposer h chaque klement de I de s a t i s f a i r e l a f o r m u l e f ( x , a ) q u i d e t e r m i n e q , e t h l ' e n s e m b l e I d ' G t r e i n d i s c e r n a b l e p a r un ensemble r k c u r s i f de c o n d i t i o n s , q u ' o n t r a n s f o r m e e n un s e u l dnonci. en a c c r o i s s a n t l e l a n g a g e ; mais comnent imposer de c e t t e manikre h l a s u i t e i n d i s c e r n a b l e d ' 6 t r e b i e n l a bonne, d ' g t r e b i e n l a s u i t e d e Morley d e q ?

11 s u f f i t , e n f a i t , de p o u v o i r imposer l e t y p e s u r @ , de c e t t e s u i t e de Morley : e n e f f e t , on s a i t que l a c l a s s e d ' u n t y p e d a n s l ' o r d r e fondamental n e depend que du t y p e s u r gf de sa s u i t e d e Morley ( v o i r (POIZAT 1 9 8 3 a ) ) , e t que l ' e x t e n s i o n non d e x i a n t e de q e s t l ' u n i q u e t y p e d a n s l a classe de q h s a t i s f a i r e f ( x , a ) . Reciproquement, n'imp o r t e q u e l l e s u i t e i n d i s c e r n a b l e e s t a s s o c i e e l a c l a s s e de son t y p e l i m i t e , s i b i e n que l ' o r d r e fondamental e s t e n b i j e c t i o n avec l e f e r mi. d e s,(gf) form6 d e s types de s u i t e s i n d i s c e r n a b l e s . L e problgme e s t donc d ' i d e n t i f i e r , f i n i m e n t ou r e c u r s i v e m e n t , une s u i t e i n d i s c e r n a b l e parmi d ' a u t r e s ; il se r6soud de lui-m6me s i l ' o r -

d r e fondamental e s t f i n i ( o u m6me s ' i l n ' y a q u ' u n nombre f i n i d e classes de t y p e s n o n - r & a l i s & s ) . D ' a i l l e u r s , on p o u r r a i t u t i l i s e r d i r e c tement l ' o r d r e fondamental e n a j o u t a n t d e s p r e d i c a t s q u i f o r c e n t l ' e x i s t e n c e d ' u n e l o n g u e s u i t e c r o i s s a n t e Mi d e r e s t r i c t i o n s B l e m e n t a i r e s Mi, s a t i s f a i s a n t f(x,Z), e t representant sur de M, a v e c ai d a n s Mi+l

-

M~ l e s f o r m u l e s de l a b o r n e d e q.

En consequence, l a c o n j e c t u r e e s t v r a i e s i T e s t omega-stable e t omega-catkgorique : m a i s ce n ' e s t p a s une t r i v i a l i t 6 de v o i r que l ' o r d r e fondamental d ' u n e t e l l e t h C o r i e e s t f i n i : c ' e s t un c o r o l l a i r e de l ' a n a l y s e d e Cherlin-Harrington-Lachlan, q u i m o n t r e n t que chaque s u i t e i n d i s c e r n a b l e y e s t d e t e r m i n e e p a r s e s premiers 616ments : j e s a i s a u s s i l a d e m o n t r e r p o u r t o u s l e s modules omega-stables

(dans

Malaise et GuCrison

161

un langage f i n i , p a r exemple s i l ' a n n e a u e s t f i n i m e n t engendre), e t a u s s i pour t o u s l e s exemples q u i m e viennent B l ' e s p r i t , par exemple pour l e s corps d i f f e r e n t i e l l e m e n t c l o s de c a r a c t e r i s t i q u e n u l l e . Anand P i l l a y , p a r un argument 2 l a f o i s simple e t ingenieux, a montrC q u ' e l l e e t a i t v r a i e pour une thCorie f i n i d i m e n s i o n e l l e . I1 s u f f i t a l o r s d ' a v o i r de longues s u i t e s de Morley pour des r e p r e s e n t a n t s pl, pn des c l a s s e s minimales de l ' o r d r e de Rudin-Keisler, e t P i l l a y remarque qu'on peut l e s c h o i s i r de maniere q u ' B pi s o i t associkes des formules f i ( x ) , g i , , ( x ) g i , k ( x ) avec parametres dans l e modele

...

,...

..

gi,k premier, t e l l e s que dans une p a i r e de modgles oh g i , l , . n'augmentent pas, t o u s l e s nouveaux k l k m e n t s s a t i s f a i s a n t f i r e a l i s e n t p . . Cela l u i permet, en i n t r o d u i s a n t des chaines de modeles comme &re c e l l e c*-dessus, h f o r c e r p a r un s e u l &once l a s u i t e de Morley de pi. Pour l e c a s g&Cral, comme j e l ' a i d i t , l a c o n j e c t u r e e s t fausse ; on peut t o u t e f o i s remarquer que 1' ordre fondamental d'une t h e o r i e omegas t a b l e e s t denombrable ( c a r il y a un modele s a t u r 6 denombrable : cons i d e r e r l e s types s u r c e modele ! ) , s i b i e n qu'on peut coder e n un s e u l r e e l r t o u s les t y p e s s u r de omega-suites i n d i s c e r n a b l e s . Par consequent, fi T e s t om6ga-stable, dans un langage f i n i , il e x i s t e un r e e l r t e l que t o u t modble de T (recursivement e n r ) - r e s p l e n d i s s a n t s o i t s a t u r 6 . C e n ' e s t pas un thCor6me b i e n s k r i e u x , mais il explique pourquoi t o u t e t h k o r i e s a t i s f a i t l a c o n j e c t u r e , & moins d ' & t r e sp6cialement fabriquge pour &re un contre-exemple : c ' e s t une consequence de l a Thbse de Church g & n k r a l i s & e!

-

L e r e s u l t a t v r a i l e p l u s proche de l a c o n j e c t u r e e s t de Buechler, qui

a f f i r m e que t o u t modele M r e s p l e n d i s s a n t d'une t h 6 o r i e omega-stable, dans un langage f i n i , e s t homogene ; on s a i t qu'un modble homogene e s t c a r a c t e r i s e p a r son c a r d i n a l e t p a r l e s types (de n-uples) q u ' i l r e a l i s e ; on retrouve donc l e r e s u l t a t pour l e c a s omega-catkgorique ( c a r homogene = s a t u r e ) , e t on v o i t a u s s i que dbs que M e s t r e s l e n d i s s a n t e t om6qa-sature, il e s t s a t u r 6 . Le reel r ' q u i co e tous l e s types s u r @f de n-uples s u f f i t l u i a u s s i & s a t u r e r l e mod6le ; j e ne v o i s pas t r o p q u e l - e s t son r a p p o r t avec l e r prgc6dent.

+-

L'argument de Buechler repose s u r un lemme d e l i c a t , yui, comme beaucoup de r e s u l t a t s profonds, donne 1'impression premiere de d e v o i r &tre faux, mais se trouve fLnalement & t r e j u s t e : s i I e s t une s u i t e i n f i n i e , i n d i s c e r n a b l e s u r a, dont t o u s l e s k1BmenE s a t i s f o n t f (x, 2 ) , e t dont l e type l i m i t e e s t f ( x , a ) - r k g u l i e r , a l o r s , pour que I soit ind6pendante au-dessus de Z , il s u f f i t que s e s 616ments l e s o i e n t deux-A-deux ; en d ' a u t r e s termes, sous c e s hypothbses, on e s t sfir qu'une s u i t e i n d i s c e r n a b l e est l a s u i t e de Morley qu'on c r o i t dgs qu'on en c o n n a i t l e s d e w premiers klPments ; en cherchant des contreexemples dans une t h k o r i e comme c e l l e des corps d i f f 6 r e n t i e l l e m e n t c l o s , on c o n s t a t e r a que l ' h y p o t h e s e de r k g u l a r i t g f o r t e du type l i m i t e e s t n e c e s s a i r e ; il ne s u f f i t pas q u ' i l s o i t de poids m. Ce r e s u l t a t de Buechler t r i v i a l i s e l a c l a s s i f i c a t i o n des mod6les resp l e n d i s s a n t s d ' u n e t h e o r i e omega-stable ; p a r un argument de Lowenheim, un modele r e s p l e n d i s s a n t de c a r d i n a l lambda a une r e s t r i c t i o n Qlkment a i r e denombrable r e s p l e n d i s s a n t e q u i r k a l i s e l e s m&mes types ; e t pour a l l e r de omega & lambda, on u t i l i s e l e lemme d ' g t i r e m e n t de Schmerl ; c o m e un modele homogene e s t c a r a c t k r i s k p a r les types q u ' i l r k a l i s e , on v o i t q u ' i l y a l e m&me nombre de modeles r e s p l e n d i s s a n t s

162

B. POIZAT

en t o u t c a r d i n a l . Pour une t h C o r i e omega-stable, un modble r e s p l e n d i s s a n t , c ' e s t un modble homogbne e t r6cursivement saturC ; il y a un exemple, i n s t a b l e , de modble homogbne e t recursivement s a t u r 6 q u i n ' e s t pas r e s p l e n d i s s a n t ; il e s t a u s s i de J u l i a Knight. J e c r o y a i s que c e nombre de modbles r e s p l e n d i s s a n t s 6 t a i t un ; J u l i a Knight m'a envoy4 une l i s t e de contre-exemples oh il y a un, deux, n, w , ou 2 @ t e l s modbles ( l a c o n j e c t u r e de Vaught e s t triv i a l e pour une c l a s s e pseudo-616mentaire oh t o u s l e s modbles s o n t homogbnes ! ) ; c e s contre-exemples o n t 6 t 6 e n s u i t e modifies p a r Daniel Lascar, de manibre 2 S t r e dimensionels (''non multidimensional", comme d i r a i t S h e l a h ) , avec, b i e n sfir, une i n f i n i t e de dimensions, pour ne pas c o n t r a r i e r P i l l a y : i l s o n t m e m e d e s rangs U Ggaux 2 un : c e l a indique une t h e o r i e d e s modbles absolument t r i v i a l e . Pour l e s premiers de l a s e r i e , l e rang de Morley e s t 4 , mais l e s t y p e s n o n - r g c u r s i f s s o n t de rang de Morley 2, c e q u i , dans une t h e o r i e r & u r s i v e , e s t l e minimum p o s s i b l e .

....

...

Note technique pour ceux q u i s a v e n t ( e t pour donner e n v i e de s a v o i r S o i t M r e s p l e n d i s s a n t non-denomb r a b l e , dont nous voulons montrer l'homog&Git4 ; s o i e n t A, b , A ' dans M, A e t A ' & a n t de m6me t y p e , e t de c a r g i n a l i n f e r i e u r & c e l u i de M : nous voulons b ' dans M t e l que AAb e t A ' b ' a i e n t m6me t y p e : c o m e il y a des modbles premiers, nous pouvons supposer que A e t A ' s o n t d e s modbles de T ; en procddant p a r &apes, on s e rambne au c a s oh tp(b/A) e s t RK-minimal, e t m6me f ( x , a ) - r G g u l i e r , avec en o u t r e tp(b/A) unique e x t e n s i o n non-deviante de s a r e s t r i c t i o n & a. S o i t a ' l e correspondant de a : p a r omkga-homog&n6it6,-on p e u t t r o u v e r b ' r e a l i s a n t s u r a ' l e t y p e correspondant & t p ( b / a ) : s i gn p e u t en t r o u v e r un deuxibme, b", b ' e t b" k t a n t independants s u r a ' , on a l e s deux premiers elements de n o t r e s u i t e de Morley, e t , g r s c e au lemme de Buechler, on f o r c e une longue s u i t e i n d i s c e r n a b l e & 6 t r e b i e n l a bonne. Sinon, pour t o u t c de A ' , on p e u t t r o u v e r b- avec t p ( b ' f i I ) = ct p ( b - L ' ) , b,- e t 6 t a n t independants au-dessus de a ' (par-omegahomo&&kit&) ; comme b-et b ' s o n t dependants au-dessus de a ' , e t que

'a ceux q u i ne s a v e n t p o i n t e n c o r e )

c

t p ( b ' / z ' ) e s t de poid: un, il f a u t que b ' e t donc AAb e t A ' l b ' o n t m6me type. 5

-

c

s o i e n t independants ;

LE SOULAGEMENT

Je commence p a r d e c r i r e une innocente p e t i t e t h e o r i e to, oh s e s e n t l e d o i g t de Daniel Lascar ; c ' e s t c e l l e d e l a s t r u c t u r e A formee d e s e n t i e r s n a t u r e l s munis de l e u r f o n c t i o n s u c c e s s e u r s, O ' c e r t a i n s & a n t c o l o r e s en b l a n c , d ' a u t r e s en n o i r , de l a manibre s u i v a n t e : - s i n e s t p a i r , t o u s l e s x t e l s que n2 4 x c ( n + l ) ' s o n t b l a n c s s i n e s t impair, t o u s l e s x t e l s que n 2 c x 4 ( n + 1 l 2 s o n t n o i r s .

-

Un modble de to e s t form6 de Ao, p l u s un c e r t a i n nombre de c o p i e s de l a fonction-successeur des e n t i e r s r e l a t i f s , qui sont s o i t tous blancs, s o i t t o u s n o i r s , s o i t b l a n c s p u i s n o i r s , s o i t n o i r s p u i s b l a n c s . Voil& une t h g o r i e s a n s mystbre : e l l e e s t de r a n g U un, d i m e n s i o n e l l e , avec 4 dimensions.

si r e s t un ensemble n i f i n i n i c o f i n i de nombres e n t i e r s , on l u i a s s o c i e de m e m e l a t h 6 o r i e tr de l a s t r u c t u r e Ar d 6 f i n i e c o m e A

0'

en remplacant "n e s t p a i r " p a r "n e s t dans r", 'In e s t impair" p a r "n n ' e s t pas dans r" ; c ' , e s t une facon t r b s 6conomique d e coder un r e e l dans une s t r u c t u r e dont l a t h e o r i e d e s rnodhles e s t t r i v i a l e !

