Linear Orderings

LINEAR ORDERINGS This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMU...

Author: Author Unknown

32 downloads 938 Views 25MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

LINEAR ORDERINGS

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume.

LINEAR ORDERINGS

JOSEPH G.ROSENSTEIN Department of Mathematics Rutgers-The State University of New Jersey New Brunswick, New Jersey

@

1982

ACADEMIC PRESS A Subsidiary of Harcourt Brace Jovanovich, Publishers

New York London Paris San Diego San Francisco

Sao Paulo

Sydney

Tokyo

Toronto

COPYRIGHT @ 1982, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF mis PUBLICATION MAYBE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDlNG PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

1 1 1 Fifth Avenue, New York,New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road,

London N W l

7DX

Library of Coryress Cataloging i n Publication Data Rosenstein, Joseph G. Linear orderirys. (Pure and applied mathematics ; 98) Bibliography: p. Includes index. 1. Linear oroerirys. I. Title. 11. Series: Pure and applied mathematics (Academic Press) l&3.P8 vOl. 98 [QA2k8] 510s [510.3'3] 80-2341 ISBN 0-12-597680-1 AACR2

W S (MOS) 1970 Subject C l a s s i f i c a t i o n : 06A05, 06-01, 03-01, 06-02, 03-02

PRINTED IN THE UNITED STATES OF AMERICA 82 83 84 85

9 8 7 6 5 4 3 2 1

For Judv and,for Mira and Ariela and now for Dalia too

In memory of my father’s mother Dvorah (Kupritz)Rosenstein and my mother’s brothers and sisters and their families Moshe Aaron Kaganowicz, his wife, Gitel, and daughter. Sonya Blume (Rubinstein)and her children, Reuven and Shifra Mordechai (Motel),his wife, Esther, and daughter, Aliza Miriam (Mira),her husband, Simcha, son, Zalman, andanother child Batye (Baranchik), her husband, and three or four children Malkah (Kravetz) and many other members of’ my Jamily who died in the Holocaust

Contents

...

Prqface Acknowledgments

PART I

1.

xvii

INTRODUCTION TO LINEAR ORDERINGS

Chapter 1 2. 3. 4. 5.

Xlll

Introduction

Basic Definitions Relations between Linear Orderings Characteristics of Linear Orderings Operations on Linear Orderings Countable Linear Orderings

Chapter 2

1.

14

23

Dense Linear Orderings

1. Dense Linear Orderings 2. Countable Dense Linear Orderings; Properties of q 3 . Scattered Linear Orderings 4. Uncountable Dense Linear Orderings; Completeness; Properties of h and 9

Chapter 3

3 6 10

25 26 32 33

Well-Orderings and Ordinals 41

Well-Orderings

2. Ordinals and Cardinal Numbers 3 . Ordinals and Induction 4. The Arithmetic of Ordinals 5. Continuous Functions 6 . The Arithmetic of Cardinal Numbers References

vii

47 52 59 62 64 66

...

CONTENTS

Vlll

PART II

Chapter 4 I. 2. 3. 4.

COMBINATORIAL ASPECTS OF LINEAR ORDERINGS Condensing Linear Orderings

Condensations and Homomorphisms The Condensation cF The Condensation cw The Condensation eS References

Chapter 5

Hausdorff’s Theorem

1. Introduction 2. Iterated Condensations 3. Hausdorff’s Theorem 4. H and Its Powers References

Chapter 6

93 105 109

Ramsey’s Theorem and Ehrenfeucht Games

1. Ramsey’s Theorem 2. The Theorem of Lauchli and Leonard for Scattered Linear Orderings 3. Dense Partitions of q; The Shuffle Operation 4. The Theorem of Lauchli and Leonard References

Chapter 8

76 79 84 90 92

The Ehrenfeucht-Fraisse Game

I . The Play of the Game 2. Games and Ordinals References

Chapter 7

69 71 72 74 75

110 1 I3

I I5 1 I8

120

Transitive Linear Orderings

1. Transitive Linear Orderings 2. The Order Types of Ordered Abelian Groups 3. Lexicographic Products and the Order Type of the Irrationals

121 1 24

127

CONTENTS 4. Transitivity and the Exponentlation of Linear Orderings 5 Automorphisms of Linear Orderings 6. Almost Transitive Linear Orderings References

Chapter 9

Partition Theorems

References and Bibliography on Partition Theorems

PART 111

Chapter 12 1. 2. 3. 4. 5.

145 152 155 163 171

Embeddings of Linear Orderings and Fraisse’s Conjecture

1, Indecomposable Order Types 2. Fraisscs Results and Conjectures 3. Well-Quasi-Ordering 4. Better-Quasi-Orderings 5. Additively Indecomposable Linear Orderings and Ftai’sse’s Conjecture References

Chapter 11

i30 132 136 143

Uncountable Dense Linear Orderings

1. Diagonal Arguments and the Theorem of Dushnik and Miller 2. The Theorems of Sierpinski 3 . Suborderings of i, Continued 4. r\,-Orderings References

Chapter 10

ix

174 177 183 188

196 203

205 216

LOGICAL ASPECTS OF LINEAR ORDERINGS Linear Orderings and Formal Languages

Introduction First-Order Languages Infinitary Languages Second-Order Languages Algorithms and Decision Procedures References

223 229 236 240 24 1 246

CONTENTS

X

Chapter 13

The First-Order Theory of Linear Orderings

1, The Ehrenfeucht-Fraisse Games and First-Order Theories 2. The Ehrenfeucht Games and the First-Order Theory of Linear Orderings 3. Decidability of First-Order Theories of Linear Orderings 4. Model Theory and Linear Orderings 5. The Number of Countable Models of a Complete Theory 6. Finitely Axiomatizable Linear Orderings References

Chapter 14 1.

2. 3. 4. 5.

344 349 358 363 361 3 69

The Second-Order Theories of Linear Orderings

I , The Generalized Ehrenfeucht Games 2. The Generalized Ehrenfeucht Game and the Weak Second-Order Theory of Linear Orderings 3. D.:cidability of the Weak Second-Order Theory of Linear Orderings 4. The Monadic Second-Order Theory of Linear Orderings References

Chapter 16

253 261 273 296 325 342

The lnfinitary Theories of Linear Orderings

The Expressive Power of L,,, and Lm,u The Karp Game Scott Sentences Completely Characterizable Linear Orderings Additional Results References

Chapter 15

241

370 378 383 396 399

Linear Orderings and Recursion Theory

1. Introduction 2. Looking at Q Effectively 3. Recursive Linear Orderings 4. The Arithmetical Hierarchy 5. Recursive Linear Orderings and the Arithmetical Hierarchy

40 I 405 409 4 14 421

CONTENTS

6 . Diagonal Arguments in Recursion Theory 7. Effective Versions of Combinatorial Theorems References

Complete Bibliography of Linear Orderings

xi 435 438 454

456

471 479 482

This Page Intentionally Left Blank

Preface

A book about linear orderings? You mean total orderings? What can you possibly say about them? After all, besides the natural numbers, the integers, the rationals, and the reals, what linear orderings are there? These questions, usually unspoken, were common. It is my hope that the reader will find this book a satisfactory response. My interest in linear orderings was first aroused by an old paper of Dushnik and Miller. Perhaps you too will find the following question interesting : Given a well-ordering A , it is easily verified that any order-preserving map from A to A (that is, f ( a ) < f ( b ) whenever a < b ) is also nondecreasing (that is,J’fa) 2 a for all a in A . ) Is the converse true? That is, given that any order-preserving map from A to A is non-decreasing, does it follow that A is a well-ordering? They answered this question using a simple, elegant technique. I observed that a number of theorems about linear orderings (including one of my own) were proved using variations of this technique, which I have called “condensing linear orderings.” That many different facts about linear orderings share an underlying theme suggeFted that there is a subject called h e a r orderings which is more than just a collection of isolated facts. As I examined the literature, I became more convinced of the usefulness of presenting the extensive material on linear orderings in a unified manner. I found that the various groups of people who studied linear orderings were generally unaware of other people’s work. I also found that the lack of a comprehensive treatment resulted in theorems being re-proven (like Hausdorff’s Theorem 5.4) and techniques being rediscovered (like that of Dushnik and Miller’s Theorem 9.1). My study of the literature also convinced me that there was too much material for one book. It took a while for me to become convinced of this, and I have persisted in incorporating more and more material into this book. However, I have learned (with apologies to Koheleth) that of writing a book there is no end-and so this book is now, thank God, completed, and it contains just what it contains. ...

XI11

xiv

PREFACE

My original goal was to include everything known about linear orderings not already in Sierpinski's Cardinal and Ordinal Numbers. At a later time, my more modest goal was that the book should contain what every logician would want to know about linear orderings. Still later, although I no longer had a clearly formulated goal, I knew exactly what material the book would contain. In retrospect one might say that, like any author, I put into the book precisely the material that most interested me. The reader who wishes to look further will find bibliographies attached to each of the later chapters and a rather complete bibliography of all articles on linear orderings (including those not referred to in the text) at the very end. What I have included can be seen from the table of contents. The book divides naturally into three parts : Introduction to Linear Orderings (Chapters 1-3), Combinatorial Aspects of Linear Orderings (Chapters 4- 1I), and Logical Aspects of Linear Orderings (Chapters 12-16). The introductory part contains some material with which every mathematician should be familiar and other material which, though introductory, is less familiar. By including this material the book becomes essentially selfcontained and can be used as a textbook for a course on linear orderingsat either an undergraduate level or graduate level, for either combinatorialists or logicians. Combinatorialists will find that the first two parts of the book are completely self-contained. For those combinatorialists who would like to read on and see why logicians are interested in linear orderings, I have provided an introductory chapter on mathematical logic. Having said that the book is self-contained, I must immediately backtrack a little bit. The natural context, from a mathematical point of view, for discussing ordinals is axiomatic set theory, but presenting that context would make this book too long and, more seriously, would deflect the reader from getting into the subject of linear orderings. I therefore avoid introducing axiomatic set theory and instead assume a naive familiarity with notions of set theory (like cardinal numbers) and a naive acceptance of the axiom of choice. (When a less naive point of view is appropriate, I invoke the context of Zermelo-Fraenkel set theory with choice (ZFC).) One consequence of this position is that many results which are more set-theoretic in character unfortunately are not included in this book. This position also leads to some difficulties when, in Chapter 3, each ordinal is viewed as the set of all smaller ordinals and each cardinal number is viewed as an ordinal; but these difficulties can be resolved by any reader whose interests are foundational or overlooked by any reader whose interests lie elsewhere. My interests in model theory also provided some incentive for writing this book. While teaching model theory, I felt that there was a dearth of concrete examples illustrating the basic notions, and so I was led to investigate various classes of structures. I found that linear orderings and partial

PREFACE

xv

orderings served well as testing grounds for rnodel-theoretic concepts and conjectures. The examples and observations I found illustrative I have included in the appropriate chapters. As a result, Chapter 13 can be used, together with certain earlier material, for an introductory course in model theory. Similarly, Chapter 16 can be used for an introductory course in recursive function theory. Essentially none of the non-introductory material has ever appeared in a book; much of the research reported is relatively recent. I have tried to attribute concepts and results to their creators or discoverers; any lapses I sincerely regret. The non-introductory sections of Chapter 16 consist largely of my own results; other unpublished material has been incorporated into various chapters. The book contains a large number of exercises. Although it is common to distinguish between easier and more difficult exercises, I have chosen not to do so-leaving that too as an exercise. All unproved lemmas and theorems in the first three chapters (and occasionally elsewhere) are indicated by a A rather than the used at the end of a real proof; these should also be treated as exercises and verified by the reader. Books are meant to be read, and mathematics books should not be exceptions. I have tried to write clearly and discursively, attempting to reveal what the notation often conceals. In this I have tried to follow Sierpinski’s example. It is for the reader to judge whether this attempt was successful.


Acknowledgments

A large part of this book was written during the spring semesters of 1976 and 1977, when I was a Member of the Institute of Advanced Study, and during the summer of 1978, when I was a visitor there. I would like to take this opportunity to thank the Institute and its staff for providing the facilities and the services which helped make this book possible. A number of people have assisted me by reading parts of this book and offering their suggestions, comments, criticisms, and encouragement. Peter Mulhall read the first two parts of the book, and his suggestions are reflected throughout the book. Charles Landraitis read a number of chapters and offered valuable comments. I would also like to thank G. Cherlin, S. Fellner, R. Fraisse, F. Galvin, A. Glass, Y. Gurevich, J.-G. Hagendorf, C. Holland, R. Laver, and J. Schmerl for reading and commenting on various chapters of the book. I suppose that those errors which remain are my responsibility, but I would rather fault some gremlins. I will be grateful if the reader would bring any evidence of their interference to my attention.

xvii


PART I INTRODUCTION TO LINEAR ORDERINGS


CHAPTER 1 INTRO DUCT10N

$1. BASIC DEFINITIONS

Let the set A be given and assume that the elements of A are arranged, or ranked, in a certain order so that, given any two distinct elements al and a2 of A, either a, is ranked lower than a2 or a2 is ranked lower than u l . The simplest examples that arise in mathematics are those in which A is the set N of natural numbers, the set Z of integers, the set Q of rational numbers, or the set R of real numbers, and one element al of A is ranked lower than another element a2 of A precisely if a, < a2 in the natural ordering of A . In each of these examples the ranking of the elements of A is easily seen to have the following properties: (1) If x is ranked lower than y and y is ranked lower than z, then x is ranked lower than z. (2) Given two distinct elements x and y, either x is ranked lower than y or y is ranked lower than x,but not both. (3) No element is ranked lower than itself.

A ranking of A can be considered as a binary relation R on A where R is determined by the condition (a,,

a2)E R

if and only if

a, is ranked lower than

a2.

This motivates the following definition. DEFINITION 1.1 : A linear ordering of the set A is a binary relation R on A satisfying the conditions

(1) i f ( x , y ) E R a n d ( y , z ) E R , t h e n ( x , z ) E R ; (2) if x # y, then either (x, J ) E R or ( y , x) E R, but not both; (3) ( x , x > $ R.

Thus the usual linear ordering of the natural numbers N would correspond to the binary relation R , = {(m,n)Im < n } on N , and the usual 3

4

1, INTRODUCTION

linear ordering of the rational numbers Q would correspond to the binary relation R, = { ( r ,s) r < s} on Q. In other texts, linear orderings of a set are referred to as total orderings, simple orderings, or chains. A given set A can be linearly ordered in many different ways. For example, there are the following distinct linear orderings of the set N of natural numbers :

I

RN = { ( m , n ) ( m < n ) ,

the natural ordering;

R 1 = { ( m , n ) ( m> n ) ,

the backwards ordering;

R 2 = { ( m ,n ) 10 < m < n or ( n = 0 and 0 < m ) } ; R3 = { ( m , n ) I(m < n and m and n both even) or (m < n and m and n both odd) or ( m even and n odd)}. It is easily verified that each of these binary relations satisfies conditions (1)-(3) of Definition 1.1. Let us arrange the natural numbers, from left to right, according to each of these linear orderings: R,: 0,1,2,3,4,5,6, . . . R , : . . . ,6,5,4,3,2,1,0 R,: 1,2,3,4,5,6,... 0 R,: 0,2,4,6,8, . . . 1,3,5,7,9, . . .

Thus, for example, the binary relation R2 ranks every positive natural number lower than 0 and ranks two positive natural numbers according to their natural order. It is useful, in dealing with linear orderings, to visualize what they look like. Although this is difficult to d o with arbitrary mathematical structures, it is possible with linear orderings, because one can imagine a linear ordering R of A as being a way of placing the elements of A along a line, with a, to the left of a2 if and only if (ul, a,) E R. Thus, for example, the reader should train himself to see the linear ordering R, of N as a non-terminating sequence of points .

.

.

. . .

...

0 1 2 3 4 5 labeled by natural numbers; and the linear ordering R, of N as a nonterminating sequence of points followed by a single point .

.

.

.

.

1 2 3 4 5

...

.

0

1.

BASIC DEFINITIONS

5

all labeled by natural numbers. He should similarly attempt to visualize, as best he can, each example of a linear ordering that he comes across. EXERCISE 1.2: In each of the following cases: (1) Verify that the relation R on N is in fact a linear ordering of N . (2) Arrange the natural numbers, from left to right, according to the linear ordering R, as was done above for R , , R 2 , and R, . ( 3 ) Attempt to visualize the linear ordering R of N and attempt to describe in words what it looks like.

(a) R

=

(b) R

=

{ ( m , n ) ((m< n and ni and n both even) or ( m > n and m and n both odd) or (m even and n odd)).

{ ( m , n ) 1 (m < n and m and n both even) or (m > n and rn and

n both odd) or (m odd and n even)). (c) R = ( ( m , n ) ( m 5 a (mod 4) and n = b (mod 4) where 0 I a, b < 4, and either il < b or (a = b and m < n)}. [Recall that m = a (mod p) where 0 I a < p means that m leaves a remainder of a when divided by p.] (d) Let p be a fixed natural number, R = ((m,n)lrn = a (modp) and n = b (modp) where 0 I a, b < p, and either a < b or (a = b and m < n)}. R = ((m,n)l(m < n and neither m nor n is a positive power of a (e) prime) or (m is not a positive power of a prime and n is a positive power of a prime) or (m is a positive power of a prime p and n is a positive power of a prime q and p < q) or (m and n are both positive powers of the same prime p but n is a higher power of d}. R = { ( m , n ) I as in (e) except the second clause is replaced by “(a (f) is not a positive power of a prime and m is a positive power of a prime)”}. [The reader may find examples (e) and (f) somewhat hard to visualize at this point. If so, he may delay these examples until after Definition 1.43.1 (4) Describe the linear orderings R of N which look like

(a) . . . ,12,8,4,0 . . . ,9,5,1 . . . ,10,6,2 . . . ,11,7,3. (b) 0,4,8,12,. . . . . . ,9,5,1,2,6,10,. . . . . . ,11,7,3. DEFINITION 1.3: The notation ( A , R ) is used to represent the structure consisting of a set A which is linearly ordered by the relation R ; the structure

6

1.

INTRODUCTION

( A , R ) is called a linear ordering. Thus R is a linear ordering of A (Definition 1.1) and ( A , R ) is a linear ordering mean the same thing. In contexts where it is clear which linear ordering of A is intended, we will sometimes speak of the linear ordering A instead of the linear ordering ( A , R ) or the linear ordering R of A-the phrases should thus be understood as being synonymous. When we use the phrase let A be a linear ordering we mean let A be a given set with a given linear ordering R of A .

Special notation is defined below for the linear ordering referred to most frequently in the early chapters of this book. DEFINITION 1.4: (1) N is used to denote ( N , R N ) . (2) Z is used to denote ( Z , R,) where R , is the natural ordering of the set Z of integers. (3) Q is used to denote ( Q , R Q ) where RQ is the natural ordering of the set Q of rational numbers. (4) Iw is used to denote ( R , R , ) where R , is the natural ordering of the set R of real numbers.

In these introductory chapters it is important to distinguish between the set N of natural numbers and the linear ordering N = ( N , R , ) . In later chapters this special notation will be used only for emphasis and N , Q, Z , and R will be used instead of N, Q, Z,and 52 whenever there is no danger of confusion. When A is a set and R is a linear ordering of A , we will often write x , x means the same as x ’, B”, B O l q 2 > 2); then A is not of the form [b,, +) since there is no first positive rational whose square is >2, and A is not of the form (b,, +), since if b 1 2< 2, then (b,, -+) contains some rational number whose square is less than 2, and if b , > 2, then (b,, +) omits some rational number whose square is greater than 2. (4) Everything said about Q also holds for R, with the exception of the last statement. That is, every proper interval of R can, in fact, be expressed in one of the given forms. This property of R, related to completeness, will will be discussed in Chapter 2. (See Lemma 2.23.) EXERCISE 1.21 : All intervals in this exercise are intervals of Q. (1) Show that

( a l , b l )1: (a,,b,), (a1,b,l (az>bzl.

[a,,b,) [a1,b,l

'"

[a2,b2),

= Ca2,bzl

for any a , < 6 , and any u, < b,. (2) Show that and

(ai,+)

(0,l)

and

(1,+) 2: (0,l).

( a i , b i ) Y (+,b2)

and

( a i , b i ) 2: (a,,+)

(+,bi)

2

(+,bz)

N

(az,+)

for any a,, a,, b,, b 2 . (3) Show that

(+,O)

Y

(4) Conclude that

for any a , < b , and any a2,6,. (5) Show that Q 1: (0,l). (6) Do these arguments work just as well if all intervals are assumed to be intervals of R?

3.

CHARACTERISTICS OF LINEAR ORDERINGS

13

(7) Can there be an automorphism f of Q which is the identity at all but a finite number of arguments (i.e., such that f(x) = x for all but a finite number of x)? NOTE: In Definitions and Lemmas 1.16-1.20 only one linear ordering R is under consideration, and so all mention of R is suppressed. In contexts where more than one linear ordering is under consideration, the R is restored in the notation, and then we speak of R-first elements, intervals [b,,bJR, etc. DEFINITION 1.22: Let A be a linear ordering and let X be a subordering of A . We say that an element a E A is an upper bound of X in A if x < a for all x E X.We say that an element a E A is a least upper bound of X in A if a is an upper bound of X in A and, if b is any upper bound of X in A, then a Ib. Note that if there is a least upper bound of X in A, then it is unique. We say that X is bounded above in >4 if X has an upper bound in A . If X is not bounded above in A, then we say that X is cofinal in A . Similarly, we say that an element a E A is a lower bound of X in A if a Ix for all x E X ; and we say that an element a E A is the greatest lower bound of X in A if a is a lower bound of X in A and, if b is any lower bound of X in A, then b I a. We say that X is bounded below in A if X has a lower bound in A. If X is not bounded below in A, then we say that X iscoinitial in A. We say that X is bounded in A if X is bounded above in A and is bounded below in A. EXAMPLES: (1) Any finite subordering X of a linear ordering A is bounded; it has a least upper bound in A, namely, its greatest element, and a greatest lower bound in A, namely, its smallest element. (2) Let A be Q. Then (0,l) is bounded in A ; it has a least upper bound 1 and a greatest lower bound 0. The same statements are correct for [0,1), (0,1] and [0,1] as well as for (0,l) and

(3) Let A be Q and let X be {.Y > OIx2 > 2). Then X has a lower bound in Q,for example, 1, but X has no greatest lower bound in Q. (4) Let A be Q and let X be N. Then X is bounded below in A and is cofinal in A .

We now observe that all the characteristics of linear orderings defined above are preserved under isomorphisms.

14

1. INTRODUCTION

LEMMA 1.23: Let ( A , R ) and ( B , S ) be linear orderings and let f : A + B be an isomorphism of ( A , R ) onto ( B , S ) . (1 ) Then ( A , R ) has an R-Jrst (respectively, R-last) element a if and only if ( B , S ) has an S-Jrst (respectively, S-last) element b, and, in that case, f (a) = b. (2) Let C be a subordering of A. Then C is an R-interval of ( A , R ) if and only i f f [ C ] is an S-interval of ( B , S ) . Furthermore C can be expressed in one of the eight forms of Lemma 1.17 if and only iff [ C ] can be expressed in the same form. ( 3 ) Let a,, a, E A. Then a , is the R-predecessor (respectively,R-successor) of a, if and only if f ( a l ) is the S-predecessor (respectively, S-successor) of f(ad (4) Let X be a subordering of ( A , R ) and let a E A. Then a is an upper (lower) bound of X in ( A ,R ) if and only if f ( a ) is an upper (lower) bound of f [ X I in ( B , S ) . Also a is the least upper bound (greatest lower bound) of X in ( A , R ) if and only if .f(u) is the least upper bound (greatest lower bound) of f [XI in ( B , S ) . A

Because of Lemma 1.23, order types represent their isomorphic equivalents also in matters pertaining to intervals, first and last elements, successors and predecessors, and least upper bounds and greatest lower bounds. This extends the remarks preceding Definition 1.15. It goes without saying that isomorphic linear orderings must have the same cardinality, so that, for example # A. Similarly, 1.$ q, so that q < A. We now return to a question raised after Definition 1.15 and give an example of two linear orderings which are not isomorphic although each is embeddable in the other. EXAMPLE: Let A be the subordering of Q consisting of the open interval (0,l)and let B be the subordering of Q consisting of the closed interval [0,1]. Then A = B although A 9 B.

$4. OPERATIONS ON LINEAR ORDERINGS DEFINITION 1.24: Given a linear ordering R of the set A we define the backwards linear ordering R* of A by

( x , y ) E R*

if and only if

and we denote ( A , R*) by ( A , R)*.

( y ,x )

E

R,

4.

EXAMPLE:

OPERATIONS ON LINEAR ORDERINGS

15

Let R be R,. Then R* = R,.

LEMMA 1.25: (1) ( A , R ) * " = ( A , R ) . (2) Assume that ( A , R) 2: ( B , S ) . Then ( A , R)*

= (B,S)*. A

By Lemma 1.25.2, the backwards linear orderings of the class of linear orderings of a given order type are all of the same order type. Thus there is a natural map which associates with every order type z the backwards order type z*. EXAMPLE:

(N, R,) has order type a*.

PROPOSITION 1.26: 3" = 3. Proof: It suffices to show that if R is the natural ordering of Q, then (Q,R) 2: (Q,R*).Definethemapf:Q -+ Q byf(x) = -x.Thenfisclearly 1-1 and onto; furthermore, x c Ry if and only if - y < R -x if and only if -x * \ UE

A},

R‘= (((a1,O>,>(ai < R a2}9 B‘= ((b,l)lb E B}, and

s’= K ( W ) . ( ~ 2 J ) ) l b *<s b21.

We will thus feel free to use + even when A and B are not disjoint, and we observe that Lemma 1.30is also correct even when + is used improperly. This permits us to define addition of order types. DEFINITION 1.31 : Let z1 and z1 be two order types. Let ( A l , R , ) and ( A 2 ,R 2 ) be linear orderings of order type z1 and z2 respectively. We define z1 z2 to be the order type of ( A l . R l ) (A2,R2).

+

+

EXAMPLES: (1) ( N , R,) + (N,R,) v Z.Hence the order type of o*+ o,so that 5 = o* + o. (2) N N = ( N , R3). The order type of ( N , R3) is w o. ( 3 ) The order type of ( N , RN) + ( N , R,) is w o*.

+

+

Z is

+

For each natural number n, there is a linear ordering which has exactly n elements, namely, { j E N j < n} with the natural ordering. Lemma 1.32 shows that this is essentially the only example; we give its proof to illustrate the ideas of this section.

I

LEMMA 1.32: Let ( A ,R) and ( B , S) be linear orderings where A and B have the same finite number n of elements. Then ( A , R) = ( B , S). Proof : By induction on n. The result is clear for n = 0 or n = 1. So assume it true for n and let A and B each have n + 1 elements. Let a be the R-largest element of A and let b be the S-largest element of B. Let A , = A - (a)and let B, = B - { b}. Let ( A , , R,) and ( B , , So) be the suborderings of ( A , R) ( { a } , rZr) and ( B , S ) generated by A , and 8,.Then ( A , R ) = ( A , , R,) and ( B , S ) = (B,,S,) + ((b],@‘).By the induction hypothesis, since A, and B, each have n elements, ( A , , R,) 2: ( B , , S , ) . Also ( ( a ) , @ ) = ( { b } , @ ) . Hence, by Lemma 1.30,( A , R ) = ( B , S ) .

+

1.

18

INTRODUCTION

Intuitively, any two linear orderings with exactly n elements each have the same picture. Of course, two linear orderings with the same infinite number of elements need not be isomorphic, as we have already seen. DEFINITION 1.33: The order type of ( A , R ) where A has n elements is denoted n. EXAMPLES: (1) The order type of ( N , R , ) is w (2) 1 + 1 = 2. ( 3 ) 1 + o = o since N Y ({O},Qr) + N.

+1

+

EXERCISE 1.34: (1) Show that n w = o for every natural number n. (2) Show that n # m if n # m. ( 3 ) Show that o n # o m if n # rn. (4) Show that o + n # o + w for every natural number n. (5) Show that n 5 m if and only if n 5 m. ( 6 ) Show that w 1 # 1 o. Thus addition of order types is not commutative. (7) Write the order types of the examples in Exercise 1.2 as sums of the order types o and o*.

+

+

+

+

LEMMA 1.35: Addition of linear orderings is associative. Proof: Let ( A , , R , ) , < A 2 , R 2 ) ,( A 3 , R 3 )be linear orderings and assume that A l , A , , and A , are pairwise disjoint. Then ( ( A I ~ R I )+ Sf,(xo). Thusf2[A,] is not an initial segment of V ,contrary to the hypothesis. Hence f , ( x ) = f , ( x ) for all x E A , n A,.

THEOREM 3.9: Let ( W , R ) and ( V , S ) be well-orderings. Then exactly one of the following is correct:

(a) ( W , R ) ( V , O or (b) ( W ,R ) is isomorphic to a unique proper initial segment of ( V , S ) or (c) ( V , S ) is isomorphic to a unique proper initial segment of ( W ,R). Proof: Let d = { A I A is an initial segment of W and there is an isomorphismf, from A onto an initial segment of V >.Note that, by Proposition 3.8, if A E d,then there is a unique such isomorphism f’, and if A E d , B E d,and A 5 B, then f,(x) = f d x ) for all x E A. Let A* = U ( A E d}. Then, by Lemma 3.7.3, A* is an initial segment of W . Furthermore f * = { f, I A E d }is an isomorphism from A* onto an initial segment of V . There are now two cases. If A* = W , then f * is an isomorphism of W onto an initial segment f * [ W ] of V . If f * [ W ] = V , then we have (a); if f *[ W ] is a proper initial segment of V , then we have (b).

u

1.

WELL-ORDERINGS

45

If, on the other hand, A* is a proper initial segment of W, then, by Lemma 3.7.5, A* = W, for some a E W. Suppose now that f*[A*] is a proper initial segment of V ;thenf*[A*] = V, for some b E V . But then f* can be extended to an isomorphism from the initial segment A* u { a ) of W to the initial segment f * [ A * ] u ( b ] of V ; hence A* u (u> E d,so that A* u (a>L A*, which is a contradiction. Thus f * [ A * ] = V ,so that (f*)-' is an isomorphism from V onto a proper initial segment of W , so that we have (c). It is easily verified, using Proposition 3.8, that it is impossible for more than one of these cases to happen. Also, if case (b) applies, then W is isomorphic to a unique proper initial segment of V and furthermore, there is a unique isomorphism mapping W onto that proper initial segment of V . H Thus there is a standard way of comparing any two well-orderings. Either they are isomorphic or one is longer than the other in the sense that it extends beyond the other. If V is longer than W, then clearly W 5 I/. We will soon see that for well-orderings W and V , V is longer than W if and only if W < V , and W N I/ if and only if W = V (Exercise 3.11). Proposition 3.8 has another consequence. Let Wand V be well-orderings and suppose that W, and V, are initial segments of W and V , respectively, and that W, and V, are isomorphic. Thus W = W, + W, and V = V, + V,. Suppose also that W N V . This gives two isomorphisms from W, onto initial segments of V : the isomorphism from W, onto V, and the restriction of the isomorphism from W onto V . By Proposition 3.6, these two isomorphisms are identical, and hence the isomorphism from W onto V carries W, to V,, and thus maps W, isomorphically onto V, . The following cancellation law for well-orderings is therefore correct. THEOREM 3.10: Assume that A, B, and C are well-orderings and that A B = A C . Then B N C . W

+

+

The dual cancellation law, A + B = C + B implies A N C , is false, even for well-orderings, since, for example, 1 o N 2 + o although 1 # 2. Theorem 3.10 is peculiar to well-orderings and is not true in general; the reader can readily supply a counterexample.

+

EXERCISE 3.11: (1) Let ( V , S ) be a well-ordering and let f be an isomorphism of V into V . Show that f is non-decreasing, that is, x I f ( x ) for every x E V . (The converse will be discussed in 994.3 and 9.2.) (2) Let ( W, R ) and ( V , S ) be well-orderings and let f be an isomorphism of W into V . Then there is an isomorphism g of W onto an initial segment of V . [ H i n t : Use Theorem 3.9 and part (1) above.] Hence, if W 5 V ,

3.

46

WELL-ORDERINGS AND ORDINALS

then either I/ is longer than W or V Y W. Deduce that V is longer than W if and only if W < V . (3) If ( W , R ) and ( V , S ) are well-orderings and ( W , R ) = ( V , S ) , then ( W ,R ) = ( V , S ) . (4) Find linear orderings A, B, and C for which A B 2 A C but B 74 C .

+

+

It is clear that all of these notions apply to order types as well as wellorderings.

LEMMA 3.12: (1) Let ( A , R ) 2 ( B , S ) be linear orderings. Then ( A , R ) is a well-ordering if and only if ( B , S ) is a well-ordering. ( 2 ) Let ( W ,R ) N ( W’,R ’ ) and ( V , S ) 2: ( V ‘ ,S ’ ) be well-orderings. Then (a) ( W ,R ) = ( V ,S ) if and only if ( W ,R’) 2: ( V , S ’ ) and (b) (W , R ) is isomorphic to a proper initial segment of ( V , S ) if and only if ( W‘,R ’ ) is isomorphic to a proper initial segment of (V’, S). A It is now appropriate to introduce the following definition. DEFINITION 3.13: The order type of a well-ordering is called an ordinal. Given two ordinals a and 8, we write a < if any well-ordering of order type a is isomorphic to a proper initial segment of any well-ordering of order type p.

By Lemma 3.5 the sum and product of ordinals are also ordinals. By Theorem 3.9, given two ordinals u and p for which u # p, either a < fi or p < c(. Moreover, < is a natural linear ordering on the collection of all ordinals, so that it makes sense to speak of one ordinal being larger than, or smaller than, another ordinal. If A is a set of ordinals, then A is linearly ordered by 0, then y 2 = w 8 .

+

+

+

EXERCISE 3.45: For any ordinal 8, let w,,(p) be the set of 8-sequences ii = ( a y l y a,, then since ma' . n , wa2. n, = wa2. n , I CI, we contradict the minimality of p, whereas if a, = a,, we contradict the choice of n. Hence C I , > CI,. As to uniqueness, if a = my1 . rn, + (uy2.m2 + . . . + . m,

+

is another appropriate representation, then w y l a < o P + ' , so that y, + 1, and hence LY, = y,. Using Theorem 3.14.3, we find that

B=

. ( n , - 1) + waz. n2 + . . + war . nk '

-

w Y l .

(m, - 1)

+ w y 2 m, . + ... +

mr.

By the induction hypothesis, this smaller ordinal has a unique appropriate representation so that k = r, cli = yi and ni = mi for all i 5 k. Hence the representation of CI is unique.

62

3.

WELL-ORDERINGS AND ORDINALS

DEFINITION 3.47: If the Cantor Normal Form for a is wal . n1 n2 + . . . + war . nk,then a , is called the degree of a.

+ wa2.

EXERCISE 3.48: (1) Given ordinals a and p, we say that /? is a tail of a if a = a1 + jfor some ordinal a , . Show that every ordinal has a finite number of tails, and determine that number from its Cantor Normal Form using Exercise 3.44.3. Is the converse true? Namely, if z is an order type with only a finite number of distinct tails, does it follow that z is an ordinal? ( 2 ) Show that no ordinal smaller than owsatisfies w . a = a. Why is # . #w = #"? (3) Show that an ordinal y satisfies 0 .y = y if and only if y has no tail smaller than ww. (4) Show that if a < w 0 and 8, < ow,then a . p < ww. Conclude that a . w m= ww for all a < ow.

85. CONTINUOUS FUNCTIONS

Each of the functions defined above by transfinite induction is continuous at limit ordinals; we explore continuity in this section. DEFINITION 3.49: An increasing function h mapping ordinals to ordinals is said to be continuous if, whenever E. is a limit ordinal, h ( l ) = lim{h(j)lp
a for all a since, even if h(P) > p for all B < 2, it does not follow that h(2) > A. For example, with the continuous function gl(p) = 1 p it is clear that for fi 2 w we have gl(P) = /3. More interesting, for the continuous function g,(P) = w p, we find that for p = w . n , + n, we get gJP) = w . ( n , + 1) + n,, which is bigger than p, but that if /?2 w . w, so that p = w . w + 6, then

+

+

(0

+ p = w + 0 '0+ 6 = w . ( 1 + w) + 6 = w

so that q,(P)

=b

for all

'

w

+ 6 = p,

p 2 w . (0.

EXERCISE 3.50: (1) Let f be a continuous function and suppose that a < f ( 2 ) for some limit ordinal A. Show that a < f ( P ) for some < 3,. (2) For each ordinal a, find the smallest ordinal ya such that for all p 2 y u we have that qm(B)= x /I= p. ( 3 ) For each ordinal a, find a limit ordinal AU such that for the function g,*(p) = p + a there is a discontinuity at La; that is, gu*(Aa) # lim{g,*(b)I /l < i.,).

+

Consider also the continuous function f(a)= w . a. For a = wn we get f(a)= w . u" = w i + n > wn= a. O n the other hand, f ( w w )= LC) . ww= w' +'J= ww,so that a = ww is a fixed point off. (By Exercise 3.48, no smaller ordinal is a fixed point.) It follows also from this discussion that f induces an order-preserving map from the ordinals i,Hence . rF(A) 2 M. On the other hand, the equation c,'(x) = c'(x) n B shows that no two elements of B can condense together in A after stage a, for if c'(xI) = c1(x2) with I > M, then cB'(xI) = cg'(x2), so C,'(X~) = cBa(xz),which implies that c'(xI) = ca(x2).Similarly, no two elements of C can condense together in A after stage a. Thus at stage a + 1 we can condense elements of B with elements of C but after stage a 1 no further condensation can take place. Hence TF(A) I M 1.

+

+

84

5.

HAUSDORFF’S THEOREM

EXERCISE 5.16: Suppose that A = A l + A , +. . . + A , where r,(Ai) = ai for each i. Let a = maxicxiand show that a I rF(A) I CI 1.

+

LEMMA 5.17: Then

Suppose that A

=

I{Ai I i < w ) where rF(Ai) = ai for each i.

iffor some j , ai I rjfor d l i, then a j 5 rF(A) I clj + 1; (2) otherwise, rF(A) = U { a i li < Q}.

(1)

Proof: Suppose that x < y, that x E A , and y E A,. By Lemma 5.13.1 and Exercise 5.16, if x and y are ever condensed together, then this happens by stage max(ci,, a,+ . . . ,a,) 1. Hence if for some j , aiI a j for all i, then rF(A)5 u j 1; otherwise r F ( A )I u { a i l i < w } . The other inequalities are proved as in earlier lemmas.

+

+

COROLLARY 5.18: Let A i < w. Then r F ( A ) a. A

= x ( A i l i< w }

where r,(Ai) < cl ,for each

Note that analogous results hold for o* and C-sums. EXERCISE 5.19: (1) Find a bound for r,(A . D)in terms of r F ( A )and rF(D).Demonstrate your claim. (2) Show that if z is a countable order type such that a 5 z for every countable ordinal a, then 4 IT (Fraisse [2] and Shepherdson [ 5 ] ) .

53. HAUSDORFF’S THEOREM A linear ordering is scattered if no subordering is dense. This definition tells us what a scattered linear ordering is only by telling us what it is not. A more positive description of the scattered linear orderings is provided by Hausdorffs Theorem, which says that a linear ordering is scattered if and only if it can be constructed from simple linear orderings in certain prescribed ways. We will at first restrict our attention to the countable case.

DEFINITION 5.20: We define the class VD of (countable) very discrete linear orderings by presenting inductively, for each countable ordinal c1, a VD,(a< wl). Thus, automatically, class VD,, and then setting V D =

u{

3.

HAUSDOFWF’S THEOREM

85

given a linear ordering L in VD, there will be a smallest ordinal ci, called the VD-rank of Land denoted rvD(L),such that L E VD,. (i) 0 , l E VD,. (ii) Given a linear ordering J of order type o,oh,6, or n, and for each i E I a l i n e a r o r d e r i n g L i E U { V D I , I P < a } , t h e n ~ { L i l i E I }E VD,. Note that if c1 < b, then V D , G V D 0 . Then, for example, the VD-rank of w is 1, of w 2 is 2, of w” is n, and of w’ is w. Considering the definition above, the result of the next proposition is not too surprising. PROPOSITION 5.21 : Every very discrete linear ordering L is scattered; jurthermore, rF(L)I rvD(L). Proof: We show, by induction on a, that if L E VDa, then L is scattered and that rF(L)I rvD(L). There is little to prove for a = 0 so we proceed to the induction step. Assume then that L E VD, and that, with no loss of generality, r“D(L) = a. Then L = ciLili E I } where each L, E u{VD,l f l < a } . By the induction hypothesis, each L, is scattered; hence by Proposition 2.17, since L is a scattered sum of scattered linear ordering, it too is scattered. Furthermore, by the induction hypothesis, r,.(Li) I rvD(Li) < a for each i. Hence, by Corollary 5.18, rF(L) I a. Hence rF(~L) I rvD(L). H

The converse to Proposition 5.21 is a bit more interesting, but before proceeding to it, we prove the following lemma. LEMMA 5.22: Every interval of an element of V D is an element of V D ; furthermore, fi L* is an interval of I,, then rvD(L*)5 rvD(L).

u{

Proof: We proceed by induction on the VD-rank CI of L. Suppose that z{LiI i E I ) has VD-rank a where each LiE VD, I /? < a } . Since L* n Li is an interval of Lifor each i E I , by the induction hypothesis L* n LiE U{VD, I P < a } for each i E I . Hence L* = c { L * n LiI i E I } E VD,, so that ‘VD(L*) r v m . rn PROPOSITION 5.23: Every wuntable scattered linear ordering L is in V D ;,furthermore, rvD(L)I rF(L).

86

5.

HAUSDORFF'S THEOREM

Proof: We proceed by induction on a = rF(L). If a = 0, then L

2: 0 or 1 and rvD(L)S rF(L).If a = /I+1, then cs[L] 1: I where I = o,o*,6, or n. Hence L = c ( L i l i E I > where, for each i E I , rF(Li)I p. Hence Li E V D and rvD(Li)5 r,(Li) 5 p for each i E I . Hence L E V D and rvD(L)< fi 1 =

+

T F W .

Assume then that ct is a limit ordinal. Let 0 = ct, < a, < ct2 < aj < . . . be an o-sequence of ordinals whose limit is M. (Theorem 3.36). Let x o E L and let cj(xo)denote caJ(x0) for each j . Then

co(x,) LL C l ( X 0 ) c c,(xo) c . . .

and

u { c j ( x o ) I< j o}= L . Now rF(cj(xo)) Ia j for each j < w by Lemma 5.13.2. Hence cj(xo)E VD and rvD(cj(x,,))I aj for each j < o.Define D j = ( c j +l(xo)- cj(x,)) n LtXo and D - j = (cj+ l(xo)- cj(xo)>n L'"" for each j < o.Since D j and K j are , from Lemma 5.22 that D j and D P j are in intervals of C ~ + ~ ( X it~ )follows VD and that rvD(Dj),rvD(D - j ) I rvD(cj+l(xo)) I a j + 1 . Now L = x { D , 1 t E Z 1 and each D,is in u { V D , (f l < a } . Hence L E VD, and rvD(L)I rF(L). Combining these two propositions, we obtain the following theorem. THEOREM 5.24: (Hausdorff) A countable linear ordering L is very discrete if and only if it is scattered. Furthermore, rvD(L)= rF(L). A We restricted our attention to the countable case at first in order to get the exact equality between the VD-rank of L and the F-rank of L . We now turn to the general case. DEFINITION 5.25: We define a class I/ of linear orderings by presenting inductively for each ordinal M a class V, and then setting V = V,.

u

(i) 0 , l E V,. (ii) Given a linear ordering I of order type y, y*, or y* + y for some ordinal y and, for each i E I , a linear ordering Li E { Vsl p < a), then Z { L \ i € I ) €V,.

u

The V-rank of a linear ordering L is the smallest ordinal a such that L E V,. Note that, since every well-ordering is already in V,, the V-rank of a countable linear ordering has little to do with its F-rank; for this reason we treated countable linear orderings separately. It would be nice if we were able to extend Definition 5.20 (and eliminate Definition 5.25), so that the VD-rank of an arbitrary scattered ordering were equal to its F-rank; we will see how this can be done in !$I.

3.

HAUSDORFF’S THEOREM

87

The proofs of the theorems above can be adapted to prove the following theorem. THEOREM 5.26: scattered. A

(Hausdorff) A linear ordering is in I/ ifund only if it is

Indeed, if we drop from the proofs above all comparisons of VD-rank and F-rank and replace VD by V everywhere, then few other changes need be made. In Proposition 5.21, instead of using the fact that 6 is scattered, we would here have to cite the more general fact that y* + y is scattered for any ordinal y. In Lemma 5.22, no further changes are necessary. In Proposition 5.23, the case where a is a limit ordinal would be dealt with by choosing a transfinite sequence of ordinals whose union is a (and whose length is no more than the cardinality of a),and using that sequence in the analogous way to obtain the corresponding result. Since the countable linear orderings in V are precisely the linear orderings in VD, we may, without confusion, call V the class of very discrete linear orderings. THEOREM 5.27: Every linear ordering can be represented as a dense sum of very discrete linear orderings; specijically, for any linear ordering L there is a dense linear ordering D and .for each i E D a very discrete linear ordering Lisuch that L = c { L i li E D}. Proof:

By Theorem 4.9 and Theorem 5.26. W

Theorems 5.26 and 5.27 permit us to prove general results by induction on rank. Thus, if we wish to prove that every scattered linear ordering has property 9,we may, in assuming the contrary, select a counterexample of minimal rank-a procedure which often facilitates contradictions. THEOREM 5.28: If A is an uncountable scattered linear ordering, then either w 1 5 A or w l * 3 A . Proof : Suppose the conclusion false and choose A to be a counter-example of minimal I/-rank. Then A is a y-, y*-, or y* + y-sum of linear orderings of smaller rank, for some ordinal y. If any of those linear orderings is uncountable, then, since each has smaller rank, it, and hence A itself, must contain a copy of w 1 or wl*. Hence each must be countable. But then y must be uncountable, for otherwise A would be a countable sum of countable linear orderings, and hence countable, contrary to hypothesis. But if y is

88

5.

HAUSDORFF'S THEOREM

uncountable, then w1 < y by Proposition 3.25. Thus, if A is a y-sum of orderings, we can, by extracting an element from each, obtain w1 5 A ; whereas if A is a y*-sum of orderings, we obtain wl* 5 A. (What happens if many of the summands are empty?) Thus either o1IA or wl* IA. W Of course, by arguing a little more carefully, this result can be strengthened. A critical fact used was that the sum of countably many countable sets is countable. More generally, a sum of K sets of cardinality S K also has cardinality K . Thus if, in the result above, KOis to be replaced by K , uncountable should be replaced by K ' . COROLLARY 5.29: I f A is a scattered linear ordering of cardinality then either K + 5 A or (K+)* 5 A . A

K',

The corollary states a result for arbitrary successor cardinals K + . What about cardinals which are not successor cardinals? The result may be false; that is, it is possible to have a scattered linear ordering of cardinality K but neither K 5 A nor K* IA . A simple example of this is afforded by the very first non-successor cardinal K I,J. Recalling that the cardinal numbers are all also ordinals, we let A

=

. . . + K, + K,

+ K, + K,. Kc,,$ A , and K,* 6 A. (In fact, if a* 5 A , then

Then A has cardinality CI I w.) Thus A is a counterexample. Corollary 5.29 would hold if the cardinality K of A were regular, that is, if the union of fewer than K sets, each with fewer than K elements, also had fewer than K elements. As we observed in Exercise 3.40.2,every successor cardinal is regular. Using Theorem 5.28, we obtain a more revealing proof c.f the result stated in Exercise 3.40.5. COROLLARY 5.30: If A is an uncountable linear ordering, then either q 5 A, o15 A , or (a1* IA . Proof:

If g $ A , then A is scattered, so Theorem 5.28 applies.

A linear ordering A for which w1 6 A and wl* 6 A is sometimes referred to as short. Thus any uncountable short linear ordering A has a subset of order type q. If instead we combine Theorem 5.27 and Corollary 5.29, we obtain a sharper result than that of Exercise 3.40.6.

3.

HAUSDOFWF'S THEOREM

89

THEOREM 5.31: Let A be CI linear ordering of cardinality K + . Then either K' 5 A or (K')* IA or 8 3 A for some dense order type 6 of cardinality K + .

Proof : By Theorem 5.27, A is a dense sum of scattered linear orderings. If any of these has cardinality K + , then we apply Corollary 5.29. Otherwise, the dense index set must have cardinality K + , since the sum of K many linear orderings, each of cardinality Iti, also has cardinality K . W Now suppose that A II and that A is uncountable. If we assume the continuum hypothesis, namely, c = N , , then, by Theorem 5.31, since A has cardinality N 1but has no subordering of order type o1or wl* (by Exercise 3.26.2), it follows that A has a dense subordering of the same cardinality as A. This result can be proved also without the continuum hypothesis. THEOREM 5.32: (1) There i~ no uncountable subordering A of R which is scattered. (2) I f A G R and A is uncountable, then A has a dense subordering D whose cardinality is the same as that of A. Proof : (1) By Corollary 5.30 and Exercise 3.26.2. (2) By Theorem 5.27, A is a sum of scattered linear orderings, each of which is countable (by Exercise 3.26.2),indexed by a dense linear ordering E. Let D consist of one element from each summand. Then ID1 = IEl = IAI and D is dense.

Can we improve on this result in the fashion of Corollary 5.30? That is, can we show that A must contain a particular uncountable dense order type? In $9.3 we will see that the answer to this question depends on the continuum hypothesis. In $9.4 we will show that there are linear orderings A of cardinality 'K such that 'K A and (IC+)*$ A, so that Theorem 5.31 cannot be strengthened by deleting the final alternative. Another question answered in $9.4 is the following: Is there a bound on the cardinality of short linear orderings? Finally, we mention that implicit in Morley [4] is a proof that under certain assumptions about K (K has to be strongly inaccessible and weakly compact), for every linear ordering A of cardinality K either K 5 A or K* IA.

6

EXERCISE 5.33: (1) Show that every countable scattered linear ordering has exactly a countable niimber of Dedekind cuts. Is the corresponding statement true for arbitrary scattered linear orderings?

90

5 . HAUSDORFF'S THEOREM

(2) Show that if 7 is scattered and 0 .w* -& 7, then 7 = c(p,*ly < a } for some ordinals c1 and (&\y < a } . (3) If A and B are scattered linear orderings and each is isomorphic to an initial interval of the other, does it follow that A 'v B? (Compare Exercise 1.45.1.)

$4.