Malaise et GuCrison

163

Considdrons maintenant une t h e o r i e T quelconque ; s o i t To l a t h e o r i e d e s s t r u c t u r e s obtenues en j u x t a p o s a n t un modPle M de T e t un modble A de to ; il e s t f a c i l e de v o i r qu'un modble ( M , A ) de To e s t r e c u r s i vement s a t u r d s i e t seulement s i M e t A l e s o n t ; j e ne s a i s s i l a chose reste v r a i e en gdnkral pour l a resplendance, mais c ' e s t v r a i en t o u t c a s s i T e s t omega-stable, & cause de l a c l a s s i f i c a t i o n de Buechler. On f a b r i q u e de &me l a t h d o r i e T d e s s t r u c t u r e s q u i s o n t j u x t a p o s i r t i o n d ' u n modele M de T e t d ' u n modhle B de tr ; pour l a resplendance de ( M , B ) , il f a u t c e t t e f o i s l a resplendance r e c u r s i v e en r de M. I1 e s t une facon t r h s simple de p a s s e r d ' u n modble de To & un modhle changer l e s c o u l e u r s d e s dlkments de Ao, e t b de Tr, q u i c o n s i s t e ne r i e n t o u c h e r d ' a u t r e i on o b t i e n t a i n s i un f o n c t e u r b i j e c t i f , q u i preserve t o u t , l e s c a r d i n a l i t d s , l e s extensions dlementaires, l e s ensembles de parambtres, l e s t y p e s , l a s a t u r a t i o n , l'homogkn8it6, etc... t o u t sauf l a resplendance !

c e t exemple montre c l a i r e m e n t que l a s p l e n d e u r ne p e u t &re r d d u i t e b d e s p r o p r i d t e s s t r u c t u r e l l e s , q u ' e l l e r e s t e r a pour t o u j o u r s un melange incongru d ' i n g r d d i e n t s d i s p a r a t e s : d e s c o n t r a i n t e s s t r u c t u r e l les ( c o m e l t h o m o g 4 n d i t 4 ) , e t d ' a u t r e s ( c o m e l a s a t u r a t i o n r 6 c u r s i ve) s a n s v a l e u r modhle-thdorique. C e n ' e s t r i e n d ' a u t r e qu'un gadget, avec l e q u e l l e s l o g i c i e n s , les r d c u r s i v i s t e s , les t h d o r i c i e n s d e s ensembles, ou l e s gens comme Fa p o u r r o n t f a i r e joujou, mais q u i n'aur a jamais de s i g n i f i c a t i o n pour un mathematicien normal. REFERENCES (BUECHLER 1984)

Steven Buechler, Expansions of models of omegas t a b l e t h e o r i e s , J. Symb. Logic, 4 9 ,

( BUECHLER 198?)

Steven Buechler, p r e p r i n t

(KNIGHT 1982)

J u l i a Knight, Theories whose r e s p l e n d a n t models are homogeneous, I s r a e l Journ. Math., 4 2 , 151-161

(KNIGHT 198?)

J u l i a Knight, p r e p r i n t s

( PILLAY 1984

m a n d P i l l a y , Regular t y p e s i n nonmultidimensional omega-stable t h e o r i e s , J. Symb. Logic, 4 9 , 880-891

(FQIZAT 1983)

BrWO

P o i z a t , Beaucoup d e modgles & peu de f r a i s Paris i n T h e o r i e s S t a b l e s 111, ed. P o i z a t , I. H . P. ,

(POIZAT

1983a)

Bruno P o i z a t P o s t - s c r i t u n b "Thgories i n s t a b l e s " J. symb. Logic, 48, 239-549

(POIZAT

1987)

Bruno P o i z a t ,

Cours de Theorie d e s modbles (600 p.)

LOGIC COLLOQUIUM '84 J.B. Paris, A . J . Wilkie. and G.M. Wiliners (Editors) 0 ElseL>ierScience Publishers 0. V. (North-Holland). I986

I65

On t h e l e n g t h of p r o o f s o f f i n i t i s t i c c o n s i s t e n c y s t a t e m e n t s i n f i r s t order theories

t

Pave1 Pudlgk Mathematical I n s t i t u t e Czechoslovak Academy o f S c i e n c e s Prague

1.

Introduction By t h e second i n c o m p l e t e n e s s theorem o f Gb;del, a s u f f i c i e n t l y r i c h t h e o r y

cannot prove i t s own c o n s i s t e n c y .

5

whose l e n g t h i s

T

i f one can

of t h e s t a t e m e n t , s a y , " t h e r e is no proof of

f i n d a f e a s i b l e proof in T falsehood i n

T h i s l e a v e s open t h e q u e s t i o n ,

We s h a l l show some bounds t o t h e

l e n g t h of such p r o o f s i n some f i r s t o r d e r t h e o r i e s . The main r e s u l t s (Theorems 3.1 and 5.5) can be roughly s t a t e d a s f o l l o w s : Let

diction in E

>

be a r e a s o n a b l e f o r m a l i z a t i o n of " t h e r e i s no proof of c o n t r a -

Con ( x ) T

0

and

whose l e n g t h i s

T k

E

x

."

Then f o r r e a s o n a b l e

T

there exist

such t h a t

w

(1)

any proof of

ConT(c)

(2)

t h e r e e x i s t s a proof of

in

T

2 nE

has l e n g t h

Con,(;)

in

T

;

with length

(n

k

I t had been known t h a t some lower bounds c o u l d be d e r i v e d . t t

. In f a c t we

were i n s p i r e d by a paper of M y c i e l s k i [ l o ] and we u s e an i d e a of h i s .

The

p r e s e n t knowledge of fragments of a r i t h m e t i c , which is mainly due t o P a r i s and W i l k i e , e n a b l e d u s t o reduce t h e a s s u m p t i o n s a b o u t mere containment of Robinson's a r i t h m e t i c

.

in t h e lower bound t o

The upper bound is based on a

It u s e s a l s o a t e c h n i q u e of w r i t i n g s h o r t

p a r t i a l d e f i n i t i o n of t r u t h . formulas, c f .

Q

T

[ 5 ] , C h a p t e r 7.

' T h i s paper was f i n i s h e d w h i l e t h e a u t h o r was s u p p o r t e d by t h e NSF g r a n t 1-5-34648 a t t h e U n i v e r s i t y of Colorado, B o u l d e r , CO U.S.A. ' + A f t e r t h e paper had been t y p e d , I l e a r n e d t h a t H. Friedman had proved a lower bound of t h e form

nE

,

E

>

0

.

P.PUDLAK

166

Our r e s u l t s may be i n t e r e s t i n g b e c a u s e of t h e f o l l o w i n g r e a s o n s . lower and t h e u p p e r bound a r e o n l y p o l y n o m i a l l y d i s t a n t . o f t h e lower bound,

(see Section 4 ) .

( 1 ) The

( 2 ) Some c o r o l l a r i e s

( 3 ) R e l a t i o n t o some problems i n P e r h a p s t h e most

c o m p l e x i t y t h e o r y ( s e e Theorem 3.2 and S e c t i o n 6 ) .

i n t e r e s t i n g a p p l i c a t i o n o f t h e lower bound is a more t h a n e l e m e n t a r y speed-up f o r t h e l e n g t h o f p r o o f s in GB r e l a t i v e t o ZF, (Theorem 4 . 2 ) . S e v e r a l i m p o r t a n t p a p e r s t h a t are r e l a t e d t o o u r p a p e r a r e l i s t e d i n references.

The p a p e r s E h r e n f e u c h t and M y c i e l s k i

Mostowski [ 9 ] ( l a s t c h a p t e r ) , M y c i e l s k i

141, Gandy 161, GEdel 171,

[ l o ] , P a r i k h 111) , [ 1 2 ] , S t a t m a n

[ 1 6 ] , [ 1 7 ] and Yukami 119) d e a l w i t h q u e s t i o n s a b o u t t h e l e n g t h o f p r o o f s . Esenin-Volpin

In

[ 3 ] , Gandy [ 6 ] , M y c i e l s k i [ l o ] and P a r i k h [ l l ] t h e r e a d e r c a n

f i n d t h e o u t l i n e s o f some f i n i t i s t i c p r o j e c t s . I n t h i s p a p e r w e c o n s i d e r a measure which is d i f f e r e n t from t h e m e a s u r e s used i n most o f t h e p a p e r s m e n t i o n e d above. number o f f o r m u l a s ( i . e .

Instead of counting j u s t t h e

p r o o f l i n e s ) , we i n c l u d e t h e l e n g t h of f o r m u l a s i n t o

More p r e c i s e l y , w e assume t h a t p r o o f s a r e coded by s t r i n g s i n

t h e complexity.

a f i n i t e a l p h a b e t and t h e l e n g t h of a p r o o f is t h e l e n g t h o f t h e c o r r e p o n d i n g string.

T h i s is t h e m o s t r e a l i s t i c measure.

We d o not know w h e t h e r a s i m i l a r

lower bound h o l d s a l s o f o r t h e number o f f o r m u l a s i n t h e p r o o f .

Recently J.

K r a j i r e k g a v e a n i d e a f o r a l o w e r bound o f t h e number o f f o r m u l a s i n t h e p r o o f T

which is of t h e form

of

ConT(;)

2.

Fragments o f a r i t h m e t i c

in

constant-log n

.

The w e a k e s t f r a g m e n t o f a r i t h m e t i c t h a t w e s h a l l u s e is R o b i n s o n ' s arithmetic

Q

.

The l a n g u a g e of

S(x) = S(y) + x = y ; S(x+y) ; x - 0 =

0#

0 ; x.S(y)

Q

0 x .

s(y); x # = x*y

+

c o n s i s t s of -b

0,S,

+,

; t h e axioms a r e

By ( x = S ( y ) ) ; x + 0 = x ; x + S(y) = IAo

denotes

Q

p l u s t h e scheme of

Proofs and Finitistic Consistency Statements in First Order Theories induction for bounded arithmetical formulas, i.e. quantifiers are of the form

Q

3x

t

, Vx

(it is sufficient to assume that plus an axiom expressing

IAo

,

i t

formulas where all some term in the language of

t

is just a variable).

t

vxy 3 z ( z = xy)

167

.

LA

+ exp

is

Exponentiation can be

introduced naturally without using a function symbol for it (namely, Bennett [l] has shown that exponentiation can be defined by a bounded formula).

All

the standard theorems o f number theory and finite combinatorics are provable in IA

+ exp

,

(cf.

"21).

Syntax can be arithmetized in a natural way even in

some weaker theories, see [131. information.

The reader can consult these papers for some

Let us only remark that one can prove the scheme of induction

also for exponentially bounded formulas (i.e. formulas with quantifiers of the form

3 x c t Vx i t

IAo + exp

,t

term in the language of

S(0)

,2

- r + i.

(a1+

1) +?((.az

.

Denote by

1=

Q

1

n

Let

plus exponentiation) in

,

a. 6 {0,1) and

Then the term

will be denoted by

and called the

the numeral as a term of the form a term is too long. length of a numeral

1.1

SS

+ 1)

+Lo(...))

nth numeral.

...S(2)

The usual definition of

is not suitable here, since such

denotes the integral part of

2 is proportional to

symbols for the formalizations of

+,

0 ,

In

(...I

. ,

xy

logz(n+l)

.

Hence the

We shall not introduce new etc.

If such a symbol is

not in the language of the theory in question, the terms constructed from them should be understood as abbreviations.

P. PUDLAK

168

it is i m p o r t a n t t o

When we c o n s i d e r t h e l e n g t h of p r o o f s in some t h e o r y , s p e c i f y t h e set o f axioms. axiomatization

A

T h e r e f o r e w e s h a l l d i s t i n g u i s h two c o n c e p t s :

i s an a r b i t r a r y set of s e n t e n c e s , while a t h e o r y

d e d u c t i v e l y c l o s e d set o f s e n t e n c e s .

T

an

is a

The d i s t i n c t i o n is more i m p o r t a n t i f

d o e s not have a f i n i t e a x i o m a t i z a t i o n ,

since i f

T

T

has a f i n i t e axiomatiza-

t i o n , t h e n t h e l e n g t h s of t h e s h o r t e s t p r o o f s i n f i n i t e a x i o m a t i z a t i o n s d i f f e r o n l y by an a d d i t i v e c o n s t a n t (and we u s u a l l y u s e o n l y f i n i t e a x i o m a t i z a t i o n s ) . We s h a l l w r i t e

t o d e n o t e t h a t t h e r e e x i s t s a p r o o f of

+

in

A

whose l e n g t h ( i n c l u d i n g t h e

i n .

l e n g t h of f o r m u l a s ) is

The aim o f t h i s s e c t i o n is t o show t h a t , in s p i t e o f t h e f a c t t h a t much weaker t h a n p r o v a b l e in

IAo + e x p + exp

IA

,

every numerical i n s t a n c e of a

h a s a s h o r t p r o o f in

.

Q

n1

is

(1

sentence

This i s r o u g h l y t h e c o n t e n t

of t h e f o l l o w i n g lemma. Lemma 2.1 For e v e r y e x p o n e n t i a l l y bounded f o r m u l a +), t h e r e e x i s t s a polynomial

v a r i a b l e of

IAo + exp then,

for every

m E w

then

CutI

If

x

is t h e o n l y f r e e

such t h a t i f

, d +(m)

F i r s t we prove a n o t h e r u s e f u l lemma.

.

p

(where

+ Vx +(XI

Qi

x, ZX, 22x,...

+(x)

I(x)

Let

. ,:2

2 7 , 2;

,...

denote the

is a f o r m u l a w i t h t h e s i n g l e f r e e v a r i a b l e

denotes the following sentence

x

,

Proofs and Finitistic Consistency Statements in First Order Theories

If

A I - CutI

,

t h e n we s a y t h a t

I

is a

CUt

in

I69

A

Lemma 2.2 I

Let

be a c u t i n

such t h a t , f o r every

and

A

k,n

E

Q C A

.

Then t h e r e e x i s t s a p o l y n o m i a l

p

w

.

A ,*Ice)

Proof:

I

Given a c u t

one c a n c o n s t r u c t a n o t h e r c u t

c l o s e d u n d e r a d d i t i o n and f o r e v e r y [13].

x

I'

from

In f a c t it is p o s s i b l e t o f i n d a f o r m u l a

a r i t h m e t i c p l u s a unary p r e d i c a t e QICutR

(i)

S t a r t i n g with

I

+

[Cut

JR

R

6 Vx(JR(x)

i n s t e a d of

R

I'

ZX

such t h a t

I'

is

e x i t s and i s i n

J,(x)

I

,

cf.

i n t h e l a n g u a g e of

such t h a t +

6 3 y ( y = 2'

JR(2.x)

and a p p l y i n g

J

k-times

.

6 R(y)))] we get a cut

Ik

such

(ii) (iii U s i n g a t e c h n i q u e f o r w r i t i n g s h o r t f o r m u l a s which is d e s c r i b e d in [5], C h a p t e r 7, we can f i n d

of

Ik

JR s u c h t h a t

R

o c c u r s in i t e x a c t l y o n c e .

w i l l increase only l i n e a r l y with

k

.

Now i t f o l l o w s from ( i ) t h a t t h e

l e n g t h s o f p r o o f s o f ( i i ) and ( i i i ) w i l l be o n l y p o l y n o m i a l i n fact that

Ik

is a c u t and ( i i i ) we c a n c o n s t r u c t a p r o o f of

whose l e n g t h is p o l y n o m i a l i n ( i i ) we o b t a i n t h e lemma.

0

In(

.

Thus t h e l e n g t h

k

.

I (n) k -

Using t h e in

A

Combining t h i s p r o o f w i t h t h e proof of

P.PUDLAK

170 P r o o f o f Lemma 2 . 1 : If

i s a bounded f o r m u l a , t h e n by C o r o l l a r y 8.8 o f [ 1 3 ]

+(x)

I A o + e x p k Px + ( x )

i f f f o r some c u t

I

c l o s e d under

+

and

*

By an i n e s s e n t i a l m o d i f i c a t i o n of t h e proof we g e t t h e same theorem a l s o f o r e x p o n e n t i a l l y bounded f o r m u l a s . I(;)

.

2.2,

(where we set

Thus to p r o v e

+(E) i t is s u f f i c i e n t t o p r o v e

The l a t t e r one h a s a p r o o f w i t h l e n g t h p o l y n o m i a l i n k = 0)

.

(ml

by Lemma

0

I n o r d e r t o be a b l e t o a r i t h m e t i z e s y n t a x i n some t h e o r y , w e have t o assume t h a t t h e t h e o r y c o n t a i n s some fragment o f a r i t h m e t i c . us to reduce t h i s assumption t o

Q

.

Lemma 2 . 1 e n a b l e s

This i s b e c a u s e ( 1 ) t h e u s u a l s y n t a c t i c a l

c o n c e p t s a r e n a t u r a l l y f o r m a l i z e d by e x p o n e n t i a l l y bounded f o r m u l a s , b a s i c p r o p e r t i e s o f them a r e p r o v a b l e i n

IA

+ exp

,

(2) t h e

( 3 ) the sentences t h a t

we s h a l l c o n s i d e r w i l l be e x p o n e n t i a l l y bounded s e n t e n c e s o f t h e form

+(_I)

P u t o t h e r w i s e , t h e b a s i c p r o p e r t i e s of f o r m u l a s , p r o o f s e t c . whose l e n g t h is assumed t o be

n

axiomatization interprets and

GB)

.

Q

A

,

have p r o o f s p o l y n o m i a l i n contains

Q

(nl

.

The a s s u m p t i o n t h a t a n

can be weakened by assuming t h a t

( t h i s is r e a l l y n e c e s s a r y i n c a s e o f set t h e o r i e s ,

A

only e.g.

ZF

.

Proofs and Finitistic Consistency Statements in First Order Theories 3.

171

The lower bound

In this section we shall prove the lower bound on the length of proofs of finitistic consistency statements.

The main theorem will be stated using

finitistic counterparts of the well-known derivability conditions for the Znd G"oe1

incompleteness theorem.

Then we shall argue that they are met by natural

The relation that we shall consider is

arithmetizations. proof of length

5x

axiomatization.

In this section, however,

'I.

It will be denoted by

, where

PA(x,y)

some standard countradiction, say denoted by

ConA(n)

0

=

A

is not determined by

PA

an arbitrary formula satisfying the derivability conditions. A

in Theorem 3.1.

1.

r i

stress this fact, we omit the subscript

is provable by a

"y

Thus

P(n,

-I

1)

is an A

,

it is

In order to Let

1

, which

denote will be

later, is a finitistic consistency statement.

It is convenient to assume that formulas and proofs are strings in the element alphabet

{O,l}

.

The GEdel numbers of formulas and proofs are the

numbers with corresponding diadic expansions.

This allows us to use

also to denote the length of formulas and proofs. GEdel number

n

,

is proportional to

(01

.

,

(thus

x1

.

,...,xk

I.-I

a formula with the

Again the length of

0

We shall also use the notation

for an arithmetization of the function

...,I&)

0 is

If

will be denoted by '0'

then

tWO

are free in

(nl,

...,n.k)

3(:,,...,

this arithmetization has the following property:

CC

Gsdel number of

+(n-1 '

We shall assume that

there exists a polynomial

such that

Why we can make such an assumption will be explained later.

p

P.PUDLAK

172 Theorem 3.1 Let and l e t

A

be a c o n s i s t e n t a x i o m a t i z a t i o n ,

p1 , p 2 , p 3 , q 1 , q 2

A I-

E

>

,

let

P(x,y)

be a f o r m u l a

b e p o l y n o m i a l s such t h a t

(0)

Then t h e r e e x i s t s

Q E A

0

x

5 x'

6 P(x',y)

+

P(x,y) ;

s u c h t h a t f o r "0 n E w

Proof: I n o r d e r t o s i m p l i f y n o t a t i o n , we s h a l l write

... +

t o d e n o t e t h a t f o r some p o l y n o m i a l

...

1 . .

p

,

... .

p(n)

By D i a g o n a l i z a t i o n Lemma, t h e r e e x i s t s a f o r m u l a Q I-

Thus

(i)

Q

D(x) ++

&

D(m) - ++

h e n c e t h e same is t r u e a l s o f o r A

A

we c a n e a s i l y d e r i v e f o r e v e r y

(ii)

1

not

.

P(x,'D(x)'

1

D(x) )

such t h a t

.

P(E,~D(I$~ )

,

Now, from ( l ) , ( i ) and t h e c o n s i s t e n c y of

m A*

~ ( 5,)

Proofs and Finitistic Consistency Statements in First Order Theories Let

S(m)

denote

P(m, D(m) )

.

173

Since

is a p r o p o s i t i o n a l t a u t o l o g y , we g e t from ( 0 ) and (1)

Now, s e v e r a l a p p l i c a t i o n s of ( 3 ) and (0) y i e l d

f o r some p o l y n o m i a l and t h e d e f i n i t i o n o f

q3

,

J s ( ~ ) ( is p r o p o r t i o n a l t o

(since

S(2)

(m))

.

BY ( 2 )

we have

which t o g e t h e r w i t h ( i ) i m p l i e s

A p p l y i n g (1) t o an i m p l i c a t i o n of ( i ) w e g e t

f o r a polynomial

q4

i m p l i c i t l y d e t e r m i n e d by ( i )

c i e n t l y l a r g e , we have by ( 0 )

By ( 3 ) and by t h e d e f i n i t i o n of

Hence,

for

m

S(m)

sufficiently large,

.

Thus, i f

m

is s u f f i -

P. PUDLAK

174

Thus we get, for m

sufficiently large, A&

(V)

-'P(q2(m),

rvS(m)'

+

D(m)

.

Now ( i i i ) , (iv) and (v) implies that for some polynomials

p4

and

q5

and

every sufficiently large m

Thus by ( i i )

does

not hold

for any sufficiently large m

easy computation and condition (0)

.

.

The theorem now follows using an

0

There are several ways in which one can argue that the natural arithmetization meets the conditions ( 0 ) - ( 3 ) . particular arithmetization.

We shall not construct any such

( F o r some fragments of arithmetic such an

arithmetization is constructed i n [ 1 3 ] and can easily be generalized for other axiomatizations).

Instead we shall describe some more general properties which

look natural and imply the conditions of the Theorem 3.1. We start by observing that from the finitistic point of view it is too little to know that an axiomatization is recursive.

Therefore we shall

consider here NP axiomatizations (which means that the set of axioms can be accepted by a nondeterministic polynomial time Turing machine). every finite axiomatization is NP

.

In particular

Now we shall introduce a finitistic

counterpart of the concept of numerability.

175

Proofs and Finitistic Consistency Statements in First Order Theories Definition Let R s wk

p(xl,

.

...,xk)

We say that

polynomial p

be a formula, let p

and every

A

be an axiomatizat i o n and let

polynomially numerates

...,nk A1

0

in A

if for some

E w

nl,

R(n l....,nk)

R

P((nl(

,... , Ink])

Phi,...,n+) .

Theorem 3 . 2 Let R

S

wk

A

.

be a consistent NP axiomatization such that

Q

S

A

and let

Then the following are equivalent:

(1)

R

is

(2)

R

is polynomially numerable in Q ;

(3)

R

is polynomially numerable in

NP ;

A

.

Now it is clear that the additional property of the formula Y =

4(:l,...,:k)

is just the polynomial numerability. By Theorem 3 . 2

a formula exists, since the

such

k+l-ary relation

m = "the number of

$(z1,.

..,n-k ) "

is NP. The proofs of

( 2 ) => (1)

and

( 3 ) => (1)

are trivial.

To prove the

converse implications we need first to arithrnetize the concept of an NP set.

In [13] this was done using another possibility.

so

called

R;

formulas. Here we briefly sketch

P. PUDLAK

176 Theorem 3.3

T h e r e e x i s t s a n e x p o n e n t i a l l y bounded f o r m u l a e v e r y NP s u b s e t

of

w

polynomially numerates

R

compute

R

for a given

k

,

Q

in

NP

k E w

there exists

.

U N P( t , x )

such t h a t

such t h a t , f o r UNP(k,x)

(More ove r, t h e r e is a f a s t a l g o r i t h m t o

T u r i n g m a c hine d e f i n i n g

R .)

P r o o f - s k e t c h: First consider

+ e xp

IA

i n s t e a d of

Q

.

I n t h i s t h e o r y we f o r m a l i z e

t h e c o m p u t a t i o n s of a u n i v e r s a l n o n d e t g r m i n i s t i c T u r i n g m a c h i n e .

w i l l mean t h a t t h e u n i v e r s a l n o n d e t e r m i n i s t i c T u r i n g m a c h i n e w i t h t h e

IJNP (t , x) p ro g ra m "clock"

Thus

a c c e p t s t h e i n p u t word

t

so t h a t

Turing machines.

x

5

i t runs i n t i m e

.

We a l s o augment t h e m a c h i n e w i t h a

This enables us to take

The i d e a is r o u g h l y a s f o l l o w s . there exists a matrix

M

and s t i l l i t is u n i v e r s a l f o r

(xlt + t

A word

UNP(t,x) x

NP

e x p o n e n t i a l l y bounded.

is a c c e p t e d w i t h a p r o g r a m

t

if

in some f i n i t e a l p h a b e t s u c h t h a t

(1)

t h e f i r s t row c o n s i s t s o f

t , x a n d a s t r i n g of

(2)

M

local

(3)

i-1,j-1' m.1 - 1 , j ' m i - l , j + l ) ; "'ij to t h e l a s t row c o d e s some a c c e p t i n g c o n f i g u r a t i o n ( s a y d e t e r m i n e d by t h e

s a t i s f i e s f i n i t e l y many

0's ;

c o n d i t i o n s (which d e s c r i b e r e l a t i o n of

o c c u r r e n c e of some p a r t i c u l a r s y m b o l ) . F i n a l l y , t h e m a t r i x is c ode d by some Let (i)

k

e x p a n s i o n o f a n a t u r a l number.

be t h e number whic h c o d e s a n NP T u r i n g m a c h i n e f o r

IAo + e x p f U N P ( k , n )

p

R

.

Then

=> R ( n )

since every sentence provable i n polynomial

11 a d i c

IAo + e x p

is t r u e .

To p r o v e , f o r some

,

( i i ) R(n) => IAo + e x p Ip(Jn0 UNP(k,") w e h a ve t o p r o v e t h e e x i s t e n c e o f a n a c c e p t i n g c o m p u t a t i o n ( t h e m a t r i x

M) v i a

Proofs and Finitistic Consistency Statements in First Order Theories a p o l y n o m i a l l y long p r o o f .

1

.

m

which codes t h e a c c e p t i n g

M) and check t h e c o n d i t i o n s ( I ) , ( 2 ) , ( 3 ) f o r t h e

computation ( t h e m a t r i x numeral

It is enough t o t a k e

1I1

1.

There a r e p o l y n o m i a l l y many i n

(i.e.,

a l s o i n t h e l e n g t h of

i n p u t ) such c o n d i t i o n s , hence w e a r e done. The proof f o r

Q

can be o b t a i n e d by a n a l y z i n g t h e above proof and

W e o m i t t h e d e t a i l s s i n c e t h e proof f o r

a p p l y i n g Lemma 2 . 1 .

+ exp

IA

was

0

only sketched.