Z AND ITS

POWERS

We noted earlier that, in some sense, it is possible to extend the notion of VD-rank in such a way that the rank of any scattered linear ordering equals its F-rank. To do this, we first discuss Z and its powers. In Chapter 3 we defined the ordinal power ap for arbitrary ordinals c( and fi by extending the sequence a, c1', g 3 , .. . into the transfinite. Here we would like to define Z p for an arbitrary ordinal p by extending the sequence Z,Z2, Z3,. . . into the transfinite. Our first problem occurs with Z'". When we defined w"', we took it to be simply lim{o"In < w ) = (o"ln < o},a well-defined, though yet unnamed, ordinal. Do we have a well-defined limit of the sequence Z,Z2, Z 3 , . . .? We first observe that 7"" is constructed from Z"by adding w copies of 7" at its right and w* copies of Z"at its left. Thus, by induction ~ n + = l

. w* + . . . + Z2.o*+ Z . w * + z. 0 + Z2 . w + . . . + zn-' . w + Z". w

p .0*

+p - 1

+ w* + 1 + w

and so we can define Poto be

. . . + Z". w* + . ' . + Z2 w* + Z . a*+ o* + 1 + 0 + Z ' 0 + Z 2 . 0 + ... + Z".0 + ... . '

This motivates the following inductive definition of Zp, for arbitrary YI y < p}. terms of { Z

p, in

DEFINITION 5.34: (1) Zo = 1. (2) Z b + l = Z P . o * + Z f i + Z f l . w . (3) Z2 = (XtZ? wly < Z.j)* + 1 + c{ZY- w l y < A} forlimit ordinals 1. +

In Chapter 3 we presented an alternative and equivalent definition of the ordinals ua (Exercise 3.45) which can be adapted for our purposes. DEFINITION 5.35: Given an ordinal B, let Zo(8)consist of all p-sequences ii = ( a T [ y< p) of elements of Z which have only finitely many non-zero

4. Z

AND ITS POWERS

91

entries. We order Z,(p) by stipulating that 6 '], IFb] 2 2" - 1, so that by induction hypothesis G,(A'", B'b) E 11. If the given element a E A is one of the last 2" - 1 elements of A , we let b be the corresponding element of B and proceed as above. If the given element a E A has at least 2" - 1 predecessors in A and at least 2" - 1 successors in A , then we choose b to be any element of B with at least 2" - 1 successors and at least 2" - 1 predecessors; since IBI 2 2"" - 1, such an element exists. Then by induction hypothesis, G,(A '", E I1 and G,(A'", B ' b ) E 11. Hence (i), and similarly (ii), holds. Thus, by Theorem 6.6 B ) if A and B each have above, PLAYER 11 has a winning strategy in G , , cardinality 2 2"+ - 1. This completes the proof. W

100

6.

THE EHRENFEUCHT-FRAYSS~

GAME

Note that a special case of this corollary is that PLAYER 11 has a winning strategy in G3(7,8)-which is included in Exercise 6.1. That no further improvement is possible in case n = 3 is seen from the fact that PLAYER I has a winning strategy in G3(6,7). EXERCISE 6.10: (1) Show that PLAYER I has a winning strategy in G,(h,k)ifk b or B C b(or both) is again of the form w + t; . a' + w* for some order type a'; and see the proofs of the corollary above and the corollary below. Note, however, that, since Corollary 6.12 uses the result of this exercise, it would be quite inappropriate to use Corollary 6.12 here.]

+

COROLLARY 6.12: PLAYER 11 hus a winning strategy in G,(w,w + ( . a ) jbr any order type a and any n 2 1.

Proof: We proceed by induction on n to show that G,(w, w + t; . a ) E I1 for every a. The case where n = 1 is obvious. Assume that the statement is true for n and proceed, as in Corollary 6.9, to use Theorem 6.6. Thus suppose that a E w = A ; then let b be the corresponding element of w + 4 . a = B. Then, since A'" c B < b ,G,(A'",B'b) E 11; and, by induction hypothesis, since B'b 2: w + t; . a, G,(A'",B'b) E 11. Suppose, on the other hand, that we are given b E B. If b is in the initial w of B, then choose a to be the corresponding element of A and proceed as above. If, on the other hand, b is not in the initial w of B, then B'b 2: w + 4 . a' and B'b c w + t; . a" + w*, where a' and a" are particular order types. Let a be the 2"th element of A . Then IA'"I = 2" - 1, so that, by the exercise above, G,(A'", B b) E 11. Hence conditions (i) and (ii) of Theorem 6.6 are satisfied, so that PLAYER 11 has a winning strategy in G,(o,w + ( . E) for every a and for every n.

+

In particular, PLAYER II has a winning strategy in G,(w,w t;) for every n. Thus no difference between w and w t; can be discerned through the games.

+

1.

101

THE PLAY OF THE GAME

These games were used by Ehrenfeucht in his investigations of the logical equivalence (or elementary equivalence) of mathematical structures and, more specifically, of well-orderings. Informally, two mathematical structures are said to be logically equivalent if there is no statement in the first-order predicate calculus which distinguishes them. Ehrenfeucht showed that two structures are logically equivalent if and only if they are Gequivalent, and used this characterization of logical equivalence to obtain interesting results about well-orderings. Similar results were obtained by Fraisse, using different terminology. Thus the logical interpretation of Corollary 6.12 is that o is logically equivalent to w 5 . c( for any order type CI.Thus, for example, no statement of the first-order predicate calculus is true for w but false for w (. One consequence of this fact is that no first-order statement and, more generally, no set of first-order statements, can be true precisely for those linear orderings which are well-orderings. This is usually expressed by saying that the notion of well-ordering is not first-order definable or that the class of wellorderings is not an axiomatizable class. We will return to well-orderings shortly. The logical uses of the Ehrenfeucht games will be further discussed at the end of this chapter and in Chapter 13. The equivalence relation -"partitions the class of linear orderings into a number of equivalence classes. We will now prove that for each n there are only a finite number of equivalence classes; this contrasts strongly with the equivalence relation -, which, as we will see, has infinitely many (and, in fact, has c ) distinct equivalence classes.

+

+

THEOREM 6.1 3 : For each n there are only a Jinite number of equivalence classes modulo -,,.

Proof: We proceed by induction on n to show that there are only a finite number f ( n ) of equivalence classes modulo n. For n = 1, as we already know, there are but two equivalence classes. Assume the claim for n. Let A be a linear ordering. Then each element a E A determines an ordered pair ( [ A ' " ] , [A'"]) of equivalence classes modulo -,,, where [B] is the -,,equivalence class of B. Let I ( A ) = { ( [ A ' " ] , [A'"])la E A } . We note that Theorem 6.6 can be understood as saying that A - n + l B if and only if I(A) = I(B). Hence the number of equivalence classes modulo is at most the number of subsets of the set of ordered pairs of equivalence classes modulo -,; but this number is 2J(")'s(").Hence f ( n + 1) 5 2f""rcn), and in particular f ( n 1) is finite.

+

102

6.

THE EHRENFEUCHT-FRA~’SS~GAME

The reader might naturally inquire at this point what the values of f ( n ) actually are, and he may wonder whether the function f actually attains the upper bound given in the proof and therefore satisfies the recursion f(1) = 2, f ( n + 1) = 2f(“)

The value for f ( 2 )would then, for example, be 16. In point of fact, however, .fV) = 7. EXERCISE 6.14: (1) Show that f ( 2 ) = 7 by giving a complete set of representatives for the equivalence relation m 2 . (2) Find f ( 3 ) . [Caution: This is not an exercise, but a major project.]

The fact that there are only a finite number of G,-equivalence classes for each n suggests the possibility that there is a “nice” countable set A of order types such that for every n E N and every linear ordering A there is an M E .4 such that A -, M. We will see in the next chapter that this is indeed the case. THEOREM 6.15: ulo -.

There are at least c different equivalence classes mod-

We define, as in the proof of Proposition 1.48, for each subset X G N a linear ordering L ( X ) such that if X I # X , , then L ( X , ) is not C-equivalent to L ( X 2 ) . We define A , ( X ) to be of order type q + (a + 2) if a E X and of order type 1 if a $ X ; and we let L ( X ) = C { A , ( X ) l a< w ) . Let X , Y G N such that X # Y . Let a be an element in X - Y (or, alternatively, in Y - X ) . We claim that PLAYER I has a winning strategy in C,+.(L(X),L ( Y ) ) .Indeed, on his first a + 2 turns, he chooses in order the final a + 2 elements of A,(X). He then considers PLAYER 11’sfirst a + 2 choices b , , . . . , b,+2 in L( Y ) . If, for some i. there is an element of L ( Y ) between bi and bi+ then PLAYER I chooses that element on his (a + 3)rd turn and forces PLAYER 11’s resignation. Otherwise, since there is no maximal discrete sequence of exactly a + 2 elements in L( Y ) ,either there is an element b beyond b a + 2in L( Y )with nothing of L( Y )in between, or there is an element b before b , in L( Y )with nothing of L( Y )in between. PLAYER I chooses such an element for his ( a + 3)rd turn; suppose that he chooses such a b and suppose, without loss of generality, that b,, < L l y ) b. Then PLAYER II presumably chooses an element c , + ~of L ( X ) after the last element c of A , ( X ) in L ( X ) . But then Proof:

1.

103

THE PLAY OF THE GAME

PLAYER I chooses an element between c and c,+ ,-and since PLAYER II cannot find an element between and b, he loses. Hence PLAYER I has a winning strategy in G,+,(L(X), L ( Y ) ) .Thus if X and Y are different subsets of N , then L ( X ) is not G-equivalent to L,(Y). Hence there are at least c distinct equivalence classes of countable linear ordering modulo .

-

-

Can there be more than c diflerent equivalence classes modulo ? We will show first that every G-equivalence class contains a countable representative. (This is the combinatorial version of the Lowenheim-Skolem Theorem for the predicate calculus; this method of proof was suggested by J. Schmerl.) Since there are but c distinct countable order types, this shows that there are exactly c different G-equivalence classes.

-

THEOREM 6.16: (Lowenheim-Skolem) Let A be a linear ordering. Then there is a countable linear ordering B A such that B A .

Proof: We first note that, as in the proof of Theorem 6.13, given an interval C E A and given nE N , each element U E Cdetermines an ordered pair S(C,a, n) = ([C'"],, [ C > " ] , )of equivalence classes modulo -,, where [D], is the n-equivalence class of D. Furthermore, by Theorem 6.13, for each n there are but a finite number of possible S(C, a, n). Thus it is possible to select a finite subset S(C, n) E C so that for each U E C there is a unique b E S ( C , n) with S(C, a, n) = S(C, b, n). The countable subset B of A IS defined to be the union of an ascending sequence

$8 = B, s B , G B ,

S.-._C B,G

B,+l E ' . .

of finite subsets of A, defined by induction on n. Assume that B, has already been defined; the k = k(n) points of B, partition A - B, into k + 1 intervals C,, C , , . . . , C, (some of which may be empty.) Define

B,+l= B, u U{S(Ci,n)IO5 i C:" B

I k}.

Thus for any b E A and any n, either b E B, or b E C iand for some a E Ci, -,C 0. From this it will follow, using the induction hypothesis, Lemma 6.3.2, and Lemma 6.5.2 on sums of 2(n - 1)-equivalent orderings, that Gz,(w", o". p) E 11. But it is clear that PLAYER 11 can respond so that the above conditions are met by taking care that, if PLAYER I chooses a point of the form (on-'. Po) + y where y < a n - 'then , he chooses a point of similar form; and, in particular, if Po

6.

106

THE

EHRENFEUCHT-FRAYS&

GAME

is finite, then PLAYER 11 chooses the same point in the other ordering. (Note that every tail of wkhas order type w k ;this is a special case of Exercise 3.44.3.) ~ n . P)EI, we will describe the first two moves To show that G 2 n + l ( (LO" of PLAYER 1's strategy. On his first move he picks the point w" in wn p. Then, no matter which point a in (0" is picked by PLAYER 11, PLAYER I at his second turn picks a point b in w" so that (b, a) 'v wk for some k < n (unless a is a successor, in which case PLAYER I picks its predecessor.) Then, no matter which point d < w" in o"+ [I is picked by PLAYER 11, (d,0")= 0".But, by the induction hypothesis, PLAYER I has a winning strategy in Gzfl-' ( w k ,o")for any k < n. Hence Gz,+l(co",ion /?)€I for any p > 0.

+

+

+

COROLLARY 6.19: For any natural number n, if TX * 2 n + 2 w"+' and u - 2 n + 3 p.

ci

< w""

< p, then

+

Proof: Write ci = ~ " .l u , wn2. a2 + . . . + conk. uk in Cantor Normal Form. In the game Gzn+2 ( x , (fin+l), PLAYER I first picks a point a of ci which determines a tail of order type conk.Then no matter which point PLAYER 11 selects in a"+ ',the tail it determines has order type a"+ But, by the theorem, PLAYER I has a winning strategy in the game G2nk+l(wnk, w"+'), hence also ni PLAYER I has a winning strategy in in the game C Z n + l ( ~ n k , f').r JThus

'.

G 2 , + 2 ( c i , ~ 1 " + 1 ) i f< c iw " + ' .

In the game G2,r+3(ci, p) where ci < LO"+' < p, PLAYER I first picks the point w"+' in p. Then no matter which point a in a is selected by PLAYER 11, its predecessorshave order type < a"+ ',so PLAYER I can continue his strategy as in the preceding paragraph to obtain a win in Gzn+ 3(ci, p). The Exercise below shows, by example, that the results of Corollary 6.19 are the best possible. EXERCISE 6.20:

Prove each of the following for all n:

(1) W " . 2 - 2 , w f l + 1 + to". ( 2 ) W".2-2,+l w " + ' w". (3) w" . 2 *2"+2 Q"+ -tw". (4) W " . 2 - 2 n + 1 o"+I.

'

(5) (6)

W"'3h.2,+1

OJ"+'

+

+On.

3 * 2 n + 2 or"' + w" for n > 0. 1 w". (7) on 4 - 2 n + 2 O". '

+

Note that (4) and (7) show that the results of Corollary 6.19 are the best possible; this proof of that fact is due to P. Mulhall.

2.

107

GAMES A N D ORDINALS

(8) Show that every ordinal is G2,-equivalent to some ordinal in the finite set (w" . a,

+ w n - l . a,- + 1

c(j"-2

. an-,

+

'

'.

+0

'

a1

+ ao:

where each ai < 22" and a, I 1. THEOREM 6.21 : Let a and fi he ordinals. Write a = ow. a, + a2 and p = cu"' . fi, + p2 where a, < o" and [j2 < (d".Then c1 fl if and only if a2 = f12 and either r l and p1 are both 0 or are both bigger than 0.

-

-

-

Proof : Since Q"'= wn . ow,by Exercise 3.48.4, it follows from Theorem 6.18 that coo . r l ou. ,8,, for any a l and fil which are non-zero. Hence a p if the conditions are met. Now suppose that a, = 0 but PI > 0. Let a2 < on. If PLAYER I chooses . in j? at his first turn, then the initial segments a. and Po of u and p determined by the first turn satisfy a. < W" < Po, so by Corollary 6.19, PLAYER I can win in 2n + 1 further turns. Hence G2,,+ ,(a,p) E 1. Thus if u p it follows that a1 and p1 are both 0 or both non-zero. Suppose now that c1 < p < (0'". Write p in Cantor Normal Form as w n l. a , (on* . a2 . . . mnk . a k . We claim that PLAYER I has a winning strategy in Gn(a,p) where n = (a, a2 . . . + ak)- 1 + 2n,. Indeed, at his first u 1 turns, PLAYER I chooses the points wnl. 1, wnl . 2 , . . . , w"' . a , ofp; at his . 2 , .. ., next a2 turns, he chooses the points wnl . a, + mn2 . 1, wnl . a, + on2 (d"'. a , + wf12. a,; . . . ; at his nexl ak - 1 turns, he chooses the points (1)111 . a, + . . . + g n k - 1 . ak- 1 wnk' 1 , . . . , 0"'' a1 ' ' . Unk. (ak - 1). He now surveys PLAYER 11'sfirst ( a , + a2 + . . . + ak) - 1 moves. Since /3 > a, at least one of the intervals in a must be smaller than the corresponding nl, interval in p. Since the intervals of p are all of form wm for some m I PLAYER I needs at most 2n, further moves to win. Thus G,(u,p) E I as claimed. Hence if a fi and u, p < ow,then u = p. Similarly, suppose that u = cu" . a1 + a2 and p = 0") . PI + p2 where cil and fil are non-zero and u2 < p2 < w". If a, = 0, then PLAYER I can win in G2,(r,p) by selecting a point of /3. whose successors have form okfor some k < n, and then using the strategy of Theorem 6.18.2 for his remaining 2n - 1 moves. If a2 > 0, then by picking the point ow. u1 in u he leaves PLAYER 11 with three ways of losing: Either PLAYER II picks a point of p which is less than o0 . p,, in which case the successor intervals are a, and ow. -y p2 so, by Corollary 6.19, at most 2n 1 further moves are necessary; or PLAYER 11 picks the first point of p2, in which case the successor intervals are a2 and p 2 , so that PLAYER 1 can win in at most t = (a, a, + . . . + ak)- 1 2n,

-

+

+

+

+ +

+

+

+

-

+

+

+

+

108

6.

THE EHRENFEUCHT-FRAYS&

GAME

+

more moves where fl = m"' . a , . . . + wflk. ak,as in the preceding case; or, finally, PLAYER II picks a point beyond the first point of p 2 , in which case the predecessor intervals are owand w" y where 0 < y < m", and in this case PLAYER I can win in at most 2n + 2 additional moves. Thus in the case where 0 < ci2 < f12, PLAYER I can win in the game G,+,(u,fl) where m = max(2n + 3, t ) where n and t are bounds depending only on /?.This completes the proof. H

+

Theorem 6.21 enables us to conclude that the set of all ordinals less than

UP . 2 forms a complete and irredundant set of representatives of the Gequivalence classes of well-orderings. By way of contrast, although any wellordering is G-equivalent to one of a countable number of well-orderings, the analogous statement is false for arbitrary linear ordering, as we verified in Theorem 6.15. The logical interpretation of Theorem 6.21 is that the set of all ordinals less than LO"' . 2 forms a complete and irredundant set of representatives of the logical equivalence classes of well-ordering ; this fact was noted earlier by Mostowski and Tarski [4]. The first application made of the EhrenfeuchtFrai'sse analysis is the following result of Ehrenfeucht [11 and Fraisse [3].

-

THEOREM 6.22: Let O N denote the collection of all ordinal numbers, with the usual ordering. Then O N d". Proof : Since m" m Z f lo") for each n, by Theorem 6.18, it suffices to show that (d'- 2 n ON for each n. This is proved, by induction on n, making only notational changes in the proof of Theorem 6.18.1. H

Theorem 6.22 has the following logical interpretation : Given a first-order statement about the ordering of the ordinal numbers, that statement will be true if and only if it is true about the ordinal numbers below LO'). As we will see later, this interpretation makes it possible to conclude that the theory of the ordering of the ordinal numbers is decidable; that is to say, that there is an effective procedure which, when presented with any first-order statement about the ordering of the ordinal numbers, will determine whether or not that statement is true. This result was also proved by Mostowski and Tarski [4]. Ehrenfeucht [11 also proved that a first-order statement about addition of ordinal numbers is true if and only if it is true about the ordinal numbers below IO")'"; or, in the language of games, ( O N , ERINGS

167

mark, Hausdorff noted that the more general conclusion follows from his results about q,-orderings. Padmavally [26] improved Theorem 9.28 as follows. If B is a linear ordering for which o,-& B and coo,* 6 B, then B I Q,; hence the cardinality of B is at most 2”fl I fi < a } if CI is a limit ordinal. A different proof ofthis result can be found in Harzheim [ 171; generalizations of this theorem appear in Harzheini [18, 191. We observed earlier that the q,-orderings share many properties with q, and generalize the countable case since q,-orderings are precisely those whose order type is q. Let us look at Q, a little more closely. It consists of all m-sequences of 0 and 1 which have only a finite number of non-zero entries. Regarding such a sequence as a finite sequence and writing it as a decimal, we see that Q, is order-isomorphic to the set of rational numbers in the interval (0. 1) which have a finite binary representation. Now every other real number in the interval ( 0 , l )has a unique infinite binary representation; conversely, given any infnite binary sequence, there is a unique real number whose binary representation is that sequence. This correspondence between real numbers in (0,l) and infinite binary sequences is not quite exact since each rational number which has a finite binary representation also has an infinite one which ends in an infinite sequence of 1. Thus if we define R, to be the subordering of A , containing all sequences which are not eventually 1, then R , = R; we are already familiar with this from Lemma 9.12. More generally we define R, lo be the subordering of A , containing all (0,-sequences except those which are eventually 1. We now examine some of the similarities between R , and R.

c{

Proof: (1) If 2 4. Then x’ c 4 5 9, so that Q, is dense in R,. (2) Arguing as in Exercise 3.26.2, if a,+ 5 R,, then the K,, intervals of R, thus created would each contain a distinct element of Q,, so that Q, would have at least K,+ elements. But, assuming the generalized continuum hypothesis, Q, has exactly K, elements, which is a contradiction. This fact can be proved, however, without the extra assumptions. Following Sierpinski [30], assume that w,+ IR , and let (iisIp < w , + ~ ) be a sequence of elements of R , of order type a,+1. For each p < a,+ let $(p) be the least ( such that agB.jC a ! ” . Now 4(p) < o,for each f l
a, > a2 > . . . . An anti-chain of A is a set of elements which are pairwise incomparable. DEFINITION 10.14 : We say that A is a wellquasi-ordering, denoted wqo, if A has no infinite descending chain and no infinite anti-chain.

184 10.

EMBEDDINGSOF LINEAR ORDERINGS AND

FRA‘I‘SSB’S CONJECTURE

EXAMPLE : 9is not a wqo since, as we showed in Theorem 9.10, there is an infinite descending chain of 9 whose elements are the order types of dense suborderings of R.

Fraisse’s Conjecture amounts to the statement that the countable order types form a wqo under embeddability. It suffices to show that the countable scattered linear orderings .Y form a wqo since no infinite descending chain and no infinite anti-chain can have more than one non-scattered entry. We will start by proving some basic facts about wqos in general. DEFINITION 10.15: Let a‘ = {a,ln < a} be an w-sequence of elements of A . We call a’ good if there are indices n < m such that a, I a,; otherwise a‘ is bad. LEMMA 10.16 : A is a wqo if and only if every w-sequence of elements of A is good. (e)If A is not a wqo, then either there is an infinite set {b, 1 n < o} of incomparable elements of A or there is an infinite descending chain * . . < b, is bad. (=) Suppose that there is a bad w-sequence a‘ of elements of A . Define a partition of [NI2 into X I and X , by specifying that if n < m, then (n,m} E X I if a, > a, and { n, m} E X , if a, I a,. (Note that for every n < m one of these two possibilities actually occurs, since the badness of Li makes a, I a, impossible.) By Ramsey’s Theorem 7.4, there is an infinite set Y which is homogeneous for this partition. If [Y]’ G X I , then the sequence { a, I n E Y ) is descending, whereas if [Y]’ G X , , then the sequence (a,ln E Y > is an infinite anti-chain, so in either case A is not a wqo. W

Proof :

Lemma 10.16 will facilitate proving that various quasi-orderings are actually wqos. LEMMA 10.17: Let A be a wqo and let Ci be an o-sequence of elements of A . Then there is a subsequence { a , In E Y } which is either strictly increasing ( n < m implies a, < a,) or constant ( n < m implies a, a,).

-

Proof: As in the proof of Lemma 10.16, partition [ N ] ’ into four sets (where for n < m the location of {a,,a,} depends on whether a, < a, a, > a,, a, a,, or a, la,) and apply Ramsey’s Theorem. H

-

3.

185

WELL-QUASI-ORDERINGS

EXERCISE 10.18: (1) Let A and B be quasi-orderings such that A E B. We say that A is a subordering of B if, whenever a,, a, E A , a , I A a, if and only if a , s Ba,. Show that if A is a subordering of B and B is a wqo, then so is A . (2) Let A , = ( A , 11)and ,4,= ( A , 12)be two quasi-orderings of A. We say that A , is an extension of A , if, whenever a 5 , b, a b. Show that if A 2 is an extension of A , and A l is a wqo, then so is A , . ( 3 ) Show that A is a wqo if and only if {b E A a $ b } is a wqo for each aEA. (4) Verify the following induction principle for wqos. If a proposition P(Q)is true for a wqo Q wheneker it is true of Q, = {Y E Q 1q $ Y} for each q E Q and if S(@)is true, then PIQ)is true for every wqo Q. Similarly, if A is a wqo, i f P ( @ ) is true, and if.Y(Q) is true for Q G A whenever it is true of Q, for each q E Q , then S(Q)is true for all Q G A (Laver [16]). (5) Show that A is a wqo if and only if for every A' E A there are elements a,, . . . , a, E A' such that if a E ,4', then ai I a for some i I n. (6) Let A and B be quasi-orderings. A function f mapping A onto B is a homomorphism if a IA b implies that f ( a ) I B f ( b ) .Show that if f is a homomorphsm of A onto B and A is a wqo, then so is B. (7) Show that if A is a wqo and {anln < w } is a sequence of elements of A , then there is a number k such that for every m 2 k there are infinitely many n > m for which a, I a,.

I,

I

DEFINITION 10.19: Given quasi-orderings A and B, we define the quasi-ordering A x B by stipulating that ( a , , b , ) I (a2,b,) if and only if a, sAa , and b , S Bb2.

LEMMA 10.20:

I f A and B are wqos, then so is A x B.

Proof: Let {(a,,b,)ln < 0)) be an w-sequence of elements of A x B. By Lemma 10.17, we can choose a subsequence {a,ln E Y ) of (a,\n < w } so that if n, m E Y and n < m, then a, I a,. Now since B is a wqo, {b,]n E Y > is good, so that for some no, m, E Y with no < m, we have b,, I b,, . Hence (ano,brio) I ( a m o b,,,), so that { (u,, b,) In < w> is good. By Lemma 10.16, we conclude that A x B is a wqo. H DEFINITION 10.21: Given a quasi-ordering A , we define the quasiordering A whose domain is the set of all finite sequences of elements of A , by stipulating that {u,, a , . . . ,an-,) 5 {b,, b,, . . . ,b,- ,) if there is an increasing h : n + m such that ai b,,i) for all i < n.

186 10.

EMBEDDINGS OF LINEAR ORDERINGS AND FRA'ISSE'S CONJECTURE

This definition begs for motivation. Let us first look at a simple example. Let A be the (well-quasi-)ordering of the natural numbers. Then the sequence 1 2 3 4 is embeddable in the sequence 4 3 2 1 4 3 2 1 4 3 2 1 as shown in Fig. 1.

On the other hand, the sequence 1 2 3 4 5 is not embeddable in the sequence 4 3 2 1 4 3 2 1 5 4 3 2 1 since (starting from the right) the best hope for an embedding would be that shown in Fig. 2, leaving no place for the first entry. 1 2 3 4 5

JJ \ \

FIGURE

2

4 3 2 1 4 3 2 1 5 4 3 2 1

Why does this definition arise in our discussion of embeddings of linear orderings? Let { L , , L 1 , .. . and { M o , M , , . . . , M m - l be } two finite sequences of linear orderings; one way for Lo + L1 + . . . + L,_ to be embeddable in M , + M , . . . + M m - , is if { L o ,L , , . . . , & - I } 5 { M o , M I , . . M m - 13. We observe that, in general, if s 5 t are finite sequences of elements of a wqo A , then Ih(s) 5 Ih(t) [where Ih(r) is the length of r ] , and that if Ih(s) = Ih(r), then si IA ti for every i. In our particular example, where A is N , there can be only a finite number of distinct sequences s such that Ih(s) = Ih(t) and s I t , where t is a given sequence. Thus an infinite descending chain of elements of N'" would generate an infinite descending chain of natural numbers-the lengths of the given sequences-contrary to established truth. Hence N'" has no infinite descending chain. The only special property of N used in this proof is that given a finite sequence t there are only a finite number of distinct sequences s of the same length as t satisfying s 5 1. This is of course false in general. All that we need for such an argument, however, is that within the set of sequences of fixed length n there be no infinite descending chain. This is true for arbitrary wqos A , as the following shows.

+

' 3

EXERCISE 10.22: Show that if A has no infinite descending chains, then neither do the sequences of elements of A of length n for any fixed n.

3.

WELL-QUASI-ORDERINGS

187

[Hint: This can be done using either Ramsey's Theorem or a simpler pigeonhole argument.] Conclude that A 0; then we can write A as an a-sum of intervals A , A , + A , + . . * . We will write A as an o-sum of intervals B, B , + B, + . . . where each Bi is embeddable in all subsequent B j . Since each Bican be written, by Corollary 10.49, as a finite sum of ha-indecomposables, and since, by our construction, each of these ha-indecomposables will be embeddable in infinitely many subsequent ones, it will follow that A is itself ha-indecomposable. We define the intervals (B,( n < o} inductively as follows: B, is A,. Assuming B,, B , , . . . ,B, have been defined and each Bi is embeddable in all subsequent Bj so far defined, we write A = A , + A , where A , = B, + . . . B,. Then A 3 A , since A is indecomposable on the right. Now we may choose B,, to be an initial segment of A , that includes the image of B, + . . . B, under this embedding and so that B, + . . . B, B,, = A , + . . . A,,, for some m 2 n + 1. Then the sequence ( B , \ n < o}has the desired properties and A is ha-indecomposable.

+

+

+

+

+

,,

,

+ +

,

We have completed our proof of Fraisse’s Conjecture for Y .The reader will have observed that little use was made of the fact that Y consisted only of countable linear orderings. Indeed, he should verify (Exercise 10.46) that one application of the fact (not proved in this book), that if A is a bqo, then so is A“ for all ordinals a, yields the conclusion that the collection 9’of scattered linear orderings is a bqo. The only modification of subP to the appropriate stance that would be needed would be to extend i collection %+, in which case Corollaries 10.49 and 10.50 would also be provable in more general form. Another quasi-ordering on Y is the relation CI 5 p if CI is a homomorphic image of p (see Definition 4.3). Recall that if a 5 p, then a 5 p although the converse is false in general since, for example, w 5 w 1 but w $ w 1. Using Laver’s methods, Landraitis 1121 has shown that (Y,5)is also a bqo.

+

+

202 10.

EMBEDDINGSOF LINEAR ORDERINGS AND FRAYSSE’SCONJECTURE

EXERCISE 10.51 : (1) Determine for which countable linear orderings A there is a partition A = B u C of A such that A = B and A = C. (2) Prove Hagendorf’s Theorem 10.8 for A scattered using Theorem 10.48 (extended to 9’) and Exercise 10.18.5.

The final result of this chapter is the second conjecture of Fra’issC, which was also proved by Laver [16]. THEOREM 10.52 : (Laver) If t is a countable scattered order type, then, modulo the equivalence relation -, there are only a countable number of order types t’ such that t’ < t . Proof: We will say that a set T of linear orderings is big if there is an uncountable set of =-equivalence classes represented in T ; otherwise, we say that T is small. Thus we must show that [t’I z’< T} is small for every z,then, by Lemma 5.14, countable scattered order type T . IfrF(t) = a and t’ i r F ( t ‘ ) 5 a, so that t‘E Y ( a ) , the set of scattered linear orderings of rank at most a. It thus suffices to show that Y ( a ) is small for every countable ordinal a. We first confine our attention to 2 and its corresponding subsets %(a), consisting of all elements of 2 of rank at most a. For any subset T c S, we let T +consist of all o-sums A , + A , + A , + . ,and we let T - consist of all w*-sums . . . + A 2 + A , + A , where each Ai E T and each A, is embeddable in infinitely many A j . Then T + E 2 and T - c % and, setting H = u{%(fi)IB < a } , we have %(a) = H u H + u H - . To show that %(a) is small for each countable ordinal a, we proceed by induction on a. Thus, by the induction hypothesis and symmetry, we need only show that H + is small; however, the induction argument requires that we show that X + is small for every X E H. Suppose, to the contrary, that X + is big for some X L H. Now if {t’E X + It 6 z’} were small for every z E X + , then we could easily construct an uncountable sequence {t,lt( < ol} of order types in X + for which t , i zo whenever a < B. But if za = x(z,,,/n < w f for each cc, then { t a , “ \ m < q, n < o}E X , which is small by the induction hypothesis; hence there is an ordinal y < w1 such that for every a < o1and every n < o there is a B < y and an m < w for which z,,, = T ~ , This ~ . implies, however, using properties of 2, that t y+ 5 ty, which is a contradiction. Hence {T’ E X + It 6 z’} must be big for some t E X + . Suppose that t = t o+ t1+ t 2 + . . . . If z’E X + and t -$ T’ and T‘ = to’ t l ’ t2‘+ . . . , then, for some n, z,,6tm’for all m < o.Hence for some n, the set T = (t’E X + It’ = ~ { t , ‘mI < o}and t,,$5,’ for every m } is big. But if t‘E T, then t’ E (Xm)+,where X , = {d E X ( a 6 a’}.Thus

+ +

REFERENCES

203

we have shown that if X + is big, then ( X u ) +is big for some B E X ; that is, if (Xu)+is small for each B E X , then X + is small. By the induction principle of Exercise 10.18.4, X + is small for every X c H , so that, in particular, H' is small. Thus X ( a ) is small for every countable ordinal a. the set of Since &(a) is small for each x , the same is true of finite sequences of elements of %(a) ordered as in Definition 10.21. Define a m a p f : X ( a ) ' " + Ybyf(z,,z, , . . . , z k - l ) = z o + z l + . . . + z,-,.Since, by Corollary 10.49, every countable scattered linear ordering is a finite sum of elements of X , the image off includes Y(cr).Moreover, if two elements ~ equivalent, the same is true of their images. Hence Y ( a ) is of X ( U ) ' are small for each c( and the proof is complete. REFERENCES Fraisse, R.. Abritement entre relations et specialement entres chaines, in Symposia Muthrn~utica,Vol. V, New York: Academic Press, 1971, pp. 203-251. [ M R 43, # 1 lo] Fraisse, R., Sur la cornparaison des types d'ordres, C. R. Acad. Sci. Paris, SPr. A 226 (1948). 1330-1331. [ M R 10, p. 5171 Calvin, F.. and Prikry, K . , Bore1 sets and Ramsey's theorem, J . Svnibolic Logic 38 (1973), 193- 198. Hagendorf. J. G., Extensions de chaines, Thesis, Marseilles: U. Aix-Marseilles. 1975. Hagendorf. J. G.. Extensions immediates de chaines et de relations, C. R . Acad. Sci. Paris 274 (1972), A607-A609. [ M R 48,1/5927] Hagendorf, J. G., Extensions immediates et respectueuses de chaines et des relations, C. R . Acad. Sci. Paris 275 (1972). A949-A950. [ M R 41, #4797] Hagendorf, J . G., Extensions immidiates respectueuses de chaines, C. R . Acad. Sci. Puris 275 (1972). A1273-Al275. [ M R 47, 447983 Higman. G., Ordering by divisibility in abstract algebras, Proc. London Math. SOC.2 (1952), 326-336. [ M R 14, p. 2381 Jenkyns, T. A., and Nash-Williams, C St. J. A., Counterexamples in the theory of wellquasi-ordered sets, in Proof Trchniqurs in Graph Theory, New York : Academic Press, 1969, pp. 87-91. [ M R 40, 1571.561 Jullien. P., Contribution a I'etude des types d'ordres dispersees, Thesis, Marseilles; U. Aix-Marseilles, 1969. Kruskal, J. B., The theory of well-quasi-ordering : A frequently discovered concept, J . Comhinatovial Theory ( A ) 13 (1972). 297-305. Landraitis, C., A combinatorial property of the homomorphism relation between countable order types, J . Symbolic Logic 44 (1979), 403-41 1. [ M R 80h: 040011 Larson. J., A solution for scattered order types of a problem of Hagendorf, Pacific J. Math. 74 (1978). 373-379. [ M R 58. #5422] Laver, R., An order type decomposition theorem, Ann. of Math. 98 (1973), 96-119. [ M R 47. #8361] Laver, R., Better-quasi-orderings and ii class of trees, in Studies in foundations and comhinatorics: Advances in Marhematics. Supplementary Series 1 (l978), pp. 31 -48. New York: Academic Press, 1978. Laver, R., On Frai'sse's order type conjecture, Annals of Math. 93 (1971), 89-111. [ M R 43. k47311

204 10. EMBEDDINGS OF LINEAR ORDERINGS AND FRASSSS CONJECTURE ~ 7 1Laver, R., Well-quasi-orderings and sets of finite sequences, Proc. Cumb. Phil. SOC.79 (1976), 1-10, 1181 Nash-Williams, C. St. J. A., A survey of the theory of well-quasi-ordered sets, in Combinatorial Structures atid Their Applications, Colgary International Conference 1969. London: Gordon and Breach, 1970. Nash-Williams, C. St. J. A . , On well-quasi-ordering finite trees, Proc. Camb. Phil. SOC. 59 (1963), 833-835. [ M R 27,k35641 Nash-Williams, C. St. J. A., On well-quasi-ordering infinite trees, Proc. Camb. Phil. Soc. 61 (1965), 697-720. [ M R 31, #90] Nash-Williams, C. St. J. A., On well-quasi-ordering transfinite sequences, Proc. Cumb. Phil. Soc. 61 (1965). 33-39. [ M R 30.#3850] Rado, R., Partial well-ordering of sets of vectors, Marhematiku 1(1954), 89-95. [ M R 16, p. 5761 Rotman, B., On countable indecomposable order types, J. London Marl7. Soc. 2 (1970). 33-39. [ M R 40,#5504] Sierpinski. W., L ~ ~ JsurR les S nombres rransfinis, Paris: Gauthier-Villars, 1928.

CHAPTER 11 PARTITION THEOREMS

If “I2 is partitioned into two sets, then, by Ramsey’s Theorem, there is an infinite subset Y C_ N which is homogeneous for the partition. Suppose that the set N is given additional structure, can one conclude anything about the structure inherited by such a homogeneous set? Thus, for example, if we consider N as a linearly ordered set of order type o,then we can conclude that there is a homogeneous subset which has the same order type. If, on the other hand, we construe N as a linear ordering of order type w’ and partition [ N ] ’ into two sets, Ramsey’s Theorem does not necessarily yield a homogeneous subset of the same order type; does such a homogeneous subset exist? (For the answer to this question see Exercise 11.2.2.) The answers to such questions are known as partition theorems and were first systematically investigated by Erdos and Rado [29] in the early 1950s. Since then a substantial body of literature on partition theorems has evolved, much of which is discussed in the recent book by Erdos, Hajnal, Matt, and Rado [19]. In this chapter we will present a few tidbits from that literature, concentrating on results about the order type of the homogeneous set. At the end of the chapter, we will say more about that literature and present a rather extensive bibliography. (Many items in the bibliography do not deal particularly with the order type of the homogeneous set; as a result, many items are not referred to in the text itself.) We begin with a discussion of which other order types have the property ascribed to o in the fist paragraph. DEFINITION 11.1 : A standard partition of A is a partition of [A]’ into two parts. A linear ordering A (or an order type z) is said to be a Ramsey ordering if, given any standard partition of A, there is a homogeneous subset for the partition which has the same order type as A (or z).

EXAMPLES: (1) w and o*are Ramsey order types. (2) L is not a Ramsey ordering; for, if we define a standard partition [Z]’ = B u C where {x,y } E B if and only if x 2 0 and y 2 0, then clearly any infinite homogeneous set Y has order type o if [ Y]’ c_ B and has order type o*if [ Y I 2 L C . 205

206

1 1.

PARTITION THEOREMS

Thus a standard partition of 2 may not have a homogeneous subset of order type (. It follows from Exercise 11.2.1, however, that any standard partition of Z does have homogeneous subsets of order type w and w * . EXERCISE 11.2: (1) Show that if A is any linear ordering such that 5 A (respectively, w* 5 A ) ,then any standard partition of A has a homogeneous subset of order type o (respectively, w*). [Hint: Apply Ramsey’s Theorem.] (2) Construct a standard partition of o o such that every infinite homogeneous set has order type 0. [Proposition 11.3 generalizes the result of this exercise, so that given any countable ordinal a, there is a standard partition of tl such that every infinite homogeneous set has order type w.]

w

-

As we will now see, Ramsey order types are quite rare, PROPOSITION 11.3: Let A be an injnite linear ordering of cardinality K . Then there is a standard partition of A such that every set Y which is homogrneous for this partition and which has cardinality K has order type K or ti*. Proof : (This is essentially Example 4B of [29].) Enumerate A = {aAl2 < K } . Partition [ A ] , by putting {a,,as} into B if a, < A as and tl into C if a, < A up and M > j.Suppose that Y is homogeneous for this partition and has cardinality K ; let Y = {aii 11 < K > where { i A / R< ti) is an increasing K-sequence of ordinals less than K. If [Y]’ G B, then aim< A aipif and only if c1 < p, so that Y has order type ti; if [ Y]’ c C , then a,* < A a,, if and only if CI > p, so that Y has order type K * . COROLLARY 11.4: The only countable Ramsey order types are w und a*. The only possible uncountable Ramsey order types are uncountable cardinals K and their reverses ti*.

It is clear that for any infinite cardinal ti, K is a Ramsey linear ordering if and only if every standard partition of K has a homogeneous subset Y of cardinality K (without regard to the ordering of Y). Any such cardinal K must be strongly inaccessible; that is, K cannot be written as a union of fewer than K sets each of cardinality less than K and also 2’ < ti whenever R < ti. (The verification of this fact is the purpose of Exercise 11.5.)The second condition excludes every successor cardinal; the first condition excludes K,, since K, can be written as KO u K, u K, u . . . ,and similarly it excludes K, for every countable limit ordinal a. The next candidate K,, is also excluded since it can be written as the union of K , cardinals each smaller than K,, .

1 1.

PARTITION THEOREMS

207

The search for strohgly inaccessible cardinals thus quickly leads to the stratosphere and, indeed, it is consistent with ZFC (Zermelo-Fraenkel set theory with choice) to assume that no strongly inaccessible cardinals existand thus that there are no uncountable Ramsey order types. [The reader should be cautioned that, in the literature, the term Ramsey cardinal is reserved for an even stronger property than what we have called a Ramsey order type.] In any case, the search for Ramsey ordering is beyond the scope of this book. EXERCISE 11.5: (1) Assume ti can be written as a union u ( A x l a< A] of fewer than K sets each of cardinality less than K . Construct a standard partition of K which has no homogeneous subset of cardinality K . (2) Assume that A < K and that 2A 2 K . Construct a standard partition of IC which has no homogeneous subset of cardinality K . [Hint : Identify subsets of J. with A-termed sequences of 0 and 1 and linearly order the subsets by first differences. Then proceed as in Exercise 7.5.2.1

Given a countable order type T different from o and o*, we know that there are standard partitions of t for which no homogeneous set has order type t. Are there any conditions on the partition which will guarantee the existence of a homogeneous set of order type z ? Theorem 11.7 gives a sufficient condition for a standard partition of q to have a homogeneous set of order type q. DEFINITION 11.6: If [A]’ = D u E is a standard partition of A and X is homogeneous for this partition, we say that X is a D-homogeneous set if [XI’ G D, and we say that X is an E-homogeneous set if [XI’ G E. THEOREM 11.7: (Erdos and Rado) Let [Ql2 = D u E be a standard partition of Q such that there is no injinite D-homogeneous set. Then there is an E-homogeneous set Y of order type q. Proof : We say that a subset I/ c Q is somewhere dense if V n I is dense in I for some interval I of Q . Suppose that given any somewhere dense set V there is an r E V such that { s E V1 {r,s} E D } is somewhere dense. Then, starting with V, = Q, we find an r, E V, so that V, = {s E V,l {r,,s} E D } is somewhere dense. Continuing inductively, assume that we have defined V,, V,, . . . , V , and r , , r , , . . . , r n - so that each r i E q and each q + ,= 1s E (ri, s} E D)is somewhere dense; we then select r , E V, so that V,, = (s E V ,I [ r n ,s} E D) is somewhere dense. Thus ( r , I n < co} would be an infinite D-homogeneous subset X E Q, contrary to hypothesis.

I

,

208

1 1.

PARTITION THEOREMS

Hence there is a somewhere dense subset V such that { s E V1tr.s) E D} is somewhere dense for no r E V. Choose I so that V n I is dense in I . Then given any r l , r 2 , . . . , rk E V and any subinterval I' E I , since each { S E V l { r i , s }E D}isnotdenseinI',neitheris U!=, {SE V l { r i , s }~ D J ; h e n c e there is a subinterval I" c I' such that if r E V n I", then { r i , r } E E for i = 1,2, . . . ,k. (Note: I/ n I" is dense in I" since I" is a subinterval of I . ) We complete the proof as follows. Choose an element r l of V in I , then choose two elements r 2 , r3 of V in the two subintervals of I , then choose four elements r 4 , r 5 , r 6 , r7 of V in the four subintervals of I , . . . , choose 2" . . . , rzn+ - of I/ in the 2" intervals formed, . . . , all the elements r Z nr2n+ , while taking care that ( r i , r m }E E for all i < rn. Then Y = {r,ln < 01 is an E-homogeneous set of order type q.

EXERCISE 11.8 : (1) Let [Q]' = D u E be a standard partition of Q, and assume that there is no homogeneous subset of order type q. Show that there are two homogeneous sets X and Y, one of order type o and the other of such that [XI' E D and [Y]' E E. [Note: In Exercise 11.2.1, order type o*, we did not conclude that [XI' and [ Y]' were included in different parts of the partition.] (2) Is it possible to partition [QI2 = D u E so that there is an Ehomogeneous set Y of order type q and so that there is an infinite D-homogeneous set X, but every D-homogeneous set is scattered? Theorem 11.7 gives a positive answer to our question only for z = q, but, as is easily seen, it is true for any countable z satisfying q 5 t. Before stating this as a corollary, we introduce some useful compact notation. DEFINITION 11.9: Given order types CJ and z we write cr + (No,z) if, given any standard partition [S]' = D u E of a linear ordering S of order type CJ,there is either an infinite D-homogeneous set or there is an E-homogeneous set of order type T. We write 0 -tr (KO, 5 ) otherwise.

COROLLARY 11.10 : I f q 5 T and z is countable, then z

+

(No,t).

A

For which countable order types is it true that z + ( K Oz)? , By Corollary 11.10,we have a positive answer unless z is scattered. By our earlier remarks, we have a positive answer if z is o or o*.Are there any other countable scattered t for which t + (No,T)? Let us first consider t = o + n, where n > 0. If we partition [TI' = D u E by putting a pair into D if and only if it has one element from o and one from n, then it is clear that there are no D-homogeneous sets with more than

1 1.

209

PARTITION THEOREMS

two elements and that any infinite E-homogeneous set must be entirely contained within w ; hence, not only is o n .ft ( K , , o n), but even o n -H (3, o 1). We are here extending the -+ notation according to the following definition.

+

+

+

+

DEFINITION 11.9: (Continued) Given order types or cardinal numbers a, and T , we write (T + (a,T ) if, given any standard partition [S]’ = D u E of a linear ordering S of order type (or cardinality) (T, either there is a Dhomogeneous set of order type (or cardinality) a or there is an E-homogeneous set of order type (or cardinality) T . We write r~ .ft (a,z) otherwise. (T,

The arrow symbolism was introduced to simplify the statements of a variety of results obtained by Erdos, Hajnal, Rado, and others concerning homogeneous sets for partitions. For example, using arrow notation, Ramsey’s Theorem is K, + ( K OKO), , Sierpinski’s example in Exercise 7.5.2 shows that 2’O -H (K,, K,),and the result of Exercise 11.5 is that if K + (K, K ) , then K is strongly inaccessible. EXERCISE 11.11 : Show that A f* (3,A), and therefore that the hypothesis that z be countable in necessary in Corollary 11.10.

To prove that T -H (K,,z)for any scattered z except o and o*, Erdos and Hajnal [ 141 proved Hausdorff’s Theorem again (independently) and were thereby able to argue by induction on the +rank ofscattered linear orderings. THEOREM 11.12: If T is a countable scattered order type diferent from w and o*, then t f* (KO, 7). Proof: This theorem is proved by induction on the eF-rank of scattered linear orderings. Actually a stronger result is proved, by induction on rank; namely, that for any countable scattered order type z, there is a standard partition [TI’ = D u E of a linear ordering T of order type T with the following properties :

(i) There is no infinite D-homogeneous set. (ii) Given any two infinite subsets R and S of T such that R < S (i.e., x E Randy E Simplyx < y), thereisanx E R a n d a y E Ssuch that (x,y) E D. We will see later that this implies the desired result. Suppose then that we have proved that all scattered linear orderings of rank less than CL have the above property and let T = To + TI + T , . . . be an o-sum of linear orderings of rank less than a. (A symmetric proof works

+

210

1 1.

PARTITION THEORElMS

for o*-sums; a simpler version works for finite sums. A (-sum can be treated as an o-sum plus an w*-sum.) Then, by the induction hypothesis, for each Tithere is a standard partition [TiI2= Di u Eiwith the required properties. Enumerate each Ti = {tni n < mi}where mi is finite or w. Define a standard partition [TI' = D u E as follows:

I

{tni,t,'}

{ tni,t , j )

D ED

E

if and only if where i < j

{tni,t,'} E D i ;

if and only if n > m.

Suppose now that R is a D-homogeneous set. Then R n Tiis a Di-homogeneous subset of Tiand hence must be finite. Suppose that t,' E R n Tiand that r,j E R n T jwhere i < j ; then, since R is D-homogeneous, m < n. Hence there are only a finite number of i for which R n Ti is non-empty, and therefore R is finite. Thus there can be no infinite D-homogeneous set. Also, if R and S are infinite subsets of T for which R < S, then there is a largest i, denoted i,, such that R n Ti is infinite. If io < j and some t,J E S n T j , then, since R n Ti,is infinite, we can choose n > m so that :r E R n Ti,; but then { t?, t , j } E D, as desired. Otherwise, S n Ti= for a l l j > i o , so that S c Ti,; but then R n Ti, and S are infinite subsets of Ti, satisfying R n Ti,< S, so that, by the induction hypothesis, there are elements x E R n Ti,and y E S for which {x, y } E Di c D. Thus the given partition also has the second property. Suppose now that z is an order type which can be written as z1 z2 where z1 and z2 are infinite. Construct the standard partition of a linear ordering T of order type t as above. Then, by (i), there is no infinite Dhomogeneous set, and, by (ii), there is no E-homogeneous set of order type z 1 + z 2 . Hence z * (K,,t). If, on the other hand, t cannot be decomposed into two infinite intervals, but then. by remarks then t must be finite, or of order type o + n or n + o*; preceding this theorem, t ft ( K O t) , unless 5 is o or o*.

+

We note that it is possible to come to a stronger conclusion at the end of the proof above. Indeed, it shows that for any countable scattered order type z, there is a standard partition [TI2 = D u E of a linear ordering T of order type z such that

(i) there is no infinite D-homogeneous set, and (ii) there is no E-homogeneous set whose order type c can be written as crl + cr2 with both crl and o2 infinite. Thus, for a countable infinite scattered order type z, the only conclusions of the form t -+ (KO, cr) which could possibly be correct are those for which cr is finite (always correct) or c is either o + n or n + o*. These possibilities are exhaustively considered in the following theorem.

11.

21 1

PARTITION THEOREMS

THEOREM 11.13: (Erdos and Hajnal) Let z and a be countable injinite a) holds precisely when z and (T satisfy one scattered order types. Then z + (KO, of the following conditions:

+

(a) a = o n for some n > 0 and w . w* 5 s; (b) a = n + o*for some n > 0 a n d o * ‘ o5 z; (c) a = w* and w* 5 7; (d) (T = o and o z.

zk and { z o , zl,. . . , zk} is D-homogeneous, we will define zk+1. Since, for each i I k, the set { y < ziI { y, zi}E E ) does not have order type w . o*, it easily follows that there is a z k + l < zk such that {zkil,zi} E D for all i Ik. The resulting set Y is clearly an infinite D-homogeneous set. This completes the verification that z + (KO, w + n) whenever o .a* Iz. We now prove the converse by showing that if w . o* 6 z, then z * (KO, o n) for any n > 0. We first observe, by Exercise 5.33.2, that T must be an ordinal sum of backwards ordinals; that is, a linear ordering T of order

+

+

+

212

1 1. PARTITION THEOREMS

type t can be written as c { T p *Ip < x j where Ta is a well-ordering for every p. Let {p(n)ln < o}be an enumeration of all ordinals less than the countable ordinal a. We define a standard partition [ T I 2 = D u E of T by stipulating that {x, y } E D, for x < y, if and only if x E T&), y E T&,, and m < n. Suppose now that Y is a D-homogeneous set. Then if X E Y and X E T,*,,,, any y E Y satisfying x < y must be in T&,) for some m < n. Hence, since no more than one element of each T&,, can be in a D-homogeneous set, x has only finitely many successors in Y. Since a is an ordinal, Y is also well-ordered, and so must be finite. Thus there is no infinite D-homogeneous set. Suppose next that Y is an E-homogeneous set and that w 1 5 Y. Then there are elements yo < y , < y , < . . . < y such that y E Y and each yi E Y . Suppose that y E T&. Then, since each { yi,y } E E, each y i E T,*,,, for some m I n. This implies that infinitely many yi are in some T&(,)for m In, and hence that o 5 T&), which is a contradiction. Hence there is no E-homogeneous set of order type o + 1. This completes the proof that t f* (No, w n) for any n > O if w . o* -$ T. H

+

+

COROLLARY 11.14: Let T be a linear ordering of order type T where o . w* 5 z (respectiuely,to* . w 5 5). Let [TI’ = D u E be a standard partition of T and suppose that for some n > 0 there is no homogeneous set of order type w n (respectively, n o*).Then there are both injnite Dhomogeneous sets and injnite E-homogeneous sets. A

+

+

EXERCISE 11.15: Determine whether the proofs of Theorems 11.12 and 11.13 remain correct if the assumption that z is countable is dropped.