Here w e were i n t e r e s t e d o n l y i n t h e f a c t t h a t n o n d e t e r m i n i s t i c polynomial But it i s c l e a r t h a t a more

t i m e c o r r e s p o n d s t o polynomial l e n g t h p r o o f s .

e x p l i c i t r e l a t i o n between t h e s e two measures can be found.

One c a n a l s o bound

t h e l e n g t h of formulas o c c u r r i n g i n p r o o f s u s i n g t h e s p a c e bound of t h e Turing machine. P r o o f s o f t h e r e m a i n i n g i m p l i c a t i o n s o f Theorem 3 . 2 :

is a d i r e c t consequence of Theorem 3 . 3 .

( 1 ) => ( 2 )

To prove

i t is enough t o show ( i ) and ( i i ) from t h e proof above f o r since

Q rA

and we have ( i i ) f o r

consistent, Q E A

,

Q

already.

A

.

(1) = > ( 3 )

( i i ) is true,

( i ) holds since for

A

e v e r y e x p o n e n t i a l l y bounded p r o v a b l e s e n t e n c e is t r u e .

0

P r o p o s i t i o n 3.4 Let

A

be an a x i o m a t i z a t i o n .

t i o n of t h e r e l a t i o n

"z

is an

Suppose

A-proof

of

is a polynomial numeration of t h e r e l a t i o n

Then

P (x,y) A

PrfA(x,y) y" i n

1.

A

5x .

is a polynomial numera-

, suppose Let

that

PA(x,y)

s a t i s f i e s t h e c o n d i t i o n s ( 0 ) and ( 1 ) of Theorem 3.1.

'ID(

be

x'

P. PUDLAK

178

The proof f o l l o w s i n m e d i a t e l y f r o m t h e d e f i n i t i o n . axiomatization,

t h e n t h e assumption t h a t

P r f A and

If

'1z(


,

and

J1

a p p e a r s a g a i n and

V'

~1

=

-

be a f i n i t e

Fp

.

QE(pn-l) r i n g o f charac(Here

a nd d e t e r m i n e t h e m u l t i p l i c a t i o n ;

B <ar> w i t h t r i v i a l m u l t i p l i c a t i o n .

of t y p e

V

-ba ,

0

I , v-a

of type

B < a l > B

V'.

J1 :

II

E

Fp, a 2

a-v

=

=

I,

V-a

=

pn-l,

0, a2

=

pn-1

V-b

=

a.V

=

a nd b2

=

i

=

or

tpn-1.

b-V

=

0,

p

E

tpn-l, pn-l,

IV.

are

Jk

F p. )

B < a l > B

=

The s e t

J1

is i s o m o r p h i c t o o n e of t h e f o l l o w i n g :

J1

is a f i x e d n o n s q u a r e i n

Lpn-l

J k - < J k - 1 , pn-k>.

t h e r e is o n l y one t y p e r e a l i z e d i n

R

J1 as a vector space over

=

=

is t h e l a s t p l a c e where a n y t h i n g

J

We f o c u s f i r s t on t h e b o t t o m l a y e r

t e r i s t i c p , p odd.

, J k '

Our a n a l y s i s s u c c e e d s by showing t h a t

a g a i n , s i n c e f o r homogeneous

ab

pi.)

Jk.

v -

=

t h e only r e l e v a n t l a y e r s f o r determining

t

pi

be t h e least i n t e g e r such t h a t

Intuitively, the

we write

b e t h e a n n i h i l a t o r of

Ji

of c h a r a c t e r i s t i c

J ( i i i ) Let V

let

1 5 i 6 n-1,

B <ar>, a i a j

=

ajai

=

0, i

*

j,

p

E

1 mod 4

3 mod 4 .

D. SARACINO and C. WOOD

210

V.

VI. b2

=

J1

-V

=

-

aibi

=

(p

3 only) J1

+3.

=

pn-l

If k > 1

-

1

aibj

-biai, 1 5 i 5 n, and =

Moveover, any

Thus if k

B <ar> B


B B B

.

xJ1

=

=

1, or - 1 ,

J ~ =x 0,

=

or (ii) (5:

xJ1

$

,

f3

(V

Pro0

2

v = 2

If

S u p p o s e now

>

v

t h e n 3).

p

Or

and

If

> 3,

p

n

pk- l x>



p)

e x i s t s mod

for i n d e p e n d e n t

Now f i x a ny

-

2(pv

while

~pn-1.

SO

T h e s e elements must c o v e r

0.

Since

2pk-1y).

a nd we g e t t h a t

By Lemma 2 ,

=

pk-2xy'

=

BY

t h e r e are a t most p v

y'); in particular

p k - 2 ~ * 2 y= 2f,pn-l.

=

x ' mod J k - 1 ,

5

x , e x a c t l y one



E

x

a nd c o u n t y" s :

and

pk-lx>.

pk-lx> t h e r e e x i s t x' a n d y '

- .

pk-2xy

J k-2.

say

3).

by Lemma 5 , t h i s shows

Suppose t h e r e is Case 1 :

+

we a r e f i n i s h e d .

i = k

< ~ k - 3 , pn-(k-2)>,

t o squares i n

J: 5 Jk-2

JZ 5 ~ k - 3 and t h e n r e p e a t down t o

E

PJi+1

and so

pJi.

+

3 , we r e p e a t t h e p r o o f s of Lemmas 1-3, u s i n g Lemma 4 f o r t h e i n d u c t i o n ,

t o go from

xyx

Ji-1 = J1

t h i s is c o n t a i n e d i n F a c t 6.

2

J1 + p J i . 2 5 i S k.

=

Suppose

onto

PJi

+

J1 + pJ2.

=

Ji+l/Ji

P > 3).

or

2

For

maps Ji

+

J1

Ji-1

UX z

But t h e n

d . v

E

X E

V

-

< p n - l , pk-lx>

y mod J k - 1

Uy mod < p n - l > .

Thus Case 1 c a n n o t o c c u r .

Choosing

and V * -UX

Finite Homogeneous Rings of Odd Characteristic F o r a l l u a n d x , ux E < p n - l , p k - l x > .

Case 2:

mod < pn- l > .

Then f o r a l l

mod < pn- l > .

Since

-!Lpk-lx

I

(-u)x

Let

with

pk-ly

y

ux

u

by

uz

(since u

-u

!Lpk-lz mod.

5

0 mod p , a nd so

5

ux

!2pk-lx

=

v , uy

=

t h i s s a y s t h a t whenever

i m p l i e s (-u)z T h u s 9.

Lpk-lx.

5

t h e r e is

V'

B u t now r e p l a c e

(-u)z d

again that

E

5 ,

uJk-1

Ilpk-lz mod < pn- l >.

uz

v

F i x u.

215

-

L Pk-l Y

, and g e t

-u)

This says

< p n - l > , showing

E

5 .

JkJk-1

0

Now we are r e a d y t o p r o v e Proposition A.

Proof:

>

(v

Case 1 :

pk-lx,

pk- l y>

xf,yl

with

2.

t h e n for a l l (pk-lx,

pk-ly,

=

v',

X'

x'y'

I

xy

mod < pn- l >.

xy

>

v

x'y'

v

or

2

E X

mod J k-1,

=

x (-y )

5

0 mod p.

a l l independent

x

y'

I

Thus

y

mod J k-1.

v'

=

v

xy

E

< p n - l , pk-lx,

B

y.

Now

I

=

~ ( p " - +~ y ) similarly T h i s s how s

E

< pn- l >

implies

xpn-k

E

s i n c e a l l m u l t i p l e s of

From x

pn-kx

lie outside

a n d so k 5 n / 2 , f i n i s h i n g Case 1 .

-apk-lx

+ E

xy

E

E

.

E



for

x

and

x + y

Likewise

This gives



hence

This

6pk-ly.

, s i n c e

x)

there exist

t h i s implies

E

.

5i.x c < p n-l>. T h u s pn-k(pn-k

J i 5 < pn-l>.

-xy

x2

xy .k < p n - l ,

v', x', y'),

(x,-y),

0 mod p , a nd so

are a l s o i n d e p e n d e n t , a n d t h i s i m p l i e s

=

pk-ly>, say

-

x(x + y)

v

by LenUIIa 8 we h a v e

(x,y) =

If

n/2.

we see t h i s is i m p o s s i b l e :

pk-lx

+

Since

NOW

s

k

pk-ly>

(pk-lx, pk-ly,

But x(-y)

Similarly and

-

v, x, y)

a p k - l x + Bpk-l(-y).

implies a

and

v' E V - .

!j

2 Jk

> 3).

p

X-Jk

5 ;

a n d so (pn-k)2

E

.

we d e d u c e t h a t pn-kx = 0 , except

0.

Thus

n

-

k 2 k,

D.SARACINO and C. WOOD

216 Case 2: v

p

If

v = 2.

t h e r e is

V'

E

> 3,

y

with

pk-ly

we c o n c l u d e t h a t f o r a l l Y

z

E

z2

=

Y2apk-1,

w = 2 , J k = < x , pn-k,

Since

=

Jk',

E

Fp*, t h e n f o r a l l

v , y 2 z a pk-ly mod

. z2

=

apk-1,

By Lemma 8

But f o r z

mod .

=

Y(crpk-lz) p a p k - l z mod

.

f o r a ny

Jk-1>

J i 2 .

follows readily t h a t

x

a n d u s i n g Lemma 8 i t

Jkl

E

k 2 n / 2 is a s i n

T he a r g u m e n t t h a t

Case 1 .

3.

YX,

=

we g e t a c o n t r a d i c t i o n :

Fp - ( 0 . 1 1

E

x2 s a p k - l x mod < p n - l > , a

0

Small V. I n what f o l l o w s we c o n s i d e r J k s u c h t h a t

v

t h e d i m e n s i o n of V

2, v

=

Pp, and show t h a t a l l p r o d u c t s from J k l i e i n t h e p r i m e s u b r i n g , h e n c e

over

< pnW k> . In p a r t i c u l a r , we p r o v e t h i s i n c a s e v = 2 , p

in

we r e a l l y a d d , i n l i g h t o f P r o p o s i t i o n A . a p p a r e n t l y no h a r d e r t h a n for sh o wi ng t h a t f o r

p

> 3

and

p

3, w h i c h is a l l

=

The p r o o f f o r a r b i t r a r y

p

is

3, a n d d o e s g i v e a n a l t e r n a t e r o u t e t o

=

2 Jk

v = 2,

5 < p n - l > , as a n e a s y c o r o l l a r y t o

J i c .

x2 = a p r x + ~ ? p " - ~ ,where

pk-r-1x2

From a(-v)

y2 y

=

pk-r

+

getting

gpn-r-l.

x2

4

pn-k>.

0 , a n d so x2

Suppose

This implies

Thus

2, a

E

Y t 0 mod p.

+ p J k ) J k = pk-r-lJIJk

mod < P " - ~ > . 0 mod p.

5 <Jk-1,

.

k I n/2.

1 S r 5 k - 1.

with

E

a , 5, Y , 6

2 Jk

2Y, hence

,



and c o n c l u d e

n - k h k , so

But t h i s c a n o n l y happen i f

By Lemma 2 . t h e n , i t follows from

J i 5

and

k I n/2. k 5 n/2

that U

Use

D. SARACINO and C. WOOD

218

4. Large V Classified.

>

w

We assume in this section that

2, and we show the Classification

Again we use the notation of Fact 3 f o r

Theorem's claims for this case. possible J1 ' s . Lemma 1

VJk

(v

>

=

0

Jkv

=

Choose x

-

x

But

y

x'y'

=

xy, so =

=

xy

=

0, giving

=

0.

JkJk-1

=

L

PJk

=

0 , by

-pk-ly, spn-l), there exist

pk-lx, pk-1~' = -pk-ly, xy

=

=

independent, and let xy = spn-l.

,.- (pk-lx,

Jk-1, hence x(y' + y)

E

x(x + y)

so

and

(pk-lx, pk-ly, spn-1)

XI, y' with pk-lxl x'

2

is of type I, then Jk

J1

by Fact 6. Therefore Jk-lJk

Proposition A . Since

If

2).

(x' - x)y

0. Now X2 = 0.

x

and

+

2

Thus Jk

This settles the picture for J1

0, giving

=

x

x'y'.

=

y

Now x'y'

y' + y =

xy'

=

and x(-y).

are also independent,

= 0.

0

of type I completely. We turn to types

I1 and 111: Lemma 2. x2

and

>

(u

2).

If J1 is of type 11, then there is x

E

Jk' with x x J1

z x J1

y x z any

=

J1

t

=

0. By adding a suitable multiple of y to z we can assume that

=

If either y2 or z2 is 0. fine.

0 also.

spn-1

for

s

+ o mod

p as w2 where w

=

ay

Otherwise we can represent +

E

Jkf for some a, 6.

BY

definability of J1, this says that all elements w of Jk' with nonzero square satisfy w (w

+

i

J1

=

0. Taking w with w2

a) x a = 0. But (w

Lemma 3 .

0

0.

=

Proof: Since w > 2 we can find y. z independent so that y =

=

(w

> 2).

if and only if

x2

+

a) x a

=

a

=

a2 we get (w

i

a

=

2a2

f

If J1 is type I1 or I11 and x =

0.

+

a)2

=

2a2

f

0 and so

0, a contradiction. E

Jk' then x x J1

=

0

0

Finite Homogeneous Rings of Odd Characteristic Case I :

Proof:

of type II. Here we know

~1

x

x

J1

y

I

J1 = 0 with y2

0 by Lemma 2 and homogeneity.

=

0. Since

f

(again by homogeneity) that y2 If

(y

Then

> 3

p

If

p

(t

=

l)a2

+

3 then y2

=

=

=

a2

and

says that +

(x

t

and

2a2

=

(y

x2

spn-1 * 0.

=

a)2

=

0, giving

Suppose there is x

(x')~ = 62spn-1

Since Jkv a2a2

+

=

VJk

fi2b2.

+

=

If

o

( x ' ) ~= x2 x2

o

=

z

+ i

p

=

3 we use

(x

-

o

have

0

imply

=

(x

+

t + 1 =

0, a contradiction.

2a2 0

Jk', x x a

are non-squares.

a x (y

=

6

x x b

=

0,

+

fib that

=

aa

+

0.

62 @ 0, 1 mod p

with

Necessarily a and 6 are not both 0. But

x , so

v

> 2 to produce x, y independent so that

=

0

=

=

and x' x b are not both

x' x a

a)

+

implies y2

-

0. Recall that here

=

Z x J1

But

-

a)2

~1

E

=

p > 3 we can find

x' x a

0 we know

f

x' x b

0, a contradiction.

=

Thus

p > 3.

x x J1 = y x J1 (x

0, 1, and so

implies x'

if

If

i

Jk* implies

E

we know for any x' = 6x

and solve for a t 6 so that (x')~ = spn-l. zero, since

x

a) x J1

+

a) x a

+

by homogeneity, again a contradiction. Thus y x J1 Case 2: J1 of type 111.

and

a non-square mod p.

ta2, t

-a2, so (y

=

aI2

+

o

=

To prove the other direction, suppose

so that both

t

we can choose

a)2

+

(x

x2

219

0,

yI2

=

+

a

+

bl2

z2

=

-3"-1.

Z2

x2

=

=

3"-l.

a2

b2

=

=

3"-'.

If x2

-3"-'

=

3"-l, so the Only elements of Jk' Similarly, if Notice that

says x x y = -x2

and

x2 = 3"-1

X f y

(x

-

E

Jk',

then z (X f

we get

which satisfy E

Jk* and

y) x J1

=

0.

y)2 = x2 says x x y = x2.

0 is the only possibility when x x J1

=

0.

This will complete our argument for Case 2 provided we can find some x

E

Jk'

This contradicts x2

with

x x J1

Jk/pJk:

0, and so x2

+

1)-dimensional subspace, and so the *-annihilator of J1

has dimension at least X

E

=

0. To do this we intersect the *-annihilators of a and b in

each is a ( v

in Jk/pJk suitable

=

i

Jk'.

v.

Since

v

>

2 , this subspace gives us 0

220

D. SARACINO and C. WOOD

then xJ1

>

(v

Lemma 4.

J1x

=

Proof:

Then f o r any v'

-

(or (v, a, pn-1) xa, (x'b

=

(v, a, pn-')).

Thus there is

xb), ( x ' ) ~= 0.

=

and so (x - x1I2

By choosing

xJ1

x2

y2

=

Proof: hence (x

>

(v

Lemma 5. with

0. Similarly

=

0, then

Take y)'

+

=

J1x

=

Notice x

-

v' e! E

Jk'. =

pk-lxl

But (x -

0

=

v',

it

XI)

J1

0,

=

-

0.

=

x 0

x 0

and

y

*

y.

x

=

xy

yx

=

=

0.

a s above and notice that Let pk-lx

=

v, pk-ly

=

(x

v'.

(v',v,x',y').

xy, ( x ' ) ~= (y'I2

+

y) x J1

Then (v,v')

5

x mod Jk-1. and so by Lemma 4 and Lemma 2.4,

x'

=

y mod Jk-1, y'

xy

=

x'y'

yx

=

0 also.

-xy.

=

Thus xy

0, and from

=

x

i

=

y

=

0,

-

(v',v),

In particular,

v, x'y'

=

=

0. But now x - x'

=

yx

J1

we can be

v', pk-ly'

=

*

(v', a, b, pn-l)

so that

x'

Thus x - x'

and so there are x', y' with (v,v',x,y) pk-'xr

= 0,

If J1 is of type I 1 or 111, and x and y are independent

2). =

pk-lx.

Moreover (x - x')J1

0 by Lemma 3 .

=

=

V' we have (v, a, b, pk-')

E

sure that x and x' are independent.

gives

Jkl such that x2

E

0.

=

Take x as in the hypothesis, and let v

by Lemma 3 .

x'a

is of type 11 or 111 and x

If ~1

2).

0. But then

=

0

it follows that 0

We now prove the Classification Theorem, Part A , for the Case I I =

Fp-dim V > 2.

For

J1

=

V , Lemma 1 is all we need.

111, we see that Jk is generated by modifying any element and

b)

to get (y

+

y aa

+

For J1

~ 1 pn-k, , and elements with square 0, by

of

Jk'

by a suitable multiple of

6b)

*

=

J1

of type I1 or

a

(or of

a

0, then applying the previous lemmas.

The verification that the resulting J's

are homogeneous is routine.

Finite Homogeneous Rings of Odd Characteristic

5. Small V

221

Classified.

In this final section we consider V

of dimension

over

2

Fp, using

results in Sections 2 and 3 to obtain the rest of our Classification Theorem. 2

Recall that from Proposition B we know that Jk f

then t h e d e s c r i p t i o n x2

( i ) is e x a c t l y a s f o r Case A , e x c e p t t h a t now

€3

From

0.

=

*3"-'

=

is

a l s o possible.

J ~ 0

{X,Y}

BIB is By the

Hence

,

n

5

L

then

an

.

.

0

a

Suppose

be n partition of

representation of

By Lemma 2.2, either

BC_ X L

L

or

.

A

.

and

BC_ Y ,

Thus a

so

is a

0

+

R(A)

n-CPP

is an n-CPP

representation, then for some finite

representation.

Proof: Use the Compactness Theorem and induction on n

.

0

We will be interested in those finite lattices which, for each n n-CPP

.

being a representation of

T I ~ ,Y respectively, is a representation of

If a : L

, C X ~ Bis

BIB

for some x

0-CPP representation.

Lemma 2.4.

(alX)(O)

representation and m

without loss of generality, let

be such that

lrlB = (alB)(x) either

u(y)IB

that

By the definition of

representation.

Proof: Clearly it suffices to consider just m A = X U Y

.

, so

X # A # Y

.

n = 0

Then there is B C_ A

and

,

[a(y) IBI # 2

,

a(x)IY

=

{X,Y)

TI =

representation of

inductive hypothesis

=

Assume

.

is a representation, yet

alY

(a(Y)(O) = a(0)IY

and consider the partition

{X,Y) , where

a(x) n

229

representations.


v(h') for P,n some h" which implies v(pn-lh' + pnh") > v(h) and therefore the contradiction v(h+pnh") > v(h). Thus vp(h') = v(h') + 1 and G satis0 fies r (v(h') + 1). PI 1

SECTION 2 : GROUPS WITH MORE THAN ONE VALUATION The purpose of this section is to show that admitting more than one p-valuation, be it for different primes or the same, yields undecidability even for very restricted classes. We first present a hereditarily undecidable auxiliary class AUX which is designed so that it can easily be interpreted in theories of valuated groups. The language of AUX contains three unary relation symbols X1,X2,X3 and three binary relation symbols 1 such that vm occurs in 'po. Then 3xqo is equivalent to:

where q; arises from 'po by replacing each occurence of a value term v(t) by the variable and likewise vm(t) by .c: Notice that x does 0 no longer occur in cp;.

ct

Lemma 3.6. In order to prove theorem 3.1 it suffices to find an TV*-equivalent formula without group quantifiers for every L*-forP V mula of the form (3.7)

3x$

where $ is a conjunction of formulas of the form vm(t(x)) =

T-

with varying m 2 2, group terms t(x) and rl a value variable or one of the constants co,c1,c2. Proof: We will in a series of reductions replace a formula of type (3.5) by a Boolean combination of formulas without group quantifiers or of type (3.7). In order to keep notation down to managable size we will arque rather informally. In particular we will

-

frequently usethe trick to replace 3x'p by the conjunction of vQ, a consequence of the axioms 3x(v&Qi) for l z i l k with $, v

...

P.H. SCHMITT

258

-

not display assumptions which do not involve the quantified variable after they have been explained once

- switch to semantic verifications without extra warning. Let us first fix in greater detail our point of departure: A v(n.x+t.) = ni & A vmi(nix+ti) = 0 . ) iEI1 iEI2 where ti are terms not involving x and q i are value variables or one

(3.8)

3x(

of the constants co,c1,c2 and I1 f 0. Using torsionfreeness, Axiom (T2) and Lemmas 1 . 1 (b), 1 . 2 (b), 1.4 (i) we may assume that for all i E I1 U I2 : ni = n. Replacing nx by x and adding vn(x) = c2 we may even assume for all i E I1 U I2 ni=l (using Lemma 1 . 2 (f), (g)). At this point we see that we may assume ni < c 1 for all i E I ~ .~f n i = c 1 occurs, we eliminate x by replacing all occurences of x by the term -ti. If n i = c2 occurs the formula is contradictory and thus equivalent to, say co # co. By distinguishing cases we may assume that some order between the ni is fixed. Let 17 = max{qi : i E I,}. For i E I1 with n i < we have by Lemma 1 . 1 (a) v(x+ti)

=

ni

iff

v(x+t)

=

n

&

v(t-ti) =

ni

.

Since v(t-ti) = n . does no longer contain x we may assume w.1.o.g. that for all i E I1 : ni = n . Using Lemma 1 .2 (b) and 1 .4 (i) again we may assume furthermore that for all i E I2 mi = m. Now the arguments of vm are terms si(x) which will in general not be of the form x + t i , but this will no longer be of importance here. It seems appropriate to give an update of the formula (3.8) after all the above reductions have been performed:

By a variable transformation x

+

x+ti we may assume that for some

i E I 1 ti = 0. Thus (3.9) is equivalent to the disjunction of the following two formulas:

.

A V(X+ti)=q& A Vm(Si(X))=ni&Vm(X)="l) iEI1 iEI2 We will show that (3.10) is equivalent to the conjunction of the following two formulas:

(3.11)

3X(V(X)=n&

Decidable Theories of Valuated Abelian Groups

259

.

A V(Si(X)) = lli & Vm(X) > ll+l) iE12 One of the implications of this equivalence is trivial. For the reverse implication we assume that g,, resp. g1 is a witness for the existential quantifier in (3.12) resp. in (3.13).

(3.13)

3X(

Since vm (go ) > n+l is true we have v(go-nh) 2 - n+1 for some h. Thus A

iEIl

v(nh+ti) =

n

&

v(nh) = n

.

Since also vm(gl) > n+l we may by Lemma 1.2 (c) assume that v(gl)

2 n+l. Thus g1 + nh will be a witness for (3.10).

Let us next deal with (3.12). We may assume that n 5 co, since otherwise (3.12) is contradictory. Since for i # j v(x+ti) = q & v ( t i - t . ) > n implies v ( x + t . ) = rl 3 7 and v(x) = n & v(tj) > n implies v(x + t . ) = n we may assume 7 (3.14) for all i,j E I1 with i # j: v(ti-tj) = & v(t.) = 0. For i E I1

v(x) =

r)

and v(ti) =

q

already imply v(x

+ t1. ) -2 n.

Thus

vm(ti) = n+1 and vm(x) > r l + l would imply vm(x + ti) = n + 1 and therefore v(x+ti) = n . In this case we would drop i from 11. Thus we may assume (3.15)

for all i E 11: vm(ti) > n + l

.

This leads us to the conclusion that (3.12) is equivalent to (3.16)

11~1 .

I G ( ~ )n mG/G(n+l) n mGI >

To prove this let g be a witness for (3.12), then we have v(g-mgo) 2 - n+l for some go and by (3.15) also elements ty for i E I1 such that v(ti-mty) 2 q+l. Using the properties of g and (3.14) we see that mgo,mtp for i E I, represent different elements in G ( n ) n mG/G(n+l) n mG. If on the other hand (3.16) is satisfied we are guaranteed to find an element "go in G(Q) n mG, mgo

B

G(n+1) such that

mgo + G(n+1) # -ti + G(n+l) for all i E I1. Thus v(mgo) = Q & A v(mgo + ti) = n & vm(mgo) = c2 > n+l iEI1 By Lemma 1.10 (3.16) can be expressed by an L:-formula quantifiers.

. without group

Now we take up the further reduction of (3.11). We may again assume (3.14). But instead of (3.15) we have this time

P.H. SCHMITT

260

(3.17)

for all i E 11: vm(ti)

=

.

q+l

For assume to the contrary that vm(ti) > n+l for some i E 1 1 , then there would be go with v(ti +mgo) 2 n+1 and v(mgo) = v(ti) = 0 . Because of vm(x) = n+l we must have v(x+ngo) = n thus v(x+ ti) = n. We would consequently drop i from 11. We finally claim that (3.11) is equivalent to (3.18)

3x(

A

iEI1

vm(x+ti)

=

n

&

A

iEI2

vm(si(x)) = n

&

vm(x)

=

q+l)

.

One implication is easy using (3.17) and Lemma 1.2 (e). If on the other hand g is a witness for (3.18) then v(g+mgo) = for some go. Since by Lemma 1.2 (c) q + mgo is also a witness of (3.18) we may have assumed v(g) = I- right away. Since v(ti) = n , we have 2 r j for all i E I1. But since v(g+ ti) > would contradict v(g+ t.) 1 vm(g + ti) = n we must have v(g + ti) = n . 0 Lemma 3.19. To prove theorem 3.1 it suffices to eliminate the group quantifier in L:-formulas (3.20)

of the form

3Xbq

where Jiq is a conjunction of formulas of the form (ms+t) = n qrs for a fixed prime q, which may be equal to or different form p and varying m,s 2 1, group terms t without x and rl a value variable or one of the constants co,c1,c2.

v

Proof: We start with a formula of type (3.7). Using the same transformations as in the first step in the proof of Lemma 3.6 and Lemma 1.2 (h) we may restrict attention to formulas (3.21)

A Jigj 021$ already meet the requirements of (3.20). where the Ji qj It remains to be shown that the existential quantifier distributes over the conjunction. Let s be the highest exponent for which v qrs appears for some prime q in (3.21). Let n be the product of all q;, j 5 k and n . = n/qs. Then there are integers m . with 7 3 7 1 = mono + + mknk. If q . are witnesses for

3X

...