Having shown that z -+ (K,,t)is rarely true, we turn to the proposition (m,z)and ask for which z this is true, when m is finite. This question was first raised by Erdos and Rado [29] and subsequently was studied by Specker [86]. We present here the answers, due to Specker, for the cases t = w 2 (positive) and t = w 3 (negative); the proof we give for z = 0)’ is basically that of Haddad and Sabbagh [41].

z

--t

THEOREM 11.16: w2 -+ (rn,w2)forevery m < a. Proof: Let [ T I 2 = D u E be a standard partition of a linear ordering

T of order type w2. We will assume, with no loss of generality, that T = {(a,b) 1 a < b < w } , ordered lexicographically. We first homogenize T so that two “similar” unordered pairs of elements of Tare either both in D or both in E ; for example, we wish to arrange matters

11. PARTITION THEOREMS

213

so that if {(a,b),(c,d ) ) and {(a',b' I, (c',d')} are similar in that both a < c < b 1, LteJine @, axiomatization of on.Moreover, qd(@,,)= 2n + 1.

Then @,, is an

=

Proof: By induction on n, as in the argument for n = 2. Moreover, since qd(@,) = 3 and qd(O) = 2, we can show by induction on n, using the fact that qd(de) = qd(4) qd(O), that qd(@,,) = 2n + 1 for all n. W

+

Note that the quantifier depth of the axiomatization a,,of W" is precisely

f ( d )as , predicted by Proposition 13.25, so that these axiomatizations are

optimal. . u2 + . . . + w " ~. a k , Turning now to the general case c( = w"' . a, + on2 we see that we want to axiomatize CY by saying that there are certain points such that the intervals between them are wk-like;for this we need to relativize to the formula ~(w,u,,u2) that is v 1 I w < u 2 . We thus define @(a) to be ( 3 x 1 ')

' ' '

(jXi,)(3X12)

A(X1'

' ' '

(jX:,)

' ' '

5 y)

(3Xik){(Vy)(X,' '

. . A @ e ( w . d 1 . x l 2 )A ae(w,x,2,x22)

nl

. . .A

(]Xik)

< " ' < xi, < X1' < . . . < XZ2 < . . * < X l k < . . < X i k )

A @e(w,x,',Xl2) A . A

' ' '

n1 @ ~ ( W , X , ~ . X ~ ~ )

nk

A"

A,

. .A ae(w%x:2.x,3) nz

"2

.A ( D o ( ~ ~ ~ : k _ A @e'(wdk) nk

nk

1,

where B'(w, vl) is u I I w. The following is then easily verified. THEOREM 13.35 : For each ordinal a < ww,@(a)is an axiomatization of

CY.

A

We conclude this section with several theorems of a technical nature specific to linear orderings that result from the equivalence between logical equivalence and G-equivalence. These theorems will be used in $813.4 and 13.5. Recall that for every k and n, Lk,,,consists of all formulas of L whose free variables are among u l , . . . , uk and whose quantifier depth is at most n. We showed that there is a finite subset @k,,, of Lk,,, such that every formula

2.

GAMES AND THE FIRST-ORDER THEORY OF LINEAR ORDERINGS

263

of Lk,,is logically equivalent to a formula of @ k , , ; thus if 4 E a,,,, then there 4. Now let A be a structure, let is a II/ E @ k , , that is logically equivalent to i a,, . . . ,ak E A , and let @ = @(A,a , , . . . ,a,) be the conjunction of all 4 E @ k , n such that A 1 b[a,, . . . ,a,]. Then A 1 @[a,, . . . ,a,] and for any B and b , , . . . ,b, E B we have (A,a,, . . . ,ak) =,,, (B, b , , . . . ,bk) if and Only if B b @ [ b , , . . . ,b,]. We can thus find a finite set {Qi(ul, . . . ,u,)l1 I i I t } of formulas of Lk,, that completely describe all of the =,,,-equivalence classes. That is, for each structure A and elements a,, . . . , ak E A, there is a unique such that A 1 Qi[a,, . . . ,a,] and, moreover,

B k a i [ b , , . . . , bk]

if and Only if

(B,b , , . . . ,bk) - k , n (A,al, . . . ,a,).

For k = 0 this says that there is for each n a finite set {Oil1 I i I t} of statements of quantifier depth at most n that describe all =,-equivalence classes of linear orderings. That is, for each structure A, there is a unique misuch that A 1 mi and, moreover,

B 1 ai

if and only if

B =,A.

This much we can say about arbitrary structures, not just linear orderings; with linear orderings we can carry this analysis much further. Note first of all that each Q imust specify the ordering of the variables. We will work with the case where @ = mi logically implies that u , < u2 < . . . < u,; any permutation can be dealt with similarly. Let A k @ [ a , , . . . , a k ] , so that, in particular, a , < a2 < . . . < a,. Let A , , , 4 , , . . . ,A,- ,, A , be the intervals (t, a,], [a,, a,], . . . , [a,- ,,a,], [a,, +) of A . By the discussion above, we can find statements Y o , 'PI,. . . ,y k that completely describe the =,-equivalence j I k - 1, let YF be the relativizaclasses of A , , A , , . . . , A , . For each j . 1 I tion of Y jto e(w, u j , ujt = ( u j I w Iu j + ,), so that B k Y F [ b j ,b j +, ] if and only if [ b j ,b j +,] 1 Y j If and only if [ b j ,b j +, ] =, [ a j , a j + , ] . Let Y o Abe the u,) and let Y / be the relativization of relativization of Y oto O(w, uL) = (w I yk to o ( w , u k ) = (0, 5 w). Let y ( u 1 , . , . ,u k ) be the formula (01 < u 2 < ' ' ' < u k ) A YoA(u1) A YIA(ul, u,) A . . . A Y f - .1(uk- u,) A y t ( u k ) . It is easily verified that Y(u,,. . . ,u k ) characterizes the =,,,-equivalence class of @,a,, . . . ,a,); that is, A k Y [al' . . . ,ak] and, for any B and b , , . . . ,b, E B,B 1 Y [ b , , . . . ,b,] if and only if (B,b , , . . . ,b,) E k , , (.4,a,, . . . ,a,). Furthermore, since each Y j E Lk,nand relativization to quantifier-free formulas does not increase quantifier rank, we conclude that Y E Lk,,. We summarize this discussion in the following lemma.

,,

LEMMA 13.36: Let A be a linear ordering and let a , < a, < . ' ' < ak be elements of A . Then there is a ,formula Y(u,, . . . ,u k ) E L,,, such that A k Y [a . . . ,a,] and such that B 1 Y [ b , , . . . , bk] if and OnlJ) if (B,b , , . . . ,bk)-kk,,(A,al,. . . ,Uk).

13.

264

THE FIRST-ORDER THEORY OF LINEAR ORDERINGS

Furthermore, Y can be chosen to be a conjunction of v1 < vz < . . ' < v k with formulas Y ~ ~ ( U Y~ l A)( v, l rvz), . . . ,Yt- ,(uk- 1,ok)r y,"(uk) so that for any linear ordering B and elements b,, . . . , b, E B we have B

YoA[b,]

if andonly if if and only if

(+,b,] =n(+,u,], [bk, +)

B 1 Y',"[bk] and fbr each j , I 5 j 5 k - 1, B =! Y P [ b , , b j + l ]

ifandonly if

=,, [a,,

+),

[ b j , b j + J =, [ a j , a j + ] ] .

This lemma is easily modified if the elements a l , a z , . . . ,ak are in some < ' ' < an(k)for order other than a, < u2 < ' . . glu E A } , where b_ is a new constant symbol. Since A has no last element, C has a model D by the Compactness Theorem and A < D by Theorem 13.57. We will assume, without loss of generality, that A is actually contained in D. Let D' = (d E Dld < a for some a E A } , so that D = D' B, where B is non-empty. To show that A < A + B, it suffices, by Lemma 13.55, to show that A B i D' B, and for this it suffices to show that A < D' by Exercise 13.53.2. Again using Lemma 13.55, to show that A < D', it suffices to show that D' < D' B, and that is what we now prove. Assume that D' B k + [ d , , . . . ,d,, b], where d,, . . . ,d, E D' and b E B. Choose a E A greater than d,, . . . ,d,. By Theorem 13.41, there is a selecting formula +#(x,u) for in the interval [a, +) of D' + B ; that is, for every d E D' B, we have D' B b +"[a,d] if and only if

+

+

+ +

+

+

+

+

d2a

and

D'

+ B b +[d,,

. . . ,d,,d].

278

13.


+

+ +

Thus since D' B t= $ [ d , , . . . , d , , b ] and b 2 a, we have D' B k $#[a,b]. But since A < D' Band a E A, there is an a' E A such that D' B k $ " [ a , a']. This implies that we have found an element a' E D' such that

+

D

Hence D'

+ B b $[d,,

. . . ,d,,a'].

< D' + B. Thus we have shown that A

i A

+ B.

Another important application of the method of complete diagrams is the following general result. THEOREM 13.59: Let A 7hen each A, < A.

=

U{A,In < o}, where A,

< A,

i A,

< ' .. .

Proof: Let C = C(A,) u C(A,) u C(A,) u . . . . Then Z has a model, by the Compactness Theorem, since any finite subset of C is completely contained within some C(A,) and hence has that A, as a model. Let B k C. Then, by Theorem 13.57,A, < B for each k . Furthermore, since A = {An[n < w}, A !E B. By Lemma 13.55, in order to show that A, i A for each k, it suffices to show that A < B. Assume that B k $ [ a , , . . . ,a,, b], where a , , . . . ,u, E A and b E B. Choose m so that a,, . . . ,a,€ A , . Since A, i B, there is an a E A , such that B k $[al, . . . ,a,, a] ; thus we have found an a E A such that B k $[a,, . . . ,a,,a], so that we can conclude that A < B and hence that A, < A for each k. 4

u

In particular, if each Ai is a model of the theory T, then A is also a model of T ; this enables us to build new models of T from old ones. If the chain {A,ln < o}is not an elementary chain, then u{A,,In < o} need not be a model of T even if each Ai is a model of T . For example, if A , = (o+ o*) .n for each n (it being understood that if n < rn, then A, consists of the first n copies of o + o* in A,), then u(A,ln<w} = ( o + w * ) . w = o + [ . o = w ,

a1though A,

+ o * ) . n = o + 6 . ( n - 1 ) + o*= o + w*

= (o

for each n. Thus if we wish to put together models of a theory T and get new models of T, it helps if suborderings are elementary suborderings. DEFINITION 13.60: Given a complete theory T, we say that a model A of T is a prime modelof T if A < B for every model B of T.

4.

279

MODEL THEORY AND LINEAR ORDERINGS

+ + + +

For example, w is a prime model of Th(w); for given any model o 5 . ct of Th(o), the map f :w + w 5 . u that maps o onto the initial w of w 5. ct is an elementary map, as is clear from a game analysis. Similarly, w w* is a prime model of its theory; for given any model w 5 . ct + o*ofTh(w a*), the map , f : o w* + o + 5 . (x + to* that maps w onto the initial o,and o* onto the terminal w*, of o 5 . (x + o* is an elementary map. If T is an arbitrary complete theory, then T need not have a prime model (see, for example, [2]), but what if T is a complete theory of linear orderings? We will return to this question later.

+

+

+

+

+

+

EXERCISE 13.61 : Let o 5 . CI and w 5 . p be models of Th(w). Show that w + [ . ct < w [ . p ifand only if ct 5 p.

+

We now introduce the notion of n-types, first discussed systematically in Vaught [22], who also provides many historical remarks on the material treated in this section. DEFINITION 13.62: Let A be a structure and let ( a l , . . . ,u,) be an ntuple of elements of A . By the n-type of ( a l , . . . ,an) we mean the set of all formulas 4 ( v l , . . . ,v,) of L such that A I= 4[ul, . . . ,a,]. We note that the n-type P of (a [ , . . . ,a,) is a consistent subset of F,(L), the set of all formulas whose free variables are among v l , . . . ,u,, and, moreover, that it is maximal among consistent subsets of F,(L) since adding any formula 4 to P would result in an inconsistency because, if 4 is not already in P, then (14) E P. (As in Chapter 12, a set Q, of formulas is consistent if no contradiction Ic/ A i ) I is deducible from Q,.) We now turn the definition around, with this observation as guide. DEFINITION 13.63: Let T be a complete theory. An n-type of T is a maximal consistent subset of F,,(L) that includes T. This definition is justified by the theorem below. THEOREM 13.64: I f P is an n-type of T then there is a countable model A of T and an n-tuple (al, . . . ,a,,> of elements of A whose n-type is P.

280

13.


Since C is consistent, it has a countable model A in which there are elements a, . . . . ,a, such that (a,, . . . ,a,) has n-type P. Note that A b T since if 4 is a statement in T, then 4 E P , so that 4 E X. H DEFINITION 13.65: Given a complete theory T, an n-type P of T , and a model A of T, we say that P is realized in A if there is an n-tuple of elements of A whose n-type is P. Otherwise we say that P is omitted in A. For example, in o each element has a different 1-type since the nth element, and only the nth element, satisfies a formula 4,(v1) that says that u , has exactly n predecessors. No element of o [ after the initial o has any of these types since any such element satisfies i$,,(vl) for every n. Do all such elements have the same 1-type? Does Th(o) have any other 1-types'? These kinds of questions are naturally discussed by connecting n-types with notions discussed earlier.

+

PROPOSITION 13.66: (1) Suppose that f is an automorphism of A such thatf(a,) = bi for 1 I iI n. Then (a,, . . . ,a,) and (b,, . . . ,b,) have the same n-type. (2) Let ( al , . . . ,a,) be an n-tuple of elements of A and ( b l , . . . ,b,) be an n-tuple of elements of B. Then the following are equivalent: (a) (al, . . . ,a,) and (b,, . . . ,h,) have the same n-type; (b) (A, ~ 1 , ... , a,) (B, b1,. . . , b,); (c) PLAYER II has the winning strategy in G,( (A, a,, . . . ,a,), (B, b,, . . . ,b,)) j o r every m. A EXERCISE 13.67 : (1) Prove Proposition 13.66. (2) Show that all elements of o 5 after the initial o have the same 1-type. (3) Show that Th(o)has no other 1-types. (4) Show that if P' is a consistent subset of F,(L), then P' s P for some n-type P of T.

+

Given a theory T , to determine all n-types of T we need only look at all n-tuples of elements in all (countable) models of T, not a small task. The following definition will simplify matters. DEFINITION 13.68: Let P be an n-type of T and let P' G P. We say that P' generates P if for each o! E P there are formulas o!,, . . . ,clk of P' such that

T k ( V v 1 ) . . . (b'o,,)[Crl(vl, . . . ,v,)

A

'

. . A a,(v,, . . . ,v,)

+ o!(vl.

. . . ,v,)].

4. MODEL THEORY AND

LINEAR ORDERINGS

28 1

Equivalently, P' generates P if whenever an n-tuple of elements of a model of T satisfies just the formulas in P', then its n-type is automatically P. If P' generates an n-type, then we may as well call P' an n-type. Thus the l-types ofTh(o) are { {4,(uo)} In < a}and ( i 4 , ( v o ) I n < w } .

EXERCISE 13.69: (1) Assume that P' G F,(L), that A 1 T , and A b $[al, . . . ,a,] for every $ E I".Assume also that whenever B b $[b,, . . . ,b,] for every $ E P', ( b l , . . . ,b,) has the same n-type as ( u l , . . . ,uJ. Show that P' generates an n-type of T . (2) Assume that P' G F,(L), A 1 T , and for some a l , . . . ,a,E A, A 1 $ [ a , , . . . ,a,]for every $ E P'. Assume also that for any two n-tuples of.elements of A that satisfy P' there is an automorphism of A mapping one n-tuple to the other. Show that if either P' is finite or T has, up to isomorphism, a unique countable model, then P' generates an n-type of T . (3) Show 'that if T = Th(q). then the n-type of ( u l , . . . , a,) is { u , < U J U , < u,} u { U L = V j l U * = L l J . We determined above all l-types of Th(w). Let us now describe the 2-types of Th(o). For each n and m, 6,,(ul) A $ m ( ~ 2 )is a 2-type P,,m;for each n, (4,(u1)} u { i 4 m ( u z ) I m< w', isa2-type P,,,;foreachm, { i 4 , , ( u l ) l n < w } u {$ m( u 2 ) } is a 2-type Pm,m.Every other 2-type contains ( i 4 , ( u l ) I n < w } u {i$m(~lz)I< m w } = P , , , but there are many distinct 2-types containing P,, . Thus if E,(x,y)says that x < y and there are exactly t elements such that x < t I y, then for each t 2 1, P,+ = P,, u { Et( u 1 , u 2 ) } and P I - = P,, ~ {E , ( u , , t~ ~ ) }a r e 2 - ty p e s . A ls=oP,, P~ u ( u l = u2)isa2-type. Finally,

p,,, u Ch f %> u ( l E , ( u l , u 2 ) 1 t < 0)u { l ~ l ( U z , u , ) ( < t 0) is also a 2-type P*,%.That these are in fact all 2-types is easily verified by a game analysis and Proposition 13.66.2. The set S,( T )of n-types of a complete theory T has a topological structure on it that sheds light on the relationship between n-types. A basic open set in the topological space S,(T) is of the form ( P E S,(T)I 4(u1, . . . ,on)E P } for a particular formula 4(ul, . . . ,on)E F,(T). Using this definition, what we now present as a definition is actually a theorem. DEFINITION 13.70: An n-type P of T is said to be principal or isolated, if P is generated by a single formula. Otherwise it is a non-principal or limit n-type.

282

13.


Equivalently, P is a limit n-type if every formula in P also belongs to some other n-type; that is, every basic open set containing P contains other n-types. Note that if P is a principal n-type, say P = {4(v1,. . . ,u,)}, then P must be realized in every model of T for, since T is complete, T I- (3v1) . . . (3vn)4(v,,. . . ,v,).

Thus, if a model of T omits an n-type P , then P must be non-principal. The converse of this is also true; that is, if P is a non-principal type of T , then there is a countable model of T in which P is omitted. This is implied by the following theorem due to Ehrenfeucht (see Vaught [22]).

THEOREM 13.71 : Ler { P j l j < 01) be a countable set of non-principal types of' a complete the0r.y T . Then there is a countable model A of T in whic3z every P , is omitted. Proof : Add to the language of T a countable set {g,ln < w } of new constant symbols; the elements of A will be precisely this set. Enumerate all ordered pairs (a,P), where a is an ordered n-tuple of the new constant symbols, P is one of { P j l j < w } , and P is an n-type. Since there are countably many such ordered pairs, we may assume that they are enumerated as an w-sequence {(ak,Pk>I k < w } . We now define by induction an expanding sequence { T k ( k< o)of theories, each involving a finite number of additional statements and a finite number of the constant symbols (@,In < w } . We define To to be T and we assume, as the induction hypothesis, that the constants appearing in Tk are g i , 1 , , g i ( 2.) ., . We also assume that all formulas of the expanded language that have the one free variable x are enumerated in an w-sequence ( + k ( X ) / k < w } . Now given Tk we proceed as follows to define T k + l .We first add to Tk a statement of form (3x)&(x)+ &(a),where g is a constant symbol appearing neither in T , nor in 4k.X). (Such a statement says that if anything satisfies &, then some & does; this guarantees that A, consisting of {fzk I k < w } , will be a model of T . This method of proof, used in proving the Completeness Theorem, is discussed in detail in Enderton (see [31, Chapter 12). Note that this statement can be added consistently to Tk since otherWise k 1 [ ( 3 X ) b k ( X ) + 4 k ( g ) ] , so that Tk k ( 3 X ) 4 k ( X ) and Tk k 14k(g); since a appears neither in Tk nor in &(X), we can pass from T , t i4,(g) to Tk t ( v x ) i d k ( x ) , contradicting the consistency of T,. We next find a formula $ ( x l , x 2 , . . . ,x,) in Pk (assuming Pk is an n-type) such that i $ ( a k ) is consistent with Tk u { ( h ) + k ( X ) -,&(a)}. If this task is impossible, then Tk

{(3x)4k(x)

$k(!d}

4.

MODEL THEORY A N D LINEAR ORDERINGS

283

for all $ in Pk. Since T , has only a finite number of statements not already in T, we can write this as

T u O(ak,b) t $(ak), where b are the constants appearing in Tk u ((h)&(x) + &(a)) that are different from those in ak.This implies that

T t ()(ak,b) + $(ak), which in turn implies that

T

( 3 Y ) W k , Y)

+

$(ak)

for all $ E Pk. But this says that Pk is a principal n-type generated by

(3y)0(x 1 9 x2 . . . > x k , y), 3

contrary to the hypothesis. We now define Tk+1 to be Tk U { ( j X ) d k ( X ) + 4k(a)}U { 1$(ak),>.By putting i $(ak) into T k +1, we guarantee that the n-type of ak will not be Pk. If, as advertised, A consists of just the constant symbols {@,In < a), then no n-tuple of elements of A will have n-type Pk,so that this Pk,and all the others, will be omitted in A. Now let T* = Tklk < o) and let T** be an arbitrary complete extension of T*. We now define a structure A whose elements are the constant symbols {a,\n < w >; more precisely, we define an equivalence relation by g, -v g , if g, = g , is in T** and WG let A be the set of equivalence classes. For each relation symbol P ( x , , . . . ,xn) in the language of T , we define an n-ary relation R on A by stipulating that R(giil),. . . ,a,(,,)will hold in A if and only if P(giil),. . . , g i c E tis) in T**. It is then easily verified, by induction on formulas, that for every formula $(xl, . . . ,x,) and any elements uiil),. . . ,ai(,)of A , A 1 4[gi(l),. . . ,gi(,)] if and only if ,)&’q . . . ,gi(,)) is in T**. In particular, A 1 T and each of the types {PjIj < o}is omitted in A.

u{

It follows from Theorem 13.71 that if A is a prime model of T, then the n-type of every n-tuple of elements of A must be principal; for if a nonprincipal n-type were realized in .4 and omitted in B, then A certainly could not be embedded elementarily in B since the n-type of an n-tuple of elements of A is the same in B as it is in A if A < B. At this point, it is appropriate to explain where the terms “principal” and “non-principal” come from. Just as the terms “isolated’ and “limit” arise naturally from the topological structure of S,( T ) ,the terms “principal” and “non-principal” arise from the Boolean algebra structure of F,( T ) .That is, if we identify two formulas that are equivalent in the theory T, then

284

13.


F,( T ) becomes a Boolean algebra, denoted B,(T), and an n-type becomes a maximal (dual) ideal in this Boolean algebra. An isolated type, one that is generated by a formula, is a principal maximal (dual) ideal, and a limit type is a non-principal maximal (dual) ideal in this Boolean algebra. A fundamental fact about Boolean algebras is that every infinite Boolean algebra has a non-principal maximal (dual) ideal. Thus, for example, the Boolean algebra consisting of all finite subsets of N and all complements of finite subsets of N has infinitely many principal maximal (dual) ideals, namely, I , = { X G N I n E X} for each n E N , and exactly one non-principal maximal (dual) ideal, namely, the set of complements of finite sets. In the context of model theory, this fundamental fact translates into the statement that if there are infinitely many inequivalent formulas in F,( T ) then there is a non-principal n-type of T. To prove this statement, suppose that there are infinitely many inequivalent formulas in F,(T). It is then easily verified that there are infinitely many n-types. If all of these are principal, then there are countably many of them; call them {P,lm < o}.where each P , is generated by I),(vl, v2, . . . ,v,). Since { i I),(vl, v2, . . . ,v,) I m < o>is consistent, it is contained in an n-type P of T that is different from each P,, contrary to assumption. Thus some n-type of T must be non-principal. The following proposition is now easily verified. PROPOSITION 13.72:

The following are equivalent:

(1) There is an infinite set of formulas of F,(T) no two of which are equivalent in T. (2) B,(T) is infinite. (3) S,(T) is infinire. (4) There is a non-principal n-type of T. A Armed with the topological and algebraic interpretations of the set S,( T ) of n-types of T , we return to our consideration of the n-types of Th(o); we continue using the terminology introduced earlier. The space S,(Th(w)) of 1-types of Th(w) can be pictured as a

...

a

Po PI

p2

a

p,

7

where P, is the 1-type {4,,(v1)).and P, = {i4,,(v1)1n< o}is the limit of {Pnln w>. The space S,(Th(wj j is more complicated. However, each P,,.,, is isolated, each Pn,mis a limit of ( P , , , I rn < m}, each P,,, is a limit of { Pn,mn < o}, each P,+ isalimitof{P,,,+,In < o),andeachP,- isalimitof{P,+,,,Irn < o > .

I

4.


.

285

P,'

. . . PI

I

pz

I

i l l

. . . .... . . . . . . .

...

....

0

...

........

.

p1.0

Po.0

p1.0

-

0

px,z px,l

pz,o

FIGURE1

Thus we have the picture shown in Fig. 1. Omitted in this picture is P:,, which, unlike the other 2-types of Th(o), is a limit of limit 2-types. (It This kind of analysis is also the limit of the isolated 2-types {Pt,2tlt < o}.) of S,(T), similar to the Cantor-Bendixson analysis of closed subsets of the real line (which Cantor used to show that every uncountable closed subset of R has cardinality c), was introduced by Morley [9] and has been a major tool in model theory. What about S,(Th(o))? The following result shows that, in a certain sense, the n-types of any complete theory T of linear orderings are determined by the 2-types of T. THEOREM 13.73: Let P be an n-type of a complete theory T of linear orderings and let P* be all formulas of P that have at most two free variables. Then P* generates P. Proof : If + ( u l , . . . , ~ , ) E Pthen, , by Theorem 13.37, there are formulas Il/ij such that 4 L

~ ( V I .? .