3

3xJiqj= let 9

=

monogo + V

qjns1

jX

~A E

...

Iqjrsi ~ (x + ti) =

rl.

+ mknkgk. We have by Lemma 1.2 (f), (g)

(m . n . g .+ m . n . t . )= 3 3 3

7 7

=

q.

26 1

Decidable Theories of Valuated Abelian Groups

for all j and all i E Ij. Since njo is divisible by qs for every 3 jo # j we obtain by Lemma 1.2 (c) v , (g+ti) = ni for all j and q~ rsi 0 i E I. 3'

In the remainder of this section we will eliminate the group quantifier in formulas of type (3.20) for a prime q different from p. The equal prime case will be dealt with in the next section in slightly greater generality. Starting from the formula ri (3.22) 3x A vq,si(miq x + ti) = 'li iEI with (mi,q) = 1 for all i E I we use Lemma 1 .2 (b), (f), (9) to obtain for all i E I : s . = s and mi = m. The x-free terms ti will of course change in this procedure. Now we may w.1.o.g. replace m by 1. Indeed if some element b satisr. fies for all i E I v (q b ' + ti) = ni then we have 1 = nlm + n2qs qts for appropriate n1,n2 and hence ~ ~ , ~ri (n l m bq+ t i ) = ni by Lemma 1.2 (c). We will use in the following for notational simplicity w to denote V

q,s' Thus it suffices to consider formulas of the form (3.23)

3x( A O$i<s

A

tEMi

W(qiX+t)

=

Qi,t)

.

By introducing possibly new existential value quantifiers as we did in the proof of Lemma 3.4 we may assume w.1.0.g. 0 E Mo

and

q M i s Mi+l

for all

.

i, 0 5 - i c s-1

By distinguishing cases we may assume that some linear order is fixed among the value-variables and constants ni,t. With respect to this . = 11i,to and rl i,tl < ni order we define ni = max{nilt: t E Mi}. If n 1 for to,tl E Mi then w(qix+tl) = ni,tl iff w(qix+to) = ni,to and w(to-tl) = ni,tl. Thus tl may be dropped from Mi. (3.24)

A W(qiX+t) = Qi) 3x( A O$i<s tENi where N. = {t E Mi : ni,t - ~ i } and

5

~1

5

- - *

5

ns-l

-

To see why the given chain of inequalities may be assumed, take some t E Ni for some i, 0 6 i < s-1. Because qt E Mi+l we get i+l xqt) = qi+l,qt 5 ni+l. If in the assumed ordering ni = Q1,t s w(q of the ni we have ni+l < q i then we are facing a

262

P.H. SCHMITT

contradictory formula and its group quantifier can trivially be eliminated. We also assume n i # co since otherwise ( 3 . 2 4 ) is again contradictory. To prepare the most important reduction step we first collect those i and j together for which n i - nj. Formally we define: ko

= O

{

the least element in { i

:

k ]= .< i 5 - s-1 and

k.

=

r

= the least j such that kr+l = s

K. 3

is not empty s otherwise

= [k.,k. )

5

3 I+? Furthermore we set

{O

,...,S-1)

W(qiX+t) = Il* & A A 7 i€Kj tENi for 0 n 1 and w(q'+lh+.r) = =

W(q'+lg+T)

nl+,

>

n l we get

Decidable Theories of Valuated Abelian Groups

263

w(q1+l (g-h)) > q l . By Lemma 1.4 (iv) there is an element c such that (1)

w(g-h+c) > q 1

(ii) ql+lc = qsc' for some c'. Now (3.28) and (i) yields by Lemma 1.2 ( e )

.

A A w(qi(h-c) + t ) = q i k. (b)

1

0)

3x(w(q x + to) 2 q )

if

1+1 < s

if 1+1 = s , with to an arbitrary element from N1.

Proof of Claim 3.30. Only one implication of the claimed equivalence

P.H. SCHMITT

264

is non-trivial. Let g be an element satisfying w(q'+lg+T) > I-). Let tl be an arbitrary element of Nk. From 3.29 (a) we have > q which yields w(ql+'g + ql-k+ltl) 2 n Lemma 1.4 w (ql-k+lt 1 - T ) (iv) provides us with an element go such that w(g0) 2 rl and ql-k+l(9, - tl) E ql+l (mod qs). By this last congruence we find an 1+1 element u such that qku F (go- ti) (mod qs) and also ql+lu q g (mod qs). Therefore we have w(q'+lu+~) > n . But now we have in addition w(qku+ t1) = w(go) 2 n. By 3 . 2 9 (a) this gives w(qiu+ t) 2~ for all i,k 5 i 5 1 and all t E Ni. But w(qiu + t) > 0 would give

.

Wq,i(t) >

n+l which is by Lemma 1.5 equivalent to w(qit) > q , contradicting the assumptions of Case I. Thus u is a witness for ( 3 . 2 9 ) . 0

By definition 3x(w(q1+'x + T ) > n ) is equivalent to wq,l+l ( T ) > q+1 which is by Lemma 1.5 (iv) equivalent to w(qS-'+lr) > n+1, an LC-formulas without group quantifiers. Likewise 3x(w(q1x + to) 2 0 can be replaced by w(qs-'to) T-. We may thus assume from now on Case 11. Let ko be the least number i r k 5 i 5 1 such that for some t E Nko w (qs-kot) > n . We fix some to E Nko with this property. We claim that ( 3 . 2 9 ) is equivalent to (3.31)

3x(

A A W(qiX+t) = & w(q'+'X+T*) > l) k o ~ i ~ tEN; l where of course the last conjunct is omitted for 1+1 = s l-ko+l and N* = {t-qi-koto : t E Nil, T * = T - q to .

TO verify this claim we first note that by the choice of to and Lemma 1.5 (iv) there is some t; such that w(to - qkot&) > rl which i-kOt0-q1tA) > r- for all i,ko 5 i 5 1 and implies w(q w (ql-ko+'t 0 - ql+l t;) > q. From this we see that h = g + t& is a witness for ( 3 . 3 1 ) . If on the other hand we are given h satisfying ( 3 . 3 1 ) then g = h - t & certainly satisfies (3.32)

A

ko5izl

A

tENi

W(qig+t) = ll

&

W(q'+lg+T)

> 11

.

But what about the conjuncts for i < ko. Let tl be some arbitrary element in Nk. As in the verification of claim ( 3 . 3 0 ) we find using Lemma 1.4 (iv) some element u with qkou 3 gkog (mod q s ) and w(q ku + t,) 2 q. Thus u satisfies ( 3 . 3 2 ) in place of g and as before we find

265

Decidable Theories of Valuated Abelidn Groups

Considering the way we obtained Ni from Mi (see ( 3 . 2 4 ) ) we notice that for all i,k 5 i < 1 qNi 5 Ni+l. This yields (3.33)

(i)

qN:

(ii)

o

E

5 Nr+l for all i,ko 5 i < 1

~c~

(iii) for all i,j,ko 5 i 5 j 5 1, t E NT, t ’ E NT 3 w(qJ-it - t’) 2 q. This allows us to reduce ( 3 . 3 1 ) further to 1 (3.34) 3X( A W(q X + t ) = Q & W(ql+lX+T*) > 0 ) tEN where we have used N to abbreviate N* 1 ‘ Indeed assume that g is a witness for ( 3 . 3 4 ) . Since 0 E N we have in particular w(q 1g) = I-.Using Lemma 1 . 4 (iv) we may assume w.1.o.g. w(g) = I-,which gives by ( 3 . 3 3 ) (iii) at least w(qlg+t) 2 I- for all i,ko 5 i < 1 and all t E Ni. But w(qig + t) > I- would imply 1-i w(qlg+ql-it) > I- which contradicts q t E N.

0

We may further assume (3.33)

(iv) w(t)

= I-

for all t E N, t # 0

since we have in any case w(t) r- and those t E N, t # 1 w(t) > T- is true may be dropped since w(q x + t) = 0 iff w(qlx) = Tl & w(t) > n.

0

for which

If I- = c2 then we may assume that the last conjunct in ( 3 . 3 4 ) is missing and the remaining formula is equivalent to the purely group theoretic formula

By the results in [ I ] this formula free formula Qo in a language with dicates for the Szmielew invariant q‘. But Qo is in TV* equivalent to P Lemma 1.2 (b) and Lemma 1.11.

is equivalent to a quantifier divisibility predicates and preB(G) = dim G/q‘G for all primes a quantifier-free formula by

In case n 5 c 1 ( 3 . 3 4 ) is equivalent to the formula (3.35)

q 1x #

-t & q1+’X = 7 tEN in the group H ( Q ) = G(qs,q)/G(qs,r)+l) with the bar denoting the canonical homomorphismus from G(qS,n) in H ( Q ) . 3x(

A

Again using [ I ] we know that ( 3 . 3 5 ) is equivalent in the theory of groups which are direct sums of copies of E(qS) to a quantifier-free

P.H. SCHMITT

266

formula

JI, i n t h e l a n g u a g e c o n t a i n i n g p r e d i c a t e s f o r t h e r e l e v a n t

Szmielew i n v a r i a n t s , which i n t h i s c a s e i s j u s t t h e d i m e n s i o n o f H ( q ) [ q ] . But JI1 i s i n TV* e q u i v a l e n t t o a q u a n t i f i e r f r e e f o r m u l a P s i n c e r q ( q ) 2 n i f f dim H(q)[q] n by Lemma 1 . 8 ( i ) f o r q 5 co and s i m p l y checked f o r SECTION 4 : VALUATED

L e t p be a prime, s

q =

c1.

0

GROUPS 2

1 an i n t e g e r . A p - v a l u a t e d g r o u p ( G , w )

is

c a l l e d a v a l u a t e d pS-group i f G (as a g r o u p ) i s a d i r e c t sum of c o p i e s of

(Kl)

Z(ps

(K2) For e v e r y m , 0 < m < s and e v e r y v a l u e B 5 c o , g E G i f w(pmg) = 6 t h e n t h e r e i s an element h E G such t h a t B = w ( h + m and pmh = p g . One r e a s o n f o r s t u d y i n g t h i s c l a s s of p - v a l u a t e d Lemma 4.1.

g r o u p s i s g i v e n by:

( G , v ) i s a t a m e l y p - v a l u a t e d g r o u p t h e n (G/qSG,v

If

i s a v a l u a t e d pS-group.

PIS

)

-

= (G/pSG,v ) a renaming of P,S t h e c o n s t a n t s o f Lv i s needed: cz i s o m i t t e d , it n k v e r i s a v a l u e

Remark: I n d e f i n i n g t h e s t r u c t u r e ( H , w )

cy

for v

cz and cG2 = cHl .

=

Proof: S i n c e G i s t o r s i o n f r e e (Kl) i s t r u e . 1.4

( K 2 ) f o l l o w s from Lemma

( v ) . That v

i s a p - v a l u a t i o n f o l l o w s from Lemma 1.4 ( i i ) . [3 PtS S i n c e w e d o n o t r e q u i r e t h a t i n v a l u a t e d pS-groups w ( g ) i s n e v e r a

l i m i t t h e r e are v a l u a t e d pS-groups which d o n o t d e r i v e from t a m e l y p - v a l u a t e d g r o u p s i n t h e way g i v e n by lemma 4 . 1 . Examples. (El)

V a l u a t e d p-groups

are v a l u a t e d v e c t o r s p a c e s o v e r t h e f i e l d

w i t h p e l e m e n t s i n t h e s e n s e of

[2].

L.

F u c h s ' d e f i n i t i o n i s more

g e n e r a l by a l l o w i n g a r b i t r a r y c o m p l e t e l i n e a r o r d e r i n g s a s s e t of values. (E2)

L e t g b e a g e n e r a t o r f o r Z ( p s ) . W e d e f i n e w by w(pig) = a + i

for 0 < i < s and w ( 0 ) = c l . Then ( Z ( p s ) , w ) i s a v a l u a t e d pS-group. (E3)

Direct sums of v a l u a t e d pS-groups a r e a g a i n v a l u a t e d pS-groups.

I n s t u d y i n g v a l u a t e d pS-groups w e w i l l u s e t h e e x t e n d e d l a n g u a g e L** = Lv U { c 2 1 U I d n ( ) V

: n

2

1)

Decidable Theories of Valuated Abelian Groups

261

where t h e i n t e r p r e t a t i o n of d n i n t h e v a l u a t e d pS-group

(G,w)

is

g i v e n by: G k dn(n)

d i m ( G ( q ) n p S - l G / G ( n + l ) n pS-’G)

iff

where G ( n ) i s t h e subgroup { g E G : w ( g ) W e need n o t i n c l u d e i n L *:

n 5 co

L n}.

T V ( p , s ) d e n o t e s t h e Lv-theory of v a l u a t e d pS-groups, L$*-theory.

for

TV*(p,s) t h e

names f o r t h e mappings wm

s i n c e t h e y c a n b e e x p r e s s e d i n t e r m s of w w i t h o u t u s i n g g r o u p quant i f i e r s as t h e n e x t lemma shows.

Lemma 4 . 2 .

For v a l u a t e d pS-groups

(G,w)

t h e following are t r u e f o r

a l l g E G and m, 0 < m < s : (i)

~ ~ , ~ = (w g ( g )) + 1

(ii) wn(g) = c 2

for

for all

n

n

L

s

prime t o

p

( i i i ) i f w p , m ( g ) i s a l i m i t number, c o , c l o r c 2 t h e n Wp , m ( g )

= Wp,m+l ( P g )

(iv)

if w ( g ) i s a s u c c e s s o r number < co t h e n Plm W p,m+l ( P 9 ) = Wp,m(g) + 1

(v)

~ ~ , ~+ s( -gm -) 1 = ~ ( p ‘ - ~ g )

.