' 7

vn)

* V A Il/ij, i

j

where each t,hij has at most two free variables. Since 4 E P, there must be an i such that Ajt+hijeP.But {Il/ijl j} s P*, so that P* t 4(vl,.. . ,vJ. Hence P* generates P.

286

13.


COROLLARY 13.74: Let T be u complete theory of linear orderings. I f S,( T ) is jinite, then S,( T ) is finite ,for all n. I f S2(T ) is countable, then S,( T ) is countable ,for all n. Proof : Let P be an n-type of T and let P* be as in the proof of Theorem 13.73. For each i, j I n, let P t j be the set of formulas of P* whose free variables are among ui and u j . Then each P t j is a 2-type of T and P* = iPZjI 1 I i Ij I n } . Both of the desired conclusions follow from this.

u

The conclusions of Corollary 13.74 are not true in general. For example, it is shown in Rosenstein [15] that for each n there is a complete theory T , such that S,( T,) is finite but S,( T,) is infinite for all m > n. We asked earlier whether every complete theory T of linear orderings has a prime model. If S 2 ( T )is countable, then by Corollary 13.74, S , ( T ) is countable for all n, so that, by Theorem 13.71, there is a countable model A of T in which no non-principal type is realized. Such a model is prime, as we now show. THEOREM 13.75: Let A be a countable model of a complete theory T in which no non-principal type is realized. Then A is a prime model of T . Proof: Let B k T and let A = { a l r u 2 , a 3 ,.. .) be an enumeration of the elements of A . Since for each n the n-type P, of ( u l , a,, . . . ,a,) is principal, there is a formula +,(ul, u 2 , . . . ,u,) that generates P,. Thus it suffices to find a sequence b , , b 2 ,b 3 , . . . of elements of B such that B 1 + , [ b l , b 2 , . . . ,b,] for each n, for then the map f :A + B defined by f(a,) = b, is an elementary map. Proceeding inductively, we assume that we have defined b,, b,, . . . ,bk SO that B k (f)k[b,,bz,. . . ,hk]. Since A = ! ( b k + l [ U l , U Z , . . . ,U k , U k + I], We have also A 1 (lo,+ I ) & + ,(a,,a,, . . . , a k , ,), so that, since ( b k generates the k-type p k , T 1 + k ( U 1 , . . . u k ) ( 3 t ~ k + l ) ( b k + l ( ~ 1., . . 7 uk? u k + l ) . But then, since B k + k [ b l , .. . ,bk], there is an element b k + lE B such that B b $k+ [ b . . . ,b k ,b k +,I, and so we can continue to embed A elementarily into B. W tik+

9

+

We return now to our earlier question whether every complete theory T of linear orderings has a prime model. It follows from Theorem 13.75 that that is the case if S 2 ( T )is countable. Rubin [17] showed that if S , ( T ) is countable, then S 2 ( T )is also countable, so that in this case too T has a prime model. Thus, we consider next an example of a linear ordering A for which S,(Th(A))is uncountable; this example is motivated by the analogy between Dedekind cuts and 1-types.

4. MODEL THEORY

AND LINEAR ORDERINGS

287

c(

EXERCISE 13.76: Let f : Q + N be a 1-1 correspondence between Q and N and let A have order type fxq) I q E Q}. Show that Th(A) has c l-types and that A is a prime model of Th(A).

This example fails since A is in fact a prime model of Th(A). We are, however, on the right track, as noted by G. Cherlin. THEOREM 13.77: prime model.

There is a linear ordering A such that Th(A) has no

Proof: Partition Q into two subsets Q1 and Q 2 each of which is dense in the other. Let f : Q 1 + N be a 1-1 correspondence between Q1 and N and let A have order type c(g(q)I q E QJ , where g(q) = f(q) if q E Ql and g(q) = 1 if q E Q 2 , Now every Dedekind cut X u Y of Q determines a l-type which asserts that z'l has no immediate predecessor or successor, and that u I is greater than all condensation classes of size f(q) for q E X and less than all condensation classes of size f(y) for q E Y. In every countable model of Th(A), infinitely many of these l-types must be realized, but none of these I-types is realized in every countable model. Thus Th(A) can have no prime model.

In Theorem 13.73 we saw that any n-type of a complete theory of linear orderings is determined by its constituent 2-types. Rubin [17] proved that, under certain circumstances, a 2-type is determined by its constituent l-types. The proof of this result, presented here as Theorem 13.80, involves a complicated induction (Lemma 13.79)and the notion of the k-n-type of a k-tuple. (Theorem 13.80 will be used in $5.) Recall that is the set of all formulas in the variables u 1 , u 2 , . . . ,uk whose quantifier depth is at most n. DEFINITION 13.78 : Let b = (6 (, 6, , . . . ,h k ) be a k-tuple of elements of B. By the k-n-type of b we mean the set of all formulas $ ( u l , u 2 , . . . ,u k ) in Lk., such that B k $ [ h , , b 2 , . . . ,hJ. The k-n-type of b is denoted P,(b). A k-n-typeof B is a maximal set of formulas of I ! , ~ , ~that is consistent withTh (B).

If we let P(b) denote the k-type of b, then P(b) = U{P,(b)In < w } . Note also that since any k-n-type of B is a finite set of formulas, the language being finite, any k-n-type of B is the k-n-type of a k-tuple of elements of B. If I is an interval of B, we let P,(I) denote the set of l-n-types of the elements of I . We introduce several notions which will simplify the statement of Lemma 13.79. A sequence Q1, Q 2 ,. . . ,Q,, of l-n-types of B is said to be realized in

288

13.


B if there are elements c1 < c2 < . . . < c, in B such that each cihas 1-n-type Qi.We let t(n) be the number of 1-n-types of B for each n and we define two sequences s(n) and u(n) by induction: s(0) = 0, u(0) = 0, s(n + I ) = 2 . s ( n ) .(t(n))s(”) 1, u(n + 1) = s(n + 1) + u(n).The formulation of the lemma is complicated because of the requirements of the induction argument.

+

LEMMA 13.79: Let B be a linear ordering and let b , < b, < ’ . . < b, und b , ’ < b2’ < * . . < b,‘ be two k-tuples of elements of B. Let B , = (t, b , ) and Bo‘ = (+-, bl’), let Bi = ( b i ,hi+,) and B,’ = (bi‘,bi+ 1) for each i, 1 I i I k - 1, and let Bk = (b,, +) and B,‘ = ( h k ‘ , +). Assume that (1) for every j , 1 I j I k, Pu(nl(bi) = PU(,)(b/); (2) for every j , 0 I j I k, P,,,- ,)(B,) = Pu(n-l)(B;)

i

j = 0 and

and either

‘u(nJB0)

= Pu(n)(Bo‘)

j = k und PU(,)(Bk) = P,(,,(Bi)

O < j < k a n d pu(n)((bj,bj+l))= Pu(,,)((bj’,b)+l))

or every sequence of 1-u(n - 1)-types of Pu(n-l , ( B j )of length s(n) is realized in B, and Bj’.

Then P , ( ( b i , b , , . . . , b J ) = Pn((b1’,b2’,. . . ,b,,’)). Proof : We proceed by induction on n to show that this conclusion holds for all k. To show that P,(b) = P,(b), where b = ( b , , b , , . . . ,bk) and b’ = (bl‘,b2’,. . . ,b i ) , it suffices to show, by Theorem 13.11, that G,((B,b), (B, b’))E 11, which is correct if the two members of each pair of corresponding intervals are both empty or both non-empty; this is guaranteed by (2). Assume now that the lemma has been proved for n and for all k. To show that P,+,(b) = Pn+l(b’),it suffices to show that G,,,((B,b),(B,b))~11.To do this, it suffices to show, by Theorem 13.4, that for each b E B there is a b ’ E B (and the reverse) such that G,((B,b, b), (B, b’, b‘))E11. Assuming that b , < b < b, (the other cases are similar), we will be able to use the induction hypotheses to conclude that

Pn((b,,b,b,, . . . bk)) = P,((bl’,b’,bz‘,. . . bk’)) 9

7

[and hence that G,( (B, b, b), (B,b’,b’))E 111 if we pick b so that b,’ < b‘ < b2’, ‘u(n)(’) = ‘u(n)(b’h P u ( n - 1 J ( b l , b ) )= Pu(n- 1)((b1’9bf)),Pu,n- 1)((b,b2))= Pucn-l)((b, b,‘)), either P,,(,,J(b,,b ) ) = PU(,)((hl’, 8 ) ) or every sequence of 1-u(n - 1)-types of P u ( n - l J ( b l , b )of ) length s(n) is realized in ( b l ,b) and (bl’,b), and either Pu(,,)((b, b,)) = PU(,)((b’,b,‘)) or every sequence of 1-u(n - 1)-types of l,((b,b , ) ) of length s(n) is realized in (b,b,) and (b‘.b,’).

4.

289


There are two main cases, the first if P,(,+ ,)((b,,b,)) = P,(,+ ,)((b1', b,')), and the second if every sequence of l-u(n)-types of Pu('")((bl, b,)) of length s(n 1) is realized in (b,,b,) and ( b , ' ,b,'). In the first case, where Pu(n+l)( X A

1 4 ( X , Z)),

then { d l A 1 @ [ b , d ] ) = { d l A 1 4 [ b , d ] ) ; moreover, for every c E A, { d l A I= @[c,d ] } is either empty or IS a bounded interval of A with minimum c. We may thus assume that for every c E A, {dl A != +[c,d ] } is either empty or is a bounded interval of A with minimum c. Consider the formula @(x,y) that is (3z)($(x, z) A 4(z, y ) ) v $(x. y). Then { d E A I A I= e[a, d ] } is an interval of A whose minimum is a and that includes c. It suffices to show that this interval is bounded, for then c E c,(a), as desired. But otherwise, A I= (Vy)(a I y + e(a,y)). Let C consist of C ( A ) together with v o > g and

13.

306


{ i d ( c , u o ) I Ab + [ u , c ] } . Since for each ~ E A {, d l A t= d [ c , d ] ) is either empty or is a bounded interval of A with minimum c, any finite union of such intervals is bounded, so that any finite subset of C is satisfied by some element of A . Hence C is consistent and has a model C = C , + C , , where C , = { d l d I a for some a E A } . As in the last paragraph of the proof of the previous theorem, since A < C, C,, C, < C , C , , so that A < C , ; this implies A + C , < C , C,, by Exercise 13.53.2, so that A < A C , . Since A C , b C, there is an element d E A C, such that i + ( c , d ) holds for all c such that A b +[a,c]. But since A < A C,, A C, 1 (Vv)(a5 y + &a, y)), which is a contradiction. We have thus shown that if b E c,(u) and a < b, then for all c > b, c E c,(u) if and only if c E CD(b).Analogously, if b E c,(u) and b < a, then for all c < b, c E c,(a) if and only if c E c,(b). To complete the proof that c , is a condensation we need only show that ifb E c,(u), then a E c,(b). For clearly, ifb E c,(a), then for all d, a I d Ib, we have both d E c , p ) and d E c,(b). But also, if b E c,(a) and a < b, then a E c,(b) and a < b, so by the second statement above with a and b reversed, for all c < a, c E c,(a) if and only if c E c,(b). Thus for all c E A , c E c,(a) if and only if c E c,(b). Thus suppose that a < b and b E c,(u); we will show that a E c,(b). We may assume that c,(u) is bounded above, for otherwise we could replace A by A A , , where A , Y A, and, since A < A A , , c,(a) in A A , is identical to what it is in A , so that c,(u) is bounded above in A + A , . (Recall that c,(u) in A is a union of bounded intervals). We may also assume, as we did two paragraphs earlier, that for every a' E A, { d l A b + [ a ' , d ] } is either empty or is a bounded interval of A with minimum a'. We show that A Y (Vy)(3z)(z < y A $(z, h)).Otherwise, since A is self-additive, it follows from Exercise 13.97 that there is element b' > cD(a)such that

+

+

+

+

+ +

+

+

+

+

+

A b (VY)(~Z)< ( Z Y A +(z,b')); hence for some a' < a, A != +[u',b], so that b' E c,(u'). Since a' < a < b , by the first part of the proof, we conclude that b' E c,(a') if and only if b' E c,(u), which is a contradiction. Thus A y (Vy)(3z)(z < y A $(z, b ) ) . Choose an element e E A such that A k $ [ c , b ] for no c < e. Then (dl A I= b [ b , d ] } , where +(x, uo) is the formula x 2 uo A ( 3 y ) ( y5 uo A $(y, x)), is bounded below by e and above by b and contains both a and b, so that a E c,(b), as was to be shown. W Suppose now that f : A + B is an isomorphism, where A is self-additive, and that a E A . Then, since for any a' E A and any formula 4 ( x ,y), A b 4 [a, u'] if and only if B b $[f(a),f(a')], it is clear that f maps cD(a)onto c , ( ~ ( u ) ) , so that isomorphisms preserve condensation classes. We are now ready to

5.

THE NUMBER OF COUNTABLEMODELS OF A COMPLETE THEORY

307

prove that T = Th(l), where I is self-additive, has c distinct countable models in the case where S,( T ) is infinite.

THEOREM 13.100: (Rubin) Assume that I is self-additive and that S,(T) is injinite, where T = Th(Z). Then T has c distinct countable models.

Proof : Since S,(T) is infinite, there is a non-principal type P. Choose a model I , of T in which P is omitted and another model I , of T in which P is realized by an element e and let B = I , cD(e) I , , where cD(e)is taken in I , and I, = I,. We will show that B k T, that the 1-type of e in B is P, cD(e)is a condensation class in B, and cD(e)is the only condensation class of B in which P is realized. Then if a , and u2 are order types and f is an isomorphism from B . a, onto B . a,, then, since b has type P if and only if f ( b )has type P and f [ c D ( b ) ]= c , ( f ( b ) )for every b, f maps the condensation classes of B . a , in which P is realized onto the condensation classes of B . a, in which P is realized. Thus ,f establishes an isomorphism from a1 onto a 2 , so that a1 = u,. Hence if a, and u2 are distinct order types, then B . a, and B . a, are not isomorphic and T has c distinct countable models. We claim that it suffices to show that B < I , 1, I,. For this implies that B k Tand, since I, < I , + 1 , t I , , the 1-typeofein I , I , + I , isP, so that, since B < I , 1, I , , the 1-type of e in B is also P. The condensation class cD(e) of 1, is also a condensation class of B since any formula defining over e an unbounded interval of I 2 will also define an unbounded interval of I , 1, 1, and hence also of B. Furthermore, the I-type of any element of either I , or 1, in I , + I, 1, is the same in B as it is in I , or I , and I , I , I,, as are the condensation classes of such elements. Thus cD(e)is the only condensation class in which P is realized. Hence to complete the proof, we need only show that R < II I 2 + I,. Let A = I , 1, 1 3 .We must show that if A k $ [ b , , . . . , b k ,a ] , where each bj E B, then there is a b E B such that A k $ [ b , , . . . ,b,, b ] . We may assume that a E Z 2 - cD(e) and a > e. and that b , < b , < . . . < b, with b p the first bi E I,. Choose b' E cD(e) such that b ' 2 b p P l ;then b' < a < b,. By Theorem 13.41, there is a selecting formula $" for $ in [b',b,] (relative to b , , . . . ,bk),that is, a formula $ # ( x L , x 2 v) , such that A k $"[b, b,, c] if and only if b ' < c < b p and A k $ [ b , . . . . ,b,,c]. If we find b e B such that A k $ " [ b , b , , b ] , then we will have a b E B such that A k $ [ b , , . . . ,b,, b ] . Thus we may assume that we are given A k q@" b,, a], where b < a x). Then $'(x) is a terminal formula of T unless +(x) is a terminal formula of T , and &(x) is an initial formula of T unless $(x) is an initial formula of T . If @x) is an interval formula of T , then any model A of T can be written as Cap k

4-CaN +

4C4 +b

( A

4"aIh

with the first (last) term absent if 4 is an initial (terminal) formula of T. Given a model A of T and a E A , we let @a be the set of all interval forthen either 4+(x) E or +-(x) E Oa, mulas of T satisfied by a. If +(x) $ so that 4(x) cannot be consistently added to Thus @ a is a maximal consistent set of interval formulas of T. Conversely, if Q, is a maximal consistent set of interval formulas of T , then @ s P for some 1-type P by Exercise 13.67.4 since @ G F,(T),and hence there is a countable model A k T and an element a E A whose 1-type is P, and therefore @a = @. DEFINITION 13.110: A maximal consistent subset of the set I ( T ) of interval formulas of T is called an interval type of T. The collection of interval types of T is denoted ZT(T).If @ is an interval type of T and A i= T, we let A(@)= { u E A ( @ ,= @}. Note that A(@) is an interval of A if it is non-empty. If A ( @ )# $3,we say that @ is realized in A, and otherwise we say that @ is omitted in A . We let IT(A) denote the set of interval types of T realized in A .

Note that for any model A k T, there is a natural correspondence between c,-condensation classes cE(u)of A and interval types @a realized in

5.

THE NUMBER OF COUNTABLE MODELS OF A COMPLETE THEORY

319

A ; thus b E cE(aJif and only if Qb = @a, so that A(@,) = cE(a).This correspondence induces a linear ordering on IT(A) by @, < I T ( A ) @ b if and only if cE(u)< cE(b);thus I T ( A ) = c E [ A ] . Let and be two interval types of T and assume that $ ( x ) E 0,but + ( x ) $ 0,.We may assume, without loss of generality, that $ J + ( x E) 0,. From this we see that if Ol,Q2 E I T ( A )n I T ( B ) , then < I T ( A ) Q2 if and only if O1 m2. Thus the various linear orderings induced on the subsets IT(A) of I T ( T ) by the correspondences with c E [ A ]are all compatible. Since S,,(T)is assumed to be countable for all n, by Theorem 13.82 there is a countable saturated model S of T . Since each interval type is contained in a l-type and since each l-type is realized in S, it follows that every interval type is realized in S , so that I T ( S )= I T ( T )and I T ( T )2 c E [ S ] ;in particular, I T ( T ) is countable. DEFINITION 13.111: An interval type CD is called a principal interval type if there is a formula $(x) E @ that generates 0;that is T k (Vx)[$(x) -, $(x)] for all 4(x) E 0.Otherwise, @ is called a non-principa1,or limit, interval

type. We now obtain a characterization of the principal interval types. LEMMA 13.112: Suppose that 4(x) is an interval formula of T. Then {@ E I T ( T ) )$(x) E @} has a first mid last element.

Proof: Suppose that it has no last element. Then there is a countable < < a2< . . . of elements of I T ( T ) that is cofinal in sequence {@ E I T ( T ) l 4 ( x )E 0.). For each n, choose a formula 4,,(x) with 4,,(x)E @, and $n'(x) E On+1 . Let C = {$(x)} u {$,,+(x)In < 01). This is a consistent set of interval formulas of T since any finite subset is contained in some @,,. Hence C c P for some 1-type P of T ; but the set of interval formulas of T in P forms an interval type, so that C c Q, for some @ E ZT(T). But each On< CD and 4(x) E Q, contradicting the assumption that the sequence {a,,In < w } is cofinal in ICD E I T (T )1 + ( x ) E a}. H COROLLARY 13.113: Let @ E I T ( T ) . Then @ is a principal interval type fi and only i f @ has an immediute predecessor and an immediate successor

in ZT(T). Proof : If @ has an immediate predecessor

and an immediate successor in I T ( T ) ,then there are formulas Cpl(x)E CDl and 4 2 ( x )E O 2 neither of which is in @. Then 4 1 + ( x )A $ J 2 - ( . x ) E @ and clearly generates contains infinitely many non-isomorphic linear orderings. We need the following lemma.

+

+

+

+

+

+

+

+

+

+

+

+

+ +

+

+ + +

+

LEMMA 13.119: Let JV be an infinite collection of C.C,-categorical order n o} of order types such that each types. Then there is a sequence { ~ , \ < zi is the order t y p e of an interval of some z E N and such that each z, is isomorphic to a proper interval of z,+ 1.

Proof : We define the notion of predecessor for KO-categoricalorder types by induction on rank (see Exercise 13.92). Thus if z has rank n + 1 and z = z1 T ~ where , both z1 and z 2 have rank at most n, then the predecessors of z consist of z,, z2, and the predecessors of z1 and 7,; similarly, if t has rank n + 1 and T = a(F), where each z’ E F has rank at most n, then the predecessors of z consists of F together with the predecessors of each T’ E F.

+

5.

THE NUMBER OF COUNTABLE MODELS OF A COMPLETE THEORY

323

Now given N , we let A"* consist of Jli together with all predecessors of order types in N.We now define by induction a sequence {z,(n < o) of elements of N * such that each z, has rank n. We will assume, as part of the induction hypothesis, that the set T , of elements of N* that have z, as a predecessor is infinite. We define zo to be 1 and, assuming that z, is defined and T , is infinite, we enumerate the elements ol,.. . , ( T ~of T , that have rank n + 1. This set is finite by Exercise 13.92. Moreover, if T E T,, then, as is easily verified by an induction on rank, some ( T ~is a predecessor of z. Thus if we define T"+ to be the first oithat is a predecessor of infinitely many elements of T,, then the induction hypothesis continues to hold and we obtain a sequence {z,ln < w ) , as desired. Suppose now that { S ( 0 , ) 1 n < o} contains infinitely many nonisomorphic KO-categorical linear orderings. Then, by applying Lemma 13.119, we can find a sequence (t,ln < co} of order types such that each T~ is a predecessor of some S(@,,,J; a subsequence {a,ln < w } can easily be selected so that each (T, is a predecessor of some S((Dq(,)),where the subscripts { q ( n )\ ti < o}form an increasing sequence. Let C, be an interval of S(aq(,,J of order type B, for each n. let g, map C, onto an interval of C,+, for each n, and let (T be the order type of C = U{C,[ n < o},where in this union each C , is identified with g,[C,]. In the example discussed earlier, each C, has order type n and the order type of C is either w, w*, or ( depending on the functions {gnln < ( 1 ) ) chosen. We now show that S(@) has an interval of order type rr. Indeed, if we choose elements {a,lz E Z ) G C so that z1 5 z2 implies aZlI azz and so that { a , , l n N ~ } is cofinal in C and { a , / n ~ Z N ) is coinitial in C, then C = U { [ a _ , , , a , , ] I nN~) ; since each [a-,,a,] is an interval of some S(@"J, each [a_,,,a,] is KO-categorical by Exercise 13.92, and hence is finitely axiomatizable by Theorem 13.106. Let $, be an axiomatization of [a-.,a,] for each n. Add to the language of linear orderings a (-sequence (a,] z E Z } of constant symbols and let C = C ( S ) u ($,'In < wI1 u { g z , I gz2/z1 5 z 2 } u {Cp(%)(zE 2 and Cp E @ } , where t+bn' is the relativization of $, to the interval [a-,, 4.Any finite subset of C can be realized in S by selecting elements of some S(0.,) to correspond to the constant symbols. Hence 'c is consistent and is realized in some countcontains elements { c, I z E Z ) able model s' of T. Since s' < s,we see that s(@) such that each interval [c - ,,c,] Y [ u - ,, a,] ;hence S ( 0 )contains an interval of order type rr. Now B itself cannot be KO-categoricalsince if its rank were k, then every interval of it would have rank at most 2k + 1 by Exercise 13.92; this is impossible since the sequence { rr, 1 YE < o}of intervals of (T all have different

324

13.


ranks. Since o is an interval of S(@), it follows that S(@) cannot be KOcategorical. Since by Theorem 13.116, S(@)is self-additive, it follows from Corollary 13.103 that there is a countable model I of T such that I . CI is different from I . fl whenever ct # p. Let S’(@)be a saturated model of Th(S(@))and let S’ be obtained from S by replacing S(@)by S’(@).Since S(@) is saturated, it follows that each I . a < S’(@). If we let S, be the result of replacing S‘(@)by I . c( in S‘,then each S , < S’ and { u E S,I S, k 4 [ u ] for all 4 E @} = I . a. Hence these c countable models are all distinct models of T since any isomorphism from S, to S, must carry I, onto I , . Thus if the sequence {S(@,)ln < a}contains infinitely many distinct KO-categoricallinear orderings, then T has c distinct countable models. We thus need only consider the case in which the sequence (S(@,)ln < 01) contains only finitely many distinct KO-categorical linear orderings. As an example of this case we can take A = A , + 3 + A , + 3 + A , + 3 + . . . , where A, = (q 2 * q ) . n. Here again I T ( A ) has order type w and the condensation classes have order type q, 2 . q, and 3. Thus I T ( T ) consists of @, < Q2 < Q 3 < . . . < 0,and we must again calculate S(@).We leave this calculation to the reader.

+

+ +

EXERCISE 13.120: Verify that if A = A , + 3 A , + 3 + . . . , where A , = ( q 2 . q) . n, then S ( @ ) = (3 ( q 2 . q)(w o*)) . 5 * q.

+

+ +

The phenomenon that occurs in this example, namely, the existence of c . . . , occurs in all cases in which ( S ( @ , ) l n < o)contains only finitely many different models. That is, we claim that there is an increasing sequence { i(n)ln < u}of natural numbers such that i(n 1) 2 i(n) n and such that S(Qicn,) N S(QiCO,) for all n 2 0, S(@i(n)+l) ‘vS(Qi(,)+ 1) for all n 2 1, and, more generally, S(Qi(,)+;)= S(@i(,l+,)for all n 2 j and all j < a. If we now define A , = S(@,(,,)+ S(Qi(,,)+ + . . . S(@‘i(n)+n+ ,), then each A, is isomorphic to an initial interval of A,+ 1. To prove the existence of {i(n)ln< o},we define it inductively together with a contracting sequence {Z(n)ln < w ) of infinite subsets of N . Let B,, B,, . . . ,B,-, be the finitely many distinct models in the sequence { S ( @ , ) l n < w ) , and let Ij(0)= { nI S(@,)N B j } for each j < k. Since { Ij(0)I j < p } is a partition of N , some Ij(0)must be infinite; call it I(0)and let i(0)be its smallest element. Proceeding inductively, we assume that i(0) < i(1) < . . . < i(k) have been defined, i(t + 1) 2 i(r) + t for every t < k, and for every t 5 k and every n E I(t), s(Qn) S(@i(O),S(Qn+ 1) L s ( Q i ( i ) + 11, . . - S ( Q n + t ) S(@i(i)+t). We now partition I(k) - (0, 1 , 2 , . . . , i(k) + k - 1) into { I j @ + 1)1j < p ) , where I j ( k + 1) = { n 2 i ( k ) + klS(@,+k+ ,) N Biand n E I ( k ) } .Then some Ij(k + 1) must be infinite, so we define that one to be I ( k + 1) and we define its smallest A E A, EA,

+

+

3

+

6.

FINITELY AXIOMATIZABLE LINEAR ORDERINGS

325

element to be i(k + I). The induction hypotheses continue to hold; hence the existence of (i,,1 n < w } is substantiated. We now show that the sequence {A,ln < w } of KO-categorical linear orderings is pairwise non-isomorphic. For suppose that n < m and that A, 'v A,. Now A , is isomorphic to the initial interval C of A,. Since C is a finite sum of S ( m j ) ,each of which is definable, it follows that C is definable, and, by Theorem 13.38, C is definable in A,. It follows that more 1-types are realized in A , than in C, so that C. and hence A,, cannot be isomorphic to A,. Hence { A , In < w } is pairwise non-isomorphic. Let cr be the order type of A = U{A,I n < 01, where in this union each A , is identified with the corresponding initial interval of A,, Then, as in the preceding case, S(@)has an interval of order type cr and, since cr cannot be KO-categorical,S(@)is not KO-categorical.Then, as in the preceding case, since S(@) is self-additive, T has c distinct countable models. This completes the proof of the main result of this section. THEOREM 13.121: (Rubin) Let T be a complete theory of linear urderings. Then T is either N,-cutegorical or has exactly c distinct countable models.

Using Corollary 13.103, the proof we have given of Theorem 13.121 can be modified to obtain the following result. COROLLARY 13.122: (Rubin) Let T be a complete theory of linear orderings. Then either T is KO-categorical,S,(T) is uncountable for some n, or there is a countable model A of T that has a self-additive interval I such that if A = A , + I + A,, then, whenever CI # p, the linear orderings A , + I ' C I + A , and A , + 1 . /3 + A , are not isomorphic.

56. FINITELY AXIOMATIZABLE LINEAR ORDERINGS

Our goal in this section is to prove that a statement that is true in some linear ordering is true in some finitely axiomatizable linear ordering. This was mentioned earlier as Theorem 13.23: Given any linear ordering A and M ; i f A is any n, there is a finitely axiomatizable M E A? such that A scattered, then M can be chosen from d oThis . was conjectured by Lauchli and Leonard [8] and was finally proved by Myers [12], Shelah, and Schmerl (independently); our proof is patterned on that of Amit [l].

326

13.


We restrict our attention for now to scattered linear orderings. The general case will be considered subsequently. Since we know, by Corollary 7.10, that any countable scattered linear ordering is G,-equivalent to an element of Ao,it suffices to prove, by the transitivity of -,, that every element of ,Aois G,,-equivalent to some finitary linear ordering. Thus, proceeding inductively, we could assume that A -,F, where F is finitary, conclude by Lemma 6.5.2 that A . w -,F . w, and then try to show that F . w is finitary. The difficulty with this approach is that F . w need not be finitary even when F is finitary; moreover, conditions under which F . w must be finitary are elusive. We digress for a moment from our goal and elaborate on the sentence above. An example of a finitary F for which F . w is not finitary is provided by F = w* + 5 + w (shown finitary in Exercise 13.18.3). To verify that

F.

= O*

+ 5 + (O + 0") + 5 + (w + o*)+ . . .

is not finitary, we observe that F . w (and even F o + w* in the following technical sense.

+ F) contains a copy of

DEFINITION 13.123: Let I be an interval of a linear ordering A. We say that 1 is a copy of w w* if 1 has order type w + 5 . CI + w* for some order type u, and if the first element of I has no immediate predecessor and the last element of I has no immediate successor. If A has an interval I that is a copy of w + w*, then we say that A contains a copy of w + w * .

+

+

+

Note that 5 5 does not contain a copy of w + a*,although F F does, where F = m* 5 + w. We will show later, in Corollary 13.142, that any A that contains a copy of w + w* is not finitary. For now, we will show that F . w is not finitary; a similar argument works for F + F. It follows from Exercise 6.1 1 and Lemma 6.5.2 that

+

F .

-,,

W*

+ (5 + 2").0

for any n; so it suffices to show that for every n. PLAYER I chooses, in his first two turns, the endpoints of a copy of 2" in w* + (5 + 2"). w ; PLAYER II must then choose a point of F . w with no immediate predecessor and a point of F . w with no immediate successor (or lose in two more turns). The interval of F . w thus determined must have order type w F . m w* for some m 2 0. But, by Exercise 6.10.2, PLAYER I has the winning strategy in C,(2" - 2, w + F . m + w * ) and hence in G , , 2(w* (5 Z n ) . w, F . 0).Thus F . w is not finitary. This completes the digression.

+ + +

+

6.

FINITELY AXIOMATIZABLE LINEAR ORDERINGS

327

The approach in [l] is to use the appropriate condensation map. Thus we will define a condensation c of each linear ordering A E .Aoso that each c(a)is finitary and so that c[A] has lower rank than A , and hence is G,-equivalent to a finitary F . We will then replace the sum A

=

c ( c ( a ) c[A]} E

by a sum that is indexed by F , is (3,-equivalent to A , and is finitary. In order to replace A by a sum indexed by F , we need to think of c[A] as not just a linear ordering, but as a labeled ordering-where each element c(a) E c [ A ] is labeled by the G,-equivalence class of c(a). If it turns out that c [ A ] -,F not only as linear orderings but as labeled orderings, then we can easily take a sum indexed by F that is G,-equivalent to A. As we will see, each c(a) can be taken to be G,,-equivalent to one of a j n i t e set of linear orderings, so that the labeling of c[A] determines a finite partition of c[A]. (We have been speaking of “labelings’’because of similarities to the discussion in 58.6, but it will be more convenient to talk about partitions.) Because of these considerations we need to enlarge our scope so that we consider partitioned orderings, as well as ordinary linear orderings, as fair game for the Ehrenfeucht analysis. DEFINITION 13.124: Let A be a linear ordering and let A , , A l , . . . , A,be an ordered sequence of k subsets of A that partition A , where k is a fixed natural number. The structure ( A ; & , A , , . . . ,A k - 1 ) is called a partitioned ordering. In some contexts, we will use A to denote this structure. If there is a possibility for confusing the linear ordering A with the partitioned ordering A, then we will use the notation A k for the latter. If we wish to emphasize the number k, then we will speak of A k as a k-partitioned ordering. (We will not introduce the notion of partitioned order types, although that too can be easily defined.)

Given two k-partitioned orderings A k = ( A ; A , , A , , . . . , Ak-l)and Bk= ( B ; B , , B , , . . . ,Bk-,),thedefinitionoftheEhrenfeuchtgameG,(Ak,Bk) is included in Definition 13.1. In order for PLAYER II to win, he must so select his moves that, if a, and b, denote the elements of A and B chosen at the tth turn, not only must a, < A a,

if and only if

b, < B b,

if and only if

b, E B,

for all s, t 5 n, but also a, E A j

for all s I n and all j < k. Thus PLAYER 11 must match PLAYER 1’s moves both with respect to the linear orderings and with respect to the partitions.

328

13.


If he can do so, then PLAYER 11 has the winning strategy in G,(Ak,B k ) ;if he cannot, then PLAYER I has the winning strategy. For k = 1 these notions coincide with the definitions given for ordinary linear orderings. The exercise below, involving the case k = 2, gives some indication of what happens at the next level. (Assume that G,-equivalence has already been discussed for k-partitioned orderings-it will be soon, and this exercise will thus serve as motivation.) EXERCISE 13.125 : Corollary 6.9 can be interpreted, for n = 2, as saying that every finite linear ordering with more than 3 elements is G,-equivalent to a finite linear ordering with no more than 3 elements. This is the situation for k = 1; the statements below concern the case where k = 2. (1) Show that there is a finite ( A ; A , , A , ) where A has 8 elements which is not G,-equivalent to any finite ( B ; B 1 , B 2 ) ,where B has fewer than 8 elements. (2) Show that every finite ( A ;A 1, A , ) where A has more than 8 elements is G,-equivalent to a finite ( B ; B1, B , ) , where B has no more than 8 elements.

We now proceed by reviewing the analysis of Ehrenfeucht games on linear orderings presented in Chapter 6 and determining to what extent it applies to Ehrenfeucht games on k-partitioned orderings. (We will be referring to this discussion when, later on, we speak of the "adapted" versions of the statements of Chapter 6.) If we define Ak Bk to mean that there is an isomorphism f : A -+ B such that f [ A j ] = B j for j < k, then Lemma 6.3 still holds. If we define Ak + Bk (for A and B disjoint) to be (A

+ B ; A0 U B o , . . . , Ak-1

U Bk-1)

and similarly for generalized sums (of partitioned orderings over an index set that is an ordinary linear ordering) and shuffles, then Lemma 6.5 and Exercise 7.18 still hold. If we define I k to be ( I ; A , n I , . . . ,Ak-1 n Z)

for any interval I of A, then the following form of Theorem 6.6 is correct: G , , ,(Ak,B k )E I1

if and only if (i) for every j < k and every a E A j there is a b e Bj such that C,((A'")k,(B'b)k)E I1 and G,((A and let a, be a. Consider the class of all formulas Cp(v,, . . . ,u,) of L,, of quantifier depth at most 0 such that A k 4 [ a o , . . . ,a,]. If this class of formulas were a set, and so could be indexed by some cardinal number, we could then take the conjunction over all the formulas of the class to obtain a formula O(uo,. . . ,u,) of quantifier depth 0. Then (3u,)@.(uo, . . . ,u,) would be a formula $(uo, . . . ,un- 1 ) of quantifier depth 0 + 1 and 14 b $ [ a , , . . . ,an-,],

so that Bb$[.f(ao), . .

'

,f(a,-,)I;

thus there would be an element b E B such that Bk@[f(ao), . . . ,f(afl-1), b ] ;

and hence B b 4[f(ao), . . ,f(a,'

11,

bl

for all formulas &u0, . . . ,u,) of quantifier depth at most 0 satisfying A k 4 [ a o ,. . . ,a,- 1 , a,], and therefore there would be a finite isomorphism g E F,, as required. Unfortunately, the class of formulas described above is not a set. Fortunately, however, it is equivalent to a set of formulas, in the sense that there is a set 5 < 2 ) of formulas in the class such that for every formula 4 in the class,

B~/\C4 4 A +

+

for every A < K . Thus, for every , I< K, there is an ( n 1)-tupleof elements of B satisfying A{4 a such that 6 = 6 6 . T .for any order type T .

+

Proof: Choose 6 > a so that w6 = 6. This is possible by Exercise 3.52.1; furthermore, by Exercise 3.52.2, Ci = m e . 6 for any 8 < 6. Thus, for any 8 < 6, 6 can be written as a disjoint union of 6 intervals of order type coo, which we will call the we-intervals of 6. (Note that an interval of 6 of order type m' is not necessarily an coe-interval of 6.) The same is also true for 6 6 . T for any order type T . For each 8 < 6 let I , be the set of all finite isomorphisms f from 6 to 6 6 . T such that if the domain of 1' consists of a , < a , < . . . < a,- 1, then

+ +

(1) for each i < n - 1, a j is in the same me-interval of 6 as a i + l if and only if ,f(ai)is in the same me-interval of 6 6 . T as f ( a i +'); (2) for each i < n, ai is the ccth element of its me-interval if and only if f(ai) is the ccth element of its wO-interval(where a < m e ) ; and ( 3 ) a, is in the first me-interval of 6 if and only if f ( a o ) is in the first coo-intervalof 6 + 6 . T .

+

It is sufficient to show that {Z, 18 I 8) has the ®ressive back and forth property (where we may take I b to be n(Z,I8 < 6}.) Clearly, the monotonicity condition (1) holds. We now prove (3); the verification of (2) is similar. Suppose then that ,f E I , + and b E 6 6 . z. Suppose that the domain offconsistsofa, < a, < . . . < a,- Ifhisinthewe+'-intervalofanelement of the range off, then we can clearly extend f to a y E I,+ 1, and hence in I , , satisfying the conditions. Similarly, there are no problems if b is in the first oe+'-interval of 6 6 . T or b is in an o'+'-interval that lies to the right of thew'+ '-intervals of the elements in the range off. The only troublesome case is when, for some i < n, f ( u i )< b < f ( a i + ' ) , b is not in the me+1intervals of either f ( a i )or f ( a i +') but there is no coo+ '-interval between the 1_' intervals of ai and ai+ in 6. Only in this case is it impossible to extend f to a function in I , , 1 ; it is possible, however, to extend it to a function in

+

+

358

14. THE INFINITARYTHEORIESOF LINEAR ORDERINGS

l o . For within the me+l-interval of ai there are many me-intervals that are to the right of the me-interval of ai yet lie within the me+'-interval of a , . Choose one such interval and, if b is the ath element in its me-interval in 6 + 6 .z, let a be the ath element in this chosen interval. Let g extend ,f by mapping a to b. Then g E I , , so that requirement (3) is met. This completes the proof of the theorem.

The following corollary was first proved by Lopez-Escobar [8], COROLLARY 14.30: There is no statement if and only if A is a well-ordering.

4

of L,,

such that A k

4

Proof : Suppose that 4 is a statement of L,, of quantifier depth a. Let 6 be an ordinal that is larger than a and satisfies the hypotheses of Theorem 14.29. Then 6 --6 6 6 . z for any order type z. If 4 is true in all well-orderings, then 4 is also true in 6 + 6 . t for any order type t, and so is also true in some linear orderings that are not well-orderings.

+

COROLLARY 14.31 : The scattered linear orderings are not dejinable by a statement of La,,. A COROLLARY 14.32 : (Landraitis [7]) The transitive linear orderings and the rigid linear orderings are not characterized by statements of Lx,,.

Proof: For the transitive case, use Exercise 14.28.

A

Thus there is no finite-quantifier sentence that expresses well-ordering. This is perhaps a convincing argument for the position that even if arbitrarily large conjunctions and disjunctions are possible in a language, so long as the possibilities for quantification remain unchanged, the language remains basically first-order in character. Back and forth arguments, as they apply to infinitary logics, are treated extensively in Barwise [l] and Kueker [4].

$3. SCOTT SENTENCES Although not every countable linear ordering can be completely characterized by a statement of L,,,, nevertheless, using the terminology of Chapter 13, every countable linear ordering is KO-categorical. Indeed, it

3.

359

SCOTT SENTENCES

follows from a remarkable theorem of Scott [131 that, given any countable structure M, there is a statement 4 of L,,, such that the only countable model of 4, up to isomorphism, is M. DEFINITION 14.33: Let M be a countable structure. A Scott sentence for M is a statement 4 such that M k $ and whenever N k 4 and N is countable, then N 1 M. THEOREM 14.34: (Scott) Every countable structure M has a Scott sentence. Proof : We define for each n-tuple a = ( a 1 , a 2 ,. . . ,a,) of elements of M and each countable ordinal CI a formula 4."(x1, x 2 , . . . ,x,) that describes M from the vantage point of a l , a 2 ,. . . ,a,; as CI gets larger, these descriptions will approximate a complete description of M as seen from these elements. We proceed by induction on a. For each a,, a,, . . . ,a,, (pao is the conjunction A{$(x1,x2,. . . ,x,)IM k +[al,a,, . . . ,a,] and $ is either an atomic formula or the negation of an atomic formula.} For successor ordinals, define &+ to be

'

$aa

A

A{(jxn+,)&,a,

+

1(x i 7 x2 , . .

A(vX,+1)V{4Ba.a,+,(X.1,X2,.

I

,xn+1 ) an+ 1 6 M 1 . . ,X,+l)lan+l E MI;

and for limit ordinals, define 4aAto be fj($axla < 1)-

It is easily verified by induction on M k q5a"[a] and

CI

that for every a = (u17a 2 , . . ,a,),

for all p < cx. Thus the sequence ((p."Icx < q j of formulas provides more extensive information about M, as seen from the perspective of a. We claim that for each a there is a countable ordinal y = y(a) such that

M @'x)L$aY(X)4 4aTx)l for all countable /3. For otherwise there is a sequence {y(cc)la < ol>of countable ordinals such that MY (Vx)[$$"'(x)

-+

4:("+')(x)]

for all CI.If to each cx we assign an n-tuple al(a), a,(a), . . . ,a,(a) of elements but not $JX("+", we then get uncountably many n-tuples of M satisfying 4Lca)

360

14.

THE INFINITARY THEORIES OF LINEAR ORDERINGS

of elements of a countable set, an impossibility. Thus the claim is correct. Moreover, if we let I = u{y(a)lfor all n and all a}, then 3, is a countable ordinal and

M != (W[4,"x) 4pB(x)I for all n, all a,, u 2 , . . . ,a, E M , and all countable ordinals /3. Now define 4 to be the statement +

(IX)$~'(X)AA{(VX)[~~'(X) .+ 4~+'(x)]Iforallnandalla,,a,,. . . , u , E M } ,

where a is any element of M . Clearly M 1 4. We want to show that if N k 4 and N is countable, then N cz M. This is proved by a back and forth argument as in Cantor's Theorem (Theorem 2.8). Enumerate M = {co, c,, c 2 , . . .} and N = {do, dl, d 2 , . . .}, where M 1 $,'[c0] and N != #,'[do]. We define inductively a map f : M -+ N. The induction hypothesis is that at the end of stage n, the domain {ao,a,, . . . ,a,} off' is finite and includes co, c , , . . . , c,, the range {bo,b,, . . . ,bm) o f f includes do, d,, . . . , d,, f'(ai) = bi for each i, 0 I i 5 m, and

+

N !=

4toa1.

. . a,,,[bO b1,

. . ., bm].

At stage n 1, we will extend f so that its domain includes the first element a of M not yet in the domain, and then we will extend it again so that its range includes the first element b of M not yet in the range. Since N'

4foal...u,Cbo,bl,...,bm],

N

4f,+,f...a,,,[bo,bl,. . . ,bm],

we have so that

N ' (3Xrn+1)4&a1 . . . a , a ( ~ o ~ x 1 ,.. . ,~m,xm+l)CbO,blr...,brn]. Choose d E N such that N k 4~~al....,a[bo,bl, . . . , b m , d ] and set f(a) = d. On the other hand, since N 1 4, we have N

'$t;;...a,,,a[bo,bl,.

. . ,bm,d],

so that for some c E M , N 1 4 & a l . . .a,ac[bo,bl, * . . ,bm,d,b], and thus we can define f(c) and b,+2 = b.

= b.

Finally, set a m + ,= a, a m + 2= c, b m f l = d,

3.

SCOTT SENTENCES

36 1

Clearly the induction hypotheses continue to hold, so that ultimately we have defined an isomorphism from M to N.

COROLLARY 14.35: (1) I f A and B are couiitable linear orderings and A q,,lo B, then A 2 B. (2) I f A is a countable lineur ordering, then there is a countable ordinal 6 such that if B is countable and A B, then A 2i B. A Thus even non-scattered linear orderings have Scott sentences. However, as Landraitis observes in [6], if A is a dense sum of scattered orderings, one cannot estimate the minimum quantifier depth of a Scott sentence for A in terms of the quantifier depths of Scott sentences of the scattered summands of A. Indeed, as Exercise 14.36 shows, for any countable ordinal y it is possible to construct a dense sum of 1’s and 2 s that has no Scott sentence of quantifier depth below 7 . EXERCISE 14.36: Let Z, be a subset of q of order type Z“for each ordinal a. Let A , = x { , f z ( r )Y E .d1, where f , ( r ) = 1 if r E 2, and f,(r) = 2 if r $2,. Show that A , = m . A , for all B 2 CI.

1

We will use Scott’s Theorem to prove a theorem of Makkai [9] that says that, under certain circumstances, a countable structure A is I,,,,equivalent to an uncountable structure B. Makkai’s condition is that there exists a countable structure A‘ that properly contains A and such that given any finite number of elements of A, there is an isomorphism from A onto A‘ that leaves each of those elements fixed. By Scott’s Theorem, this says that there exists a countable structure A’ properly containing A such that ( A , a i , a z , . . . ran)

(A’,al,a,, . . . , a n )

for every a l , a 2 , . . . ,a, E A . This of course is reminiscent of the notion of elementary substructure defined in Chapter 13. DEFINITION 14.37: Let A and B be structures of the same type, with B properly containing A. We say that A is an L,,,-substructure of B, written A - L , , , B, if ( A , u I , a 2 , .. . ,a,) =o,,lco (B,a1,a2,.. . ,a,) for every a l r a 2 , . . , u, E A.

By Scott’s Theorem, if A and B are countable and A is an Lm1,-substructure of B, then, given any finite set of elements of A, there is an isomorphism from A onto B that leaves each of the given elements fixed.

362

14.

THE INFINITARY THEORIES OF LINEAR ORDERINGS

Suppose that A and B are countable and that A is an L,,,-substructure of B. We wish to define a sequence {A,la < o,}of countable structures, beginning with A, = A and A , = B, such that A, is an I,,,,-substructure of A, whenever M < p. The definition will of course proceed by induction, so we assume that {A, M < y} have all been defined and that if M b(m)) E 11, and for euch i, 0 I i < m,

+

H n ( ( a ( 4 > 4 i l ) ) a , W ) b(i ,

+ 1))B) E 11.

Proof : Analogous to the proof of Theorem 6.6, using the additional fact that the union of finitely many finite sets is finite. A LEMMA 15.8: Let A , B, and C he linear orderings and let n 2 1 be a fixed natural number. I f H,(A, B) E I1 and H,(B, C ) E 11, then H,(A, C ) E 11.

Proof : Similar to the proof of transitivity of G,-equivalence (Corollary 6.7) using induction on n and Theorem 15.7 instead of Theorem 6.6. We are now ready to make the following definition. DEFINITION 15.9: We say that A is H,-equivalent to B if PLAYER 11 has a winning strategy in H,(A,B). We say that A is H-equivalent to B if A is H,-equivalent to B for every n. By Lemmas 15.4 and 15.8, these are equivalence relations.

15.

374

THE SECOND-ORDER THEORIES OF LINEAR ORDERINGS

In Chapter 6 we showed that any well-ordering was G-equivalent to a unique ordinal less than ow. 2 . The following theorems establish a similar result for H-equivalence. THEOREM 15.10: For every natural number n 2 1,

( I ) H,(w",o". p) E I1 for every ordinal p > 0. (2) If a < w" and CL < p, then H,(a, p) E I. Proof : We first prove ( 1 ) by induction on n. For n = 1 it is obvious that H,(w, w . p) E I1 for every ordinal p > 0. Assume the claim true for n - 1. If PLAYER I on his first turn in H,(w", W" . ,!l) chooses points in on,then PLAYER 11 chooses the same points in on. p. Then all corresponding intervals are isomorphic except for the right-most ones, which are, respectively, of the form o" and 0". fl; but by induction hypothesis and transitivity (Lemma 15.8), H,- l(o",(on . p) E 11, so that PLAYER 11 will win this play of the game. If, on the other hand, PLAYER I on his first turn in Hn(o",co". p) chooses points a , < a, < . . . < a, in a " .p, then PLAYER 11 chooses b, < b, < . . . < b, in co" so that if ai = w"-' . yi di, where hi < on-' for all i, then bi = con-' . gi ai are chosen so that if y i = 0, then g i = 0, so that y i = y j if and only if gi = g j , and so that g1 S g 2 I . . . Ig k . Corresponding intervals are then either isomorphic or are on- . a1 + 6 and on- . CL, + 6, where a1 and a2 are both non-zero and 6 < con-'. In all of these cases, by the induction hypothesis and Lemma 15.6, PLAYER 11 has a winning strategy in each of the games, with n - 1 turns, played with a pair of corresponding intervals. Hence by Theorem 15.7, H,(o", o". /3) E 11. We now prove (2) by induction on n. For n = 1 we have already seen that Hl(a, p) E I for a < fl and a < u.Assume that the claim is true for all k < n and suppose that a < w" and a < p. Write a = on-' . a + a,, where a l < on-' and we may assume that a > 0 [since otherwise by the induction hypothesis H,- ,(a, 8)E I.] If a1 = 0, PLAYER I picks the a points o"-'. 1, con-' . 2 , . . . , on-' . a of ,!l. Then no matter which a points of a are picked by PLAYER 11, at least one of the resulting intervals A will have order type < ( o n - ' . By the induction hypothesis, H,- , ( A , w " - l )E I, so that by concentrating on such an interval and the corresponding interval of p, PLAYER I has a winning strategy in Hn(w"- . a, p). If a1 > 0, say a1 = w"' . a , (on* . a, . . . + 0"". ak, then PLAYER I picks a a , a2 . . ' ak points of3!, appropriately and, again using the induction hypothesis, PLAYER I has a winning strategy in H,(cL,B). This completes the proof.

+

+

'

+

+ + +

+

+

+

COROLLARY 15.11 : Let CL and /3 be ordinals, Write a = ow. a1 CL, und b = o w p, . b 2 , where a, < (ow and 8, < ow.Then CL and p are H-

+

1. THE GENERALIZED EHRENFEUCHT GAMES

equivalent if and only if ci2 = p2 and either uI and bigger than 0.