Proof: ( i ) t r i v i a l s i n c e pmh = 0 f o r a l l h E G . ( i i ) t r i v i a l s i n c e g i s d i v i s i b l e by a n y n prime t o p.

B e f o r e p r o v i n g ( i i i )w e n o t e : I f w p , m ( 4 ) < Wp,m+l (Pg) t h e n w(pg + p m + ’ h ) 2 ~ ~ , ~+ 1( fgo r) some h. By axiom ( K 2 ) w e have f o r some go E G

w(go)

g o - g + p m h E pS-lG.

2

w ( g ) + 1 and pgo = p g + p m + ’ h . By ( K l ) PIm Thus ( g o - g ) E P G and w e o b t a i n t h e c o n t r a -

d i c t i o n ~ ~ , ~ = ( wpg, m)( g o ) > w(go) Wp , m ( g )

5

2 ~ ~ , ~+ 1( . gS i )n c e

W p , m +(~p g ) i s t r u e f o r a n y p - v a l u a t i o n w e t h u s have Wp , m ( 4 )

I

Wp,m+l ( P 9 )

5 Wp,m(9) + 1

-

(iii)I f w

( 9 ) = f3 i s a l i m i t number o r co t h e n t h e r e c a n be by Plm ( K 2 ) no h s u c h t h a t w(pg + pm+’h ) = B . Thus w p,m+l ( P 4 ) = Wp,m(g)* If W p , m + l ( p g ) = c 2 t h e n pg E pm+’G which i m p l i e s by ( K l ) g E P G , i . e . Wp,m(g) = c 2 . (iv) Let w ( 9 ) be R+1. Then t h e r e i s s o m e h s u c h t h a t w(g+pmh) = R Prm and t h u s w ( p g + p m + l h ) 2 B+1, i . e . w (pg) 2 B+1. p,m+l 0 ( v ) F o l l o w s from ( i i i ) & ( i v ) .

268

P.H. SCHMITT

Lemma 4 . 3 . (i)

For all

n

< co:

G(n)/G(q+l) is a direct sum of cop es of Z(p)

(ii) G(n)/G(q+l) = G(n+l) (iii) If n

=

fl

plG/G(q+l+l

O'w+r, 0 < r < 1 then G ( n )

(iv) G ( C ~ )n P~G/G(C,) n p 1G

= 10)

n plG Il

for 1 >

plG/G(n+l) n plG

o

= {O}

.

Proof: (i) clear, since w(pg) > w(g) for w(g) < co, (ii) Since p 1G(n) c_ G(n+1) n p1G the mapping g * plg is a homomorphismus from plG(q) into G(n+l) fl plG/G(n+l+l) Il plG. By (K2) it is both injective and surjective. (iii) By (K2) w(p 19) n implies already w(p 1g) 2 r- + (1-r). (iv) By (K2) w(pg)

2 co implies already w(pg)

=

c l , i.e. pg = 0.

To apply the results of this section to complete the proof of Theorem 3 . 1 we need the following observations connecting the invariants r and dn.

Pt*

Lemma 4.4. Let (H,v) be a tamely p-valuated group, (G,w) = (H/pSH,vprs).Then (i)

in (G,w) dn(n) is false for all n 2 1 and q of the form = O . w + r , 0 5 r < s or = CI;

n

(G,w) C dn(n + 1) iff (H,v) C r ( n + 1-s+l); Prn (iii) in (G,w) dn(co) is false if s > 1 and (G,w) C dn(co) is equivalent to (H,v) C rp (cl) if s = 1. ,n Proof: (i) By (K2), (ii) by Lemma 1.8 (ii), (iii) by definition. (ii) for 1

2

s:

Theorem 4.5. Every L:*-formula is in TV*(p,s) equivalent to a formula without group quantifiers. Lemma 4.6. In order to prove theorem 4 . 5 every L:*-formula of the form (4.7)

it suffices to find for

3xcp

where cp is a conjunction of formulas of the form w(t) = for group terms t involving x and tl a value variable or one of the constants co,cl an TV*(p,s)-equivalent formula without group quantifiers. Proof: same as for lemma 3 . 4 .

0

Decidable Theories of Valuated Abelian Groups

269

The elimination of the group quantifier in formulas of the form (4.7) greatly parallels the reductions performed in section 3 from ( 3 . 2 3 ) onward where the role of the prime q is now taken over by p. The main difference lies in the fact that we have now w(pg) > w(g) while in section 3 w(qg) = w(g) was possible. For this reason we will give a very sketchy proof of theorem 4.5 indicating only the major steps in the reduction and refering to the corresponding parts of section 3 for the trick to be used. Starting from (4.7) we obtain as a first reduction (4.8)

3x( A A W(piX+t) = lli,t) Oci<s tEMi with t group terms not containing x and 0 for 0 5 i < s-I

.

Mo, pMi 5 Mi+l

E

This uses only the fact that w is a p-valuation and the possible introduction of new value quantifiers. Set r)i = max{qilt (4.9)

t

:

E

Mi}. Then ( 4 . 8 )

is equivalent to

3X( A A W(piX + t) = ni) OCiCs tENi with Ni = {t E Mi : TI. = nil and l,t (see ( 3 . 2 4 ) ) .

'lo < q l
Q+1)

.

Decidable Theories of Valuated Abelian Groups

27 1

This is certainly a consequence of (4.14) 3X(

A

tEN

W(pS-lX+t)

= rl)

.

For assume g is a witness for (4.14) then p s-1-1 g would be a witness for (4.13). If on the other hand g is a witness for (4.13) then 1

w(pl+’g) > T- + 1 implies by (4.2) w p,s-l (p 9) > T- + 1, thus > q + 1 for some h and h would be a witness for (4.14). w(p1g+pS-lh) Thus (4.13) and (4.14) are actually equivalent and we are reduced to the case of (4.12) with 1

=

s-1

.

We assume therefore from now on (4.12) (C) W(T) = II + 1

.

We also note that (4.12) (d) wp,l+l ( T ) > T - + 2 is implied by (4.12). We define No = {t E N : w(pt-r) > rl+1]. Claim 4.15.

(4.12) is equivalent to dn(q) with n the smallest number

such that pn 1. I NoI + 1.

Proof: Assume first that g i s a witness for (4.12) in some valuated pS-group (G,w). For all t E No we get w(p(plg + t)) > rl + 1 . Thus there are by axiom (K2) elements c; such that pc; = p(plg+ t) and w(c;) = w(p(plg+ t)) - 1 > 0. Setting ct = ctt -pig+ t we get pct = o and w(ct) = q . By axiom (Kl) we must have ct E pS-lG. For to,tl E No with to # tl we have w(cto - ctl) = q , since w(cto - ctl) > rl would imply w(cio - cil + to - t l ) 2 q + 1 and therefore w(to- t1) > q + 1 contradicting (4.12) (a). Thus I G ( q ) fl pS-lG/G(u+l) n pS-lGI ->INo[+l which implies dn(q).

...,

If on the other hand c l , cm are m = IN0 I + 1 representatives Of different cosets in G(q) I7 pS-lG modulo H = G(u+l) fl pS-lG. Let ci E G be such that p’c; = ci. By (4.12) ( d ) we find T~ such that W(P To + T ) 2 q + 2 which gives w ( p l + ’ ~ ~ =) q + l by (4.12) (c). 1 Axiom (K2) allows us to assume w(plTo) = q. Since p T~ + H, p1(-rO+ci) + H 1 5 i 5 m are (m+l) different cosets, we find some

c in {ci : 1 5 i 5 m} U { O } such that for all t E No W(P’(T~+C) + t) = q while pl+’ ( T ~ + c = ) pl+’~, still guarantees

W(P l+’T0+T)

> ll+1.

Finally consider t E “No

for which we certainly have

P.H. SCHMITT

212

w(p 1 ( T ~ + C+)t ) w(p'+l~~+pt) > n (4.12).

rl.

+

But strict inequality would imply

1 contrary to t

Thus T O + c is a witness for

No.

A s easy consequence of theorem 4.1 we obtain

Corollary 4.16. For every L**-sentence cp there is some Lr*-sentence V $ such that for all valuated pS-groups (G,w) (G,w) k cp

iff

Val**(G,w)

.

C $

Corollary 4.17. For any two valuated pS-groups (G1,w?), (G2,w2): (G,,wl)

(G2,w21

iff

Val**(Gl,wl)-Val**(G2,w2)

.

SECTION 5: DECIDABILITY RESULTS The major problem in proving decidability of the class of all tamely p-valuated groups, namely which Lf-structures do occur as the value part of a tamely p-valuated group, is solved in the following lemma. Lemma 5 . 1 . Let a be a well-ordered set, co,c1,c2 element such that subsets of CY U {co,c1,c2) then the folloa < c < c1 < c2 and r qIn wing conditions are equivalent: (I) there is a tamely p-valuated group (G,a,v) such that ) N Val*(G,v); q,n for all q and n 2 1 and all y:

(a U {co,cl ,c2),

For every

cardinality

w

,

there are

2x

nonisomorphic ULF

x.

(vii) There exists a locally finite group H every

x 2 w1

groups of

there exists a

ULF

of cardinality

group of cardinality x

w1

such that for

in which H

does

not embed. Macintyre and Shelah used Ehrenfeucht-Mostowski models to construct their nonisomorphic ULF

groups.

Consequently we do not have a very clear idea of the

structure of these groups, and it remains an interesting problem to construct 2' nonisomorphic

ULF

groups which are nonisomorphic for simple "group-theoretic"

reasons. Hickin solved this problem for x = w1 nonisomorphic complete ULF

in [ 2 ] , where he constructed

groups. He also showed that no locally finite

278

S. THOMAS

group of c a r d i n a l i t y

o1

is inevitable.

be i n e v i t a b l e i f i t embeds i n e v e r y equal t o

IHI

.

ULF

(A l o c a l l y f i n i t e g r o u p

H

is s a i d t o

g r o u p of c a r d i n a l i t y g r e a t e r t h a n o r

Macintyre introduced t h i s n o t i o n i n [ 4 ] where, assuming

h e showed t h a t t h e r e are no i n e v i t a b l e a b e l i a n g r o u p s o f c a r d i n a l i t y Similar r e s u l t s w e r e obtained f o r

x

=

2w

by S h e l a h

0

,

wl.)

[a].

I n t h i s paper, we s h a l l p a r t i a l l y extend H i c k i n ' s r e s u l t s t o a r b i t r a r y successor c a r d i n a l s .

Our main r e s u l t is:

THEOREM Let

h z w

.

ULF

groups of c a r d i n a l i t y

(a)

if

(b)

(G.C.H.)

S

5 G5

{GSl 5
$

in K

C

G.

such that < K,a,b >

2



.

Then f can be chosen so that

PROOF Let g"

h € H

satisfy '1 = h.

Since

T

f Hp, there exists

g € H

such that

# gh. We still have freedom in the choice of coset representatives for bH,

baH, cH and

caH. We select

b, ba, c and

-1

cag

respectively. Then,

regardless of the choice of the other coset representatives, we have (bh)f(T)

-1

p(a)f(")

(,,)f(T)-'p(a)f(T)

=

bah

=

-1

7

.

0

This lemma will be used to restrict the size of soluble subgroups in the construction. The next lemma enables us to build many nonisomorphic ULF

groups.

LEMMA 4 . With the notation of lemma 2 , suppose that: is an involution.

(i)

a 6 G\H

(ii)

z c s(H)\H~.

(iii) there exists h € H Then

f

such that ah-lhT a ! , H.

can be chosen so that

PROOF.

-1 7 By hypothesis, haH, ha h h a H hah-l, hah-lhTa h-l

and H

are distinct cosets. We choose

as coset representatives. Thus

Complete Universal Locally Finite Groups of Large Cardinality

CASE 1.

hah-'hTa

h-'h"a

t?

h"a

28 1

H.

We a l r e a d y know t h a t (ha)~(a)f(")~(a)f(") =

(h" a)f(")

F h" a H.

This h o l d s r e g a r d l e s s of whether we s t i l l have a c h o i c e of c o s e t r e p r e s e n t a t i v e for

Thus w e have

h" a H.

(ha) d a ) f (7) d a ) f ( d

Case 2 .

a h - l h" a €

hah-'h"

Clearly

h" a H

h"aH = ha h-lh"

a H.

+

(ha) f ( 7 ) p ( a ) f ( T ) p(a)

h" a H.

# H and by h y p o t h e s i s Then t h e r e e x i s t s

h" a H

g F H

hah-l h" a h-l h"

# haH.

Suppose t h a t

such t h a t

= h" a g.

The assumption i n t h i s c a s e i m p l i e s t h a t h" aga € Thus

ga € H

and so

h" a H.

a € H, a contradiction.

h" a H

# hah-l h" a H.

w € H

such t h a t

A second a p p l i c a t i o n of t h e assumption y i e l d s an element

hah-l h" a h-l h" a Since

z 6 Hp,

there e x i s t s

as c o s e t r e p r e s e n t a t i v e f o r

z € H

h" a H .

=

Once a g a i n , w e have

We conclude t h a t

= h" a w .

such t h a t

z"

Thus

h" a z-lz"

.

# zw.

W e choose

h" a 2-l

282

S . THOMAS

We now u s e lemma 3 t o prove: LEMMA 5 Let

x

be a l o c a l l y f i n i t e group of c a r d i n a l i t y

G

f i n i t e isomorphic nonconjugate subgroups of f i n i t e group

3

< G,T >

(i)

F:

(ii)

I f u E < G , z >\G,

5

.

w

F1,

Let

F2

G.

Then t h e r e e x i s t s a l o c a l l y

GI1




satisfies the

w i l l be d e f i n e d i n d u c t i v e l y ,

GS

by completing diagrams a s i n lemma 3 .

By lemma 1, t h e r e e x i s t s

zo E S(GO)

such t h a t

'col Ff zo = F i .

< = q + l Let

=

I G q+l

= p..

We may assume t h a t t h e c h a i n ,

has been c o n s t r u c t e d so t h a t f o r each

Let

X = (G

\

Gq)

x

( < Gp,

+1+1 well-ordering, X = < xi

I

z >\Gp

q r l

q

0 < i < p.

i < p

Then

).

,

there exist

1x1

=

so t h a t i f

p

bi,

G

q+l ci

U H?1' i

then a C

"7.

Complete Universal Locally Finite Groups of Large Cardinality

283

We shall use lemma 3 to complete the diagram P

G

P1

:f

GT

P

> S(+

in :H

Assume inductively that coset representatives for G

T

so that, regardless of further choices, we will have

have been chosen

f(n)-'p(g)f(n)

f GG1

for each x. = < g, n > with j < i.

By lemma 3, we can choose coset

representatives for GT

that

J

xi

=

5

< a,a >

.

in l + : H

We define

so

T

y+1

!$ GW1,

where

f(Tn).

=

is a limit ordinal Assume inductively that we have defined elements

i< 5

so that if

i< j< 5

7.

then g

7 T~

P

f(U) -1p(a)f(U)

6

S ( G ) is defined by

5.

the image of

7.

g

7

= g

for a l l

zi € S(Gi) g 6 Gi.

for all Then

5

=

g

for

g 6

Gi. Notice that we have taken

under the direct limit mapping

and that every element of

< GE,

T~

>

has a preimage in some < G f , zi >

for

i < 5 . Thus

< G

T >

satisfies the requirements of the lemma, where

7 = 7

0

A '

3.

KILLING OUTER AUTOMORPHISMS Suppose that

n

is an outer automorphism of the uncountable ULF

group

In this section, we shall show that there exists a locally finite group H such that n

cannot be extended to an automorphism of H.

2

G

G.

284

S . THOMAS

LEMMA 6

Let

b e a s i m p l e g r o u p of c a r d i n a l i t y

G

b >o

.

Suppose t h a t

pi

€ Aut G

s a t i s f i e s t h e condition: t h e r e exists for a l l

H

5

and

G

g € G

such t h a t

IHI < b

xnH = g x H

and

x E G\H.

Then

x

= g x g

-1

for all

x

G.

PROOF

Choose any e l e m e n t hg # 1.

Suppose t h a t (hg)

class

Then

k

y

G

.

Since

= < H,g >.

yx

Then

x"

=

f o r some

g x h

h € H.

is s i m p l e , i t i s g e n e r a t e d by t h e c o n j u g a c y

G

Hence t h e r e e x i s t s and s o

G1

!! Go

x

y € G

such t h a t

y

-1 hg y

d

G1

=



.

!! G 1. Thus t h e e q u a t i o n xn yn = (xy)"

y i e l d s elements

ho, hl

€ H

such t h a t

g x h g Y h o = g x Y hl Hence

y

have

xz

k

-1 hg y

-1 hl ho € H

=

-1

xpi = g x g

.

Fix

z"

=

g z g

-1

,

From now o n , ie.

G1,

x

1

a contradiction. Go.

Then f o r a l l

So f o r a l l z € Go,

x

!! G o , w e

i t follows t h a t

Hence

Go.

g x z g Thus

5

x'(g)

= g

-1

=(xz)pi=gxg

as required.

@:G x

-1

+

Sym(G)

for all

=pi

.

0

w i l l denote the l e f t regular representation,

x , g c G.

LEEIMA 7

With t h e n o t a t i o n o f l e m m a 2 , s u p p o s e t h a t satisfy: (i) (ii)

"[HI = H . x

c G\H.

pi

€ Aut G

and

g € H

Complete Universal Locally Finite Groups of Large Cardinality

,7 c

s(H)\H~

(iii)

u

(iv)

xn H # g" x H. Then

f

285

can b e chosen so t h a t

PROOF Since

T

assumption x

and

#

H , XH

(g")-'

), t h e r e e x i s t s

p(l'"

and

x'

(g')-'

x'

H

h E H

xf(dP(g)"

h"

# h.la" H in

a r e d i s t i n c t c o s e t s of

as c o s e t r e p r e s e n t a t i v e s .

h

such t h a t

.

G.

By W e choose

Then

=

(g-l

=

(g")-l x" 10'

and

LEMMA

a.

Let

G

be a

ULF

an o u t e r automorphism. 7

C

w and l e t

u € S(G)

n €

Aut G

be

such t h a t f o r a l l

:

>\GP

is infinite.

PROOF Let

X = < gs

element occurs

X

I

5

< X > be a l i s t of t h e elements of

times.

Since

G

G

such t h a t every

i s simple, w e may use leuma 6 t o o b t a i n a n

S. THOMAS

286 expression,

G

=

U

E<X

, satisfying

G

c o n d i t i o n s ( a ) t o ( c ) of lemma 5 ,

together with: (d)

TT

(e)

g5 € G

(f)

[G ] = G

5

5 '

5 ' x € G

There e x i s t s

E+1

\G

5

such t h a t

xTTG

#

5

g :

The r e s u l t now follows e a s i l y from lemmas 5 and 7.

x G5

0

DEFINITION.

u € Sym(G), t h e n

If

sup(cr)

=

I g'

{g € G

# g}

.

t h e subgroup c o n s i s t i n g of t h o s e permutations s a t i s f y i n g Alt(G)

Isup(u)

I

< o

is

.

i s t h e subgroup of f i n i t e even permutations.

5 Sym(G,o)

Note t h a t

5 Sym(G)

Sym(G,w)

Sym(G,w)

5 S(G).

Hence i f

i s t h e element given by lemma 8 ,

u

then

is a l o c a l l y f i n i t e group of c a r d i n a l i t y

Thus i f T

x = 8

€ < u , Gp

T €

H

I t i s e a s i l y checked t h a t

0 E Sym(G,w)

Consequently f o r a l l

9

g € G,

Aut H

extends t h e o u t e r automorphism

theorem 11.4.6 of S c o t t [7] s a y s t h a t t h e r e e x i s t s q ( h ) = Y-l h Y

and

then

Z,

Suppose t h a t

is any element, w i t h

X.

for a l l

h E H.

Thus f o r a l l

y

g € G,

n € Aut G.

E Sym(G)

Since

such t h a t

Complete Universal Locally Finite Groups of Large Cardinality

By theorem 10.3.6 of S c o t t [ 7 ] , t h e r e exists But t h i s means t h a t

(y(g)n)-'

u y(g)

TI

g € G

such t h a t

Y

281

a contradiction.

€ H,

@(g)n

=

.

So we have

proved : LEMMA 9.

The o u t e r automorphism

H.

of

Aut G

TI €

cannot b e extended t o a n automorphism

0

The f o l l o w i n g lemma w i l l b e u s e f u l d u r i n g t h e main c o n s t r u c t i o n . LEMMA 10.

Let

XC

G

b e a subgroup of c a r d i n a l i t y

X

.

Then

NH(X) = NG(X).

PROOF.

Suppose t h a t

e

€ Sym(G,w)

and

h € H T C

normalizes

0 €

N(Xp).

[e,

Xp] = 1.

Clearly

n > 1. Then f o r

Thus

4.

[e,

Xp]

T C

N(Xp)

(3-l p(g) 0 € G p

Suppose t h a t p(x) € Xp

8 # 1. L e t

, we

Then

h = 0.7

f o r some

Applying t h e c a n o n i c a l p r o j e c t i o n ,

< Gp, cr >.

p:H + < Gp, cr >, w e s e e t h a t

Xp.

and hence

i f and o n l y i f sup(€)) = {gl,

T € Gp.

[e,

We a l s o o b t a i n

p ( g ) ] = 1. Hence

..., gn}

,

where

have

# 1, a c o n t r a d i c t i o n .

0

THE CONSTRUCTION

To c o n s t r u c t nonisomorphic

ULF

g r o u p s , w e r e q u i r e a w e l l known

combinatorial f a c t . LEMMA 11.

Let such t h a t :

x

be a r e gu la r uncountable c a r d in a l.

Then t h e r e e x i s t s

A 5 '2

288

S. THOMAS

(ii)

If

'I

z

C

A , t h e n 15 < x

I

A

In particular, there exists

+~(5))

'(5) C

"2

is stationary in

.

x

s a t i s f y i n g t h e above c o n d i t i o n s .

We s h a l l b u i l d smooth s t r i c t l y i n c r e a s i n g c h a i n s

, G;

where each r)

ULF

is a

and. t h a t n:G'I

# zE A

+

X.

group of c a r d i n a l i t y

The r e g u l a r i t y of

i s a n isomorphism.

Gz

Suppose t h a t

k

ensures t h a t

CE

+1

< X

i s a c l o s e d unbounded set i n n[Gq]

= Gg

5

X+.

1

= G;

5 < A+

By leuima 11, t h e r e e x i s t s

such t h a t

~ ( 5 .) By p u t t i n g i n " o b s t r u c t i o n s t o isomorphism",

q(5) #

and

[$I

we s h a l l e v e n t u a l l y reach a c o n t r a d i c t i o n .

A

For t h e n e x t few pages, w e w i l l f i x q € i . e . w e write

all

G

=

U G5. 5 o

and

5

is a l i m i t o r d i n a l .


\G:

I T-1p ( g )

there exists

0

Ga.

(i)

(b)

a g a = g i'

be a n e l e m e n t s a t i s f y i n g :

S(Ga)

{ p ( g ) 6 G,"

such

h a s been c o n s t r u c t e d .

i s a n o u t e r automorphism o f

fa

Let

a

u € Go

1.

Assume i n d u c t i v e l y t h a t CASE 1.

i > 0, there e x i s t s

For each

g

STEP THREE.

z y = gi.

1.

Gap

, ua

z

.

then

T

p(a) € GZ

)I

E G:

x




(iii)

'p

Suppose t h a t

n[GaI = Ga*

(i)

H

q.

cp

e(Ka

x) = Ka n ( x ) .

Ga = f a , a c o n t r a d i c t i o n .

Then

n

S. THOMAS

294

W e r e t u r n t o t h e proof o f lemma 1 7 . LEMMA 20

U
I

=

xglg € G:

.

t

.

x = n(a,) , w e have

Suppose that

L = < z g ( g € G:

z €K'

B

or

5

N(L)

z € H".

B

t

=

K.;

Thus

.

z

€Kt

B '

S i n c e K:

(ii) n ( a a )

.

Then

is a b e l i a n o f z

E Hi.

, is a maximal a b e l i a n subgroup of

So w e have shown:

Suppose t h a t t h e isomorphism n[G;]

x EK

Suppose t h a t

LEMMA 22

(i)

Let

and

a contradiction. nK []:

2 a

Then

7 c xglg € Ga t

K,:

Then i f

.

n[Gz] = G:

X , we must have

X. By lemma 21,

5 H"B

8

B

z € n[K:]\c

cardinality Then

0

b e t h e element p r e v i o u s l y d e f i n e d .

€ K:

Thus

= G :

E K;

. f o r some

2 a.

n:Gq + G"

satisfies:

Complete Universal Locally Finite Groups of Large Cardinality Then

n[K:]

.

=

p

Suppose t h a t

.

a

>

Then

T h i s i n d u c e s a n isomorphism

where

cp = p

-1

0 p'

are t h e c a n o n i c a l p r o j e c t i o n s , and

p, p'

297

etK?,x) =

Ki

n(x).

Hence

IP

satisfies:

cpIG21

(i)

"

Ga.

=

By theorem 11.4.1 of S c o t t [ 7 ] , subgroup o f (ii) Let

.H:

Thus:

cp[Alt(G:)]

X'l

a'

X"

B

i s t h e u n i q u e minimal normal n o n t r i v i a l

Alt(G:)

Alt(G").

=

B

d e n o t e t h e sets o f 3-cycles of

Alt(G:),

respectively.

Alt(G")

B

By theorem 11.4.2 of S c o t t [ 7 ] , w e have:

Let

x

Its o r b i t under c o n j u g a t i o n by e l e m e n t s o f

= (a b c ) € .X:

G:

is

CLAIM If

x , y € :X

[x, yp(g)1

y !t Ox U 0

and

X

# 1.

,

then there e x i s t s

g € G :

such t h a t

PROOF OF CLAIM Suppose t h a t yp'g)

= (a

Pa

Since

B

x = (a b c ) €

-1

x = (a b c) a

ya

-1

a)

and

y = (a

and

y = (a

If

g = a-l a ,

s a t i s f i e s o u r requirements.

> a , t h e r e e x i s t elements

$

B Y).

B

y),

a , b , c € G"\G:.

where

B a, B , Y

then 0

Let € :G

.

Then f o r a l l

298

S. THOMAS

for a l l

(b)

g E :G

,

[ x , ~ ' ( ~ ' 1= 1.

But t h i s c o n t r a d i c t s t h e c l a i m and (i), (iii) above.

W e conclude:

LEMMA 2 3 Suppose t h a t t h e h y p o t h e s i s o f lemma 2 2 h o l d s .

Then:

Thus t o p r o v e lemma 1 7 , it i s enough t o show t h a t

p

? a.

F o r t h e s a k e of c o n t r a d i c t i o n , assume t h a t

n [ < ag ( g E :G Suppose t h a t z

E Sym(GT,w).

P

z

\

E n[Kz]

N(n

Hence, l e t t i n g

2.

Sym(Gc,w)

Then

.

Then, a r g u i n g as i n t h e p r e v i o u s c a s e ,

E H.:

z

P

g C GZ

Sym(G;,w).

w e must h a v e

n ( z ) C Sym(GT,o).

There e x i s t s

5

YL?,I

[.