315

p1 are both 0 or are both

Proof: Since H-equivalence implies G-equivalence, we need only show that if a = do. a, y and p = ow. fll + y, where a1 and PI are non-zero and y < ow,then CI and p are H-equivalent. But this follows immediately from Theorem 15.10. W

+

We note that the result for H-equivalence of ordinals is exactly the same as for G-equivalence of ordinals. That is, every ordinal is H-equivalent to a unique ordinal less than ww . 2 . Furthermore, it follows that two ordinals are G-equivalent if and only if they are H-equivalent. This of course does not imply that two ordinals are G,-equivalent if and only if they are H,-equivalent-as one can see from Theorems 6.18 and 15.10. If two ordinals are not H-equivalent, and therefore also not G-equivalent, the least n for which they are not H,-equivalent will typically be much smaller than the least n for which they are not G,-equivalent. This is, as we will see, a reflection of the fact that the expressive power of the weak secondorder language is greater than the expressive power of the first-order language. The result above also should not suggest that whenever two linear orderings are G-equivalent they are also H-equivalent. We have seen, for example, that o + 5 is G-equivalent to o,but that they are not even H ,-equivalent. Consider now the number of equivalence classes with respect to H,equivalence and with respect to H-equivalence. The collection of linear orderings presented to show that there is a continuum number of G-equivalence classes (Theorem 6.15) provides the same result for H-equivalence. However, it also provides the following result. THEOREM 15.12 : There are continuum-many H,-equivalence classes.

A

Determine the number of HI-equivalence classes and the number of H,-equivalence classes.

EXERCISE 15.13:

A consequence of Corollary 15.11 is the following result, which is parallel to Theorem 6.22.

Let ON denote the collection of all ordinal numbers, with the usual ordering. Then O N is H-equiualent to om. A

THEOREM 15.14:

Consider next the structure (ON, +) consisting of the collection of all ordinal numbers endowed with a ternary relation { ( a , P, y) 1 a + /3 = y ) .

376

15.


Given any ordinal of the form ma, the sum of any two smaller ordinals is again a smaller ordinal, so that (wa, +) is a structure of the same type as (ON, +). For what a, if any, is (ma, +) H-equivalent to (ON, +)? THEOREM 15.15: (Ehrenfeucht) (wow, + ) isH-equivalent to(ON,

+).

Proof: We will show, for each n. that PLAYER 11 has the winning strategy in the game H,((ON, +), (ww"',+)). Recall that PLAYER 11 has won a play of this game if the following is correct-if for each t, 1 I t I n, A , consisting of a,(O) < a,(l) < . . . < a,(m,) is the subset of O N chosen at turn t and B, consisting of b,(O) < b,(l) < . . . < b,(m,) is the subset of w"'" chosen at turn t, then a,(i)

+ a,(j) = a,(k)

if and only if

b,(i)

+ b,(j) = b,(k)

for each r, s, t between 1 and n, and for each i, j , k, where 0 I i I m,, 0 i j 5 m,,O I k I m,. Now PLAYER 11's strategy is easy to imagine but hard to describe. He pretends that he's playing the game H,,(ON, w"'),in which he has the winning strategy (by Theorem 15.14), and that he is responding, at turn r, not to PLAYER 1's actual move C, (contained in ON or (u"'"') in the game H,((ON, +), (so'", +)) but to PLAYER I'S imagined move C,' (contained in in the game H,(ON, Q"). PLAYER 11 constructs the imaginary move O N or om) C,' from C, by writing each element ai E C, in Cantor Normal Form a. = Q P ' ~ . nil

+ wpiz . ni2 + . . . + w f l i k i . niki

and then defining

C,' = { B i j ( a Ei c,, 1 I j I ki). Note that if C, G woo, then C,' E (I)'". If his winning strategy in H,(ON, a") suggests the response

D,'= {ijijlaiE c,, 1 I j I ki}, then PLAYER 11 surreptitiously records D,' as his response to C,' in the imaginary game H,(ON,w"); what he tells PLAYER I is that his response to C, in the game H,((ON, + ), (wow, + )) is the set D,,where D, consists of all yi =

. nil + ws12 . ni2 + . . . + a 6 i k , . %k,

(one yi corresponding to each ai E C,). We now have to show that PLAYER 11 has succeeded-so assume that r l , a 2 ,and a3 are three elements of ON (or of wow)selected at various stages and that yl, y 2 , and y 3 are the three elements of wWo(or of O N ) corresponding

1.

311

THE GENERALIZED EHRENFEUCHT GAMES

to them. Then ai= wfli' . a,, + wfllz . n,, + . . . + g f l f k , . n i k ,

and yi

=

. n.11 + ,-,,&z

. niz

+

' * '

+

&)'Ikt

'

nik,

for i = 1,2, 3. Moreover, since PLAYER II has surreptitiously won in the game H,(ON, ww),Pij < Pi.j. if and only if dij < diPjr(where 1 I i, i' I 3, and 1 2 j I k i , and 1 < j , I k,.). In adding a , + cx2, we get wfl11.

where w'll

n , , + wD1z. n12 + . . . + ~

B,,

2

. n,,

I J y lIl g~+.w821

PZl and bl(p+l)< Bzl.But then in adding y1 + y 2 , we get

+

~ ' 1 2 .

n,,

+ ...+

w61,.

. nlP +

w'21

since 6,, 2 6,, and d l ( p + l ) < hz1. Thus cxl P1

+ Pz

=

. nZ1 + . . . + wBzkz. n2k2, . nzl

+ cx2

+ .. + *

= cxj

w'2k~

. nZk2

if and only if

P3.

Hence PLAYER II has the winning strategy in H,,((ON, +), (woo, +)) for every n, so that (ON, is H-equivalent to (woo, +).

+)

The reader should verify that the same proof shows that (awu', r). We define G E , k : y ( F n , k+ ) p ( F n . k ) inductively as follows for all k 2 r : G:,k(t)

=

{4"/4

t};

G i + l , k ( f )= { G i , k + l ( S ) I S E

t}.

Given a sequence c = ( c l ,c 2 , . . . ,c k ) we let c" = ( c , ( ~ c) , ( ~ ).,. . , c , ( k ) ) .

In the next lemma we consider how tn,k+l(A, c * A) can be obtained . t 5 Fn,k,we define F:k(t) inductively as follows: from f n , k ( A , c )Given

+

u { E ( o k + l ) } u [ L 1 k + l U j l 1 r j 5 k 1) u ( v j z Uk+,lE(uj)Ef); F , + , l . k ( t ) = ( G i , k + 2 ( F , & + 1 ( S ) ) I S E t ) ,where n transposes k FO+.k(f) = f

Note that F i k ( t ) c Fn,k+ for all n and k.

+ 1 and k + 2.

394

15.


Similarly, we need to know how t n , k ( A , c ) can be obtained from ,(A, c * el. Given r c F,,,k+ we define F&(t) inductively as follows: F&(t) = t

- (atomic formulas involving u

F i + l , k ( t )= ( F i , k + l ( G : , k + Z ( S ) ) / s E t > where

Note that F;+

l.k(f)

LEMMA 15.42:

G

n transposes k

+ 1 and k + 2.

Fn&) for all n and k.

tn,k(A,C)

= Fn-.k(t,.k+l(A,c

These operators allow us also to obtain undesired information. LEMMA 15.43: HAL, + ,(A))=

~ + ~ } ;

Let H,(t) =

u

* e ) ) for

t,($

every e E .9"(A).

from t,+ l(A), eliminating

(F;.,(s)Js E t } for all t

c F,+ l,o, Then

Proof:

+n,l

We now define x!(t), which will consist of all sums s1 + n , l s2 +n,l . . . sh of h elements oft, where t c F,,,,,. We proceed by induction: x;ct, = t ; C",l(t)=C!(l) u { r + n , l s I r ~ C h n ( t ) , ~ ~ t ) .

If t C F,+l,o,then x;(t)E xf(t)E . . . E Fncl,,,which is finite, so that for some rn, x y ' ( t )= z;(t)for all rn' 2 rn. For such an rn, U(x!(t)Ih < w } = xT(t).If we denote this x:(t)by x n ( t ) then , is the set of all finite sums s1 s 2 + n , l . . . + n , l s h , where each siE t. Finally, define w, and wn* inductively as follows for all t G F,,o:

3.

395

DECIDABILITY OF THE WEAK SECOND-ORDER THEORY

LEMMA 15.44:

n n . o)= o,(t,(&)) and t,(a . a*)= o,*(t,(E)).

t,(a

Proof : We prove this for w, by induction on n. The result for n = 0 is clear. Furthermore, suppose that we write a . o as A = c { A i l i < w ) , where each Ai has order type a. Then l#1+1(~= ) {tn,l(A7)IcEiP”(A)}

. + n , l t n , l ( A h , (c n ~ h < oj7A ) l c ~P Y A ) ) by Lemma 15.39, noting that c E A , u . . . u Ah for some h. Now = {tn,l(A,, F ; O ( ~ , ( { K ( t ) l f

E

Rj))).

I f F is a finite set of order types, then

LA@)) = an( jtn(A)I A E ~ 1 ) . Proof: By induction on n. The case for n = 0 is clear. Furthermore, I \ n t , + , ( a ( F ) ) = ( t f l , l ( a ( F )()c,) ] c E Y”la(F))). Since each such c is finite, we

396

15.


THEOREM 15.46: There is Lit1 effective procedure that, given will determine whether or not t = t,(A) for some A E A.

t G

F,,o,

Proof : We define an ascending sequence of subsets of F,,o by induction on n: R,' = {tn(I))

Rh+l ,

=

R,h u {s +n,o t l s , t E R ; } u fun(t),Wn*(t)JtER,hf u {on(R)IRG R ; } .

Then for some m, R;' = R," for all m' 2 m, so that U{R,,"Ih < w } = R," for some m. By Lemmas 15.39,15.44, and 15.45, R," = {tn(A)IA E A } .Thus we can effectivelyfind { [ , ( A )I A E A'} by calculating each R,," until we find one that equals its predecessor. H COROLLARY 15.47 : (Lauchli) The weak second-order theory of linear orderings is decidable. H

54. THE MONADIC SECOND-ORDER THEORY OF LINEAR ORDERINGS

In recent years a number of people have studied the monadic secondorder theory of various linear orderings and various classes of linear orderings. (Recall that in a monadic second-order language all predicate variables are set variables.) The monadic second-order theory of a linear ordering A, denoted M(A),is the set of all statements of the monadic second-

4.

THE MONADIC SECOND-ORDER THEORY OF LINEAR ORDERINGS

397

order language of linear orderings that are true in A . Similarly, if d is a collection of linear orderings, then the monadic second-order theory of d, denoted M ( d ) , is the set of all statements that are true in all structures of d . We will see that the properties of M(.d) often depend on which model of set theory we are working in. The earliest results, dealing with decidability, are those of Biichi, whose proofs use automata theory. He shows that M(o) is decidable [I], that M ( o , ) is decidable, that M(countab1e ordinals) is decidable, and that M(ordina1s below 0 2 )is decidable [2]. Rabin [16], also using automata theory, showed that M(q) is decidable, from which it follows easily that M(countab1e linear orderings) is decidable. (That the decidability of M ( o ) and of M ( u , ) do not require the axiom of choice was shown, respectively, by Siefkes [18] and Litman [lS].) By contrast, Gurevich and Shelah [13] show that M(A) is undecidable [by interpreting true first-order arithmetic in M(IZ)]. From this it follows (as observed by Litman) that M(al1 linear orderings) is undecidable (see Exercise 15.48.3). It should be noted that M(a1l linear orderings) is ambiguous; more precisely, whether a statement is in this set may depend on the model of set theory under consideration. Shelah [ 171 pointed out that there actually is a statement 0 in the monadic second-order language of linear orderings such that the truth of Q, in o2is independent of Zermelo-Fraenkel set theory with the axiom of choice (ZFC). Since o2can be defined by a statement Y in the monadic second-order language of linear orderings (Exercise 15.48.2), this means that the question whether (Y + @) E M(al1 linear orderings)

is independent of ZFC. Thus M(al1 linear orderings), and also M(al1 scattered linear orderings) and M(al1 well-orderings), is not completely determined without additional stronger assumptions; so it is to be expected that results about them may depend on which additional assumptions are made. (Shelah’s original proof [ 171 that M(al1 linear orderings) is undecidable assumed the continuum hypothesis.) Shelah has conjectured that M(o,) is decidable if V = L is assumed and that, with the additional assumption that there is no weakly compact cardinal number, M(al1 well-ordering) is also decidable. Shelah’s article [17] took the subject in a different direction. Modeltheoretic both in technique (continuing Lauchli [14]) and subject matter, Shelah considers questions such as categoricity and axiomatizability as well as decidability-and in the process re-proves most of the results of Biichi and Rabin. Gurevich’s work, some in collaboration with Shelah, continues in this direction. We summarize only some of their results here, leaving it to the reader to explore the many interesting questions raised in their work (see Gurevich [8,9, 10, 111 and Gurevich and Shelah [12]).

398

15. THE SECOND-ORDER

THEORIES OF LINEAR ORDERINGS

One question raised by Shelah is which linear orderings can be completely characterized by their monadic second-order properties. Shelah observes, by modifying Lauchli's argument, that any monadic statement true in some countable linear ordering is also true in some N E A.Thus the only countable linear orderings that can be monadically KO-categorical are those in A. Conversely, it is easily shown that any N E A is actually monadically KO-categorical (Exercise 15.48.4). Similarly, the countable linear orderings for which there is a single statement whose only countable model is the given one are precisely the linear orderings in A. An interesting observation of Litman [15] is that w1 is finitely axiomatizable and categorical (Exercise 15.48.2). It follows from this that the Lowenheim-Skolem Theorem fails for monadic second-order languages; more specifically,the result cited above, that any statement true in a countable linear ordering is also true in some N E 4,cannot be strengthened. Which of the structures in Jllf are actually monadically categorical? On the one hand, if A E ,Aand is scattered, then A is completely determined by its monadic theory (Exercise 15.48.5).On the other hand, q is not monadically categorical. Shelah [17] shows, in fact, that any uncountable dense linear ordering A without endpoints that is short (wl 6 A and wl* -$ A ) and in which no uncountable subordering of 3, can be embedded is monadically equivalent to q. (Such linear orderings can be obtained from Aronszajn trees.) There are also short linear orderings in which are embedded uncountable suborderings of A that are monadically equivalent to q. Gurevich [101gives a characterization of the models of M(q) by giving an axiomatization of M(q). The axioms say that besides being a short dense linear ordering without endpoints, q is modest; this property consists of an infinite collection of monadically definable properties called n-modesty. (Complete definitions of these notions can be found in Gurevich and Shelah [12].) The proof that every short and modest dense linear ordering without endpoints is monadically equivalent to q actually gives a kind of elimination of quantifiers; in particular, for each sentence in the monadic language of linear orderings, there is an n such that n-modesty decides the sentence in the presence of the axioms for short dense linear orderings without endpoints. This fact, together with the consequence of the continuum hypothesis that for each n there are n-modest short dense linear orderings without endpoints that are not (n + 1)-modest,enabled Gurevich to conclude that, in the presence of the continuum hypothesis, the monadic theory of q is not finitely axiomatizable, thereby disproving a conjecture of Shelah. Gurevich also uses the notion of modesty to obtain various new decidability results. That M(q) is decidable was obtained earlier by Rabin. Since any short and modest linear ordering can be embedded in a model of M(q), it follows that M(al1 modest short linear orderings) is decidable. On the other

REFERENCES

399

hand, if A is short but not modest--and hence fails to be n-modest for some n-then, using the continuum hypothesis, Gurevich and Shelah show that the proof that M(L) is undecidable can be imitated to show that M(A) is undecidable. Thus M(A) is undecidable for each non-modest short linear ordering but M(al1 modest short linear orderings) is decidable. In proving these results concerning modesty, a generalization of the techniques introduced by Lauchli (see 53 above) and by Shelah [17] is used. Shelah also makes a number of conjectures concerning the finite axiomatizability and categoricity of M(A).Gurevich [111shows that with the strong set-theoretic assumption of I/ = L , M ( L ) is both finitely axiomatizable and categorical in the monadic second-order theory of linear orderings. (He notes that stronger assumptions seem necessary in order to obtain this result and conjectures that, even assuming the axiom of choice and the continuum hypotheses, it is consistent that there exists a linear ordering that is monadically equivalent, but not isomorphic, to A.) We observed above that a countable ordinal CI is characterized by a monadic statement ifand only if c( < to‘”.What ifstatements of the full secondorder language of linear orderings are permitted? Garland [ 7 ] determines which countable ordinals are second-order characterizable ; in particular, every constructible ordinal (see Chapter 16) is second-order characterizable. EXERCISE 15.48: (1) Show that every ordinal in 0, then s can be thought of as thus if s = 2s03s1 encoding (so,sl, . . . , s,- l). We define the functiong(s) = Ih(s),thelength of s, to be n if s is as above. Thus Ih(f(n)) I n. This function is also recursive. The set of recursive functions is closed under composition and under various “recursive” definitions-for example, if h is recursive and the function f satisfies the identities f(n) = h(f(n)), then f is also recursive, for intuitively the given algorithm for h can be converted into an algorithm that will effectively compute new values for f from old ones. The set of recursive relations is closed under union, intersection, and complementation. If one thinks of a relation as a “predicate” rather than as a

nxcn

1.

INTRODUCTION

403

set of k-tuples, then what we have just said is that the recursive predicates are closed under disjunction, conjunction, and negation. Also, if R(x,y) is a recursive predicate, then the predicate (3x), 1, then e < k and n < k.) Then, at stage k = ( e , n ) , we work on requirement e a little more than before; that is, we carry out n steps of the computation of 4,(e). Before proceeding, the reader should review the notation 4e"(~) introduced in $4

+

+

+

438

16.

LINEAR OFWERINGS AND RECURSION THEORY

and the discussion of it there. [Note also that in the expression (p/(x), x is viewed as an element of Q.] Formally, the construction proceeds as follows: At stage k, where k = ( e , n ) , determine whether $ J / ( e )is newly defined (that is, 4:(e) is defined but $J:-'(e)is not). If 4 / ( e ) is newly defined and equals e + 1, find the first rational number r, in the standard enumeration, with t 2 k, that is between e and e + 1 but has not yet been put into 2 and put it into A. If 4 / ( e ) is not defined, or is not newly defined, or is newly defined and is unequal to e + 1, nothing need be done at this stage to satisfy the eth requirement. Finally, put r, into 2, unless it is already in A, in order to guarantee that A will be recursive. The reader should verify that this construction yields a recursive subset A of Q of order type ( that satisfies all the requirements. Note that Theorem 16.40 provides a counterexample to the effective version of the combinatorial fact that any linear ordering of order type ( has an automorphism mapping each element to its successor. EXERCISE 16.41: (1) Modify the proof above to obtain, for each natural number k, a recursive subset A , of Q of order type ( such that there is no partial recursive function that includes A , in its domain and maps each element of A , to its kth successor. (2) Similarly, obtain a recursive subset A , of Q of order type ( such that there is no partial recursive function that includes A , in its domain and for some i Ik maps each element of A , to its ith successor.

57. EFFECTIVE VERSIONS OF COMBINATORIAL THEOREMS If A is a recursive subset of Q of order type (, then there always is a Ill-automorphism f of A ; that is, there is a Ill-binary relation R(x, y) such that for each x E A there is a unique y E A satisfying R(x, y). Indeed, such an automorphism is obtained from the n,-binary relation x

E

AAYEA

AX

< y ~ ( V z ) ( x< z < y

+z

4 A).

Must there always be a recursive automorphism of A? DEFINITION 16.42: Let A c Q. We say that A is recursively rigid if there is no partial recursive order-preserving function that maps A onto A and is not the identity on A.

7.

EFFECTIVE VERSIONS OF COMBINATORIAL THEOREMS

439

THEOREM 16.43: There is a recursive subset A of Q of order type ( that is recursively rigid. Proof : As in the proof of Theorem 16.40, the recursive subset A of Q to be constructed will include all the integers. To guarantee that 4e is not an automorphism of A , and thereby that requirement e is satisfied, we will compute +e(e) and 4e(e + l), and if these appear to be successive elements of A, then we will put some elements of the interval (e,e 1) into A. More precisely, let A' consist of the integers, and, in general, let Ak denote the elements of Q put into A prior to stage k. At stage k of the construction, where k = (e, n), if +e"(e)and c$e"(e + 1)are both defined, 4e"(e 1) is the successor of 4e"(e)in Ak, and no elements of the interval (e,e 1) are in Ak, then we find the first e rational numbers that are in (e,e + 1) and not among {riI i I k } and place them in A . Otherwise we go on to stage k + 1. In any case, if rk is not already in A , we place it in 2. The subset A of Q that is thus constructed is recursive, as usual, since rk E A if and only if rk E Ak. It has order type since finitely many points are added in each interval (e,e + 1). Finally, 4e does not define an automorphism of A . For if +e(e) and de(e 1) are both defined and in A, then there is a least stage k' = ( e , n ) such that +:(e) and 4:(e + 1) are both defined and in Ak'.Note that the minimality condition on k' implies that Ak' contains no points of (e,e + 1). Now if $:(e + 1) is not the successor of 4@"(e)in Ak', then it will not be its successor in any subsequent Ak, so no points of ( e , e + 1) will ever be put into A ; since 4e cannot map any point of A to the points of A between 4e(e) and 4e(e + l), 4e will not define an automorphism of A. If, on the other hand, $e"(e + 1) is the successor of q5/(e) in Ak', then at stage k' we put e rational numbers of (e,e + 1) into A. Since no other points of this interval are ever put into A, in order to conclude that 4e does not define an automorphism of A, it suffices to show that the interval (4e(e),4 e ( e + 1)) does not contain exactly e points of A. There are three possibilities. If 4e(e) = e, then 4e(e + 1) = e + 1, so that if 4e is an automorphism of A, it must be the identity on A. If 4e(e)= m and c$e(e + 1) = + 1 for some natural number rn # e, then the number of points of (4e(e),4Je + 1)) in A is either 0 or vn, so that 4e cannot define an automorphism of A . Finally, if either +e(e) or de(e + 1) is not a natural number, then no points of (4e(e),+=(e + 1))will ever be put into A, so that again 4e will not be an automorphism of A .

+

+ +

+

Thus the effective version of the statement If A is a linear ordering of order type 5, then there is an automorphism of A .

440

16.

LINEAR ORDERINGS AND RECURSION THEORY

is

If A is a recursive subset of Q of order type [, then there is a Ill-automorphism of A . Thus the automorphism of a linear ordering of order type 5 is in general one level more complicated than the linear ordering in the arithmetical hierarchy. Let us return to another question that we raised in 45. Given a recursive subset A of Q such that q 3 A , is there always a recursive subset B of A of order type q? We saw that if A has order type 2 q, then there is such a subset (Exercise 16.38). If, however, A has order type 5 . q, there may not be one.

-

THEOREM 16.44: There is a recursive subset A of Q of order type 5 . q that has no recursively enumerable subset of order type q. Proof: We first explain how we will build A so that it has the right order type and then indicate what measures are to be taken so that it will have the desired property. At the beginning of stage k, Ak will consist of a finite number of elements, partitioned into intervals of Ak called blocks. At each stage, we will add an element at each end of each block, and no elements will ever be added between two elements of a block. Thus, in the course of time, each block will grow into a [. Also, at each stage, we will start a new block between any two consecutive old blocks, and new blocks at the extreme right and left. Thus between any two [ in A , there will be others, and A will have order type [ . q. To guarantee that Weis not a subset of A of order type q, we will generate We,and once we find that there are elements of We in two different blocks, then we will merge the two blocks into one block (together with all intervening blocks); thus the combined blocks will contain at least two elements of We. Since the same is true of the [ into which it grows, it follows that We will not be a subset of A of order type q. We cannot, however, proceed so simply, for it is easy to imagine that if we apply this technique indiscriminately, we will merge all the blocks and end up with a linear ordering of order type 5. Thus we must exercise a little restraint. We do this by assigning a number to each block and by combining two blocks on behalf of We only if they, and all intermediate blocks, are assigned numbers larger than e ; this will guarantee that, after some point in the construction, distinct blocks with numbers less than e will never be combined. Thus we assume that at the beginning of stage k, Ak consists of a finite number of blocks to each of which has been assigned some number t < k . If k = ( e , n), we perform n steps in the enumeration of the recursively enu-

7.


441

merable set We.If some element enumerated in We is not already in A , we put it in 2 and satisfy requirement e in that way. If two elements of We are in the same block, we need not do anything. If there are two elements a and b in different blocks, and if these blocks, and all the intervening blocks, are assigned numbers bigger than e, then we combine these blocks into one block to which we assign the smallest number assigned to any of the component blocks. We then add an element at each end of each block and we start new blocks between any two consecutive old blocks and at the extreme right and left. All new blocks are assigned number k and are begun with k elements. Finally, if rk is not already in A, we place it in A. Now let us examine what happens to a particular block in the course of the construction. If it starts out with number t, it may be combined with some other blocks in order to satisfy a requirement e < t and then perhaps assigned a lower number, but since each requirement e is satisfied only once, this particular block will eventually be left alone to expand in peace. Thus from some point on, it will be assigned the number t’, and will never again expand by merging; let us then assign the number t‘ to the copy of 5 in A into which it grows. To show that A consists of q copies of [, suppose that Z , and Z , are two copies of 5 in A to which have been assigned the numbers t , and t , , respectively. For each e, either requirement e is never satisfied or requirement e is satisfied at some stage k(e). Thus if k = max { k(e)1 e I min(t,, t,)}, then after stage k, the blocks between those growing into Z , and Z , will grow into 5’s that will be wholly between Z , and Z , . Similarly, at the right and left of any 5 in A, there will be other 5‘s in A . Hence A has order type 5 . q. Finally, if We L A has order type 3, then there is at most one element of We in each 5 of A . In particular, since there are only a finite number of 5’s that are assigned numbers up to e, we can find two elements of We in different 5‘s such that those cs, and all 5’s between them, are assigned numbers bigger than e. Choose n so that the two elements of We have appeared after n steps of the enumeration of We and let k = ( e , n ) . Then at stage k the two blocks, which grew into the two separate Cs, should have been combined. Since they were not, the assumptions about We are invalid and We cannot be a subset of A of order type q. What then is the effective version of the statement If A is a linear ordering of order type 5 . q, then A has a subset B of order type q. According to our earlier discussion, it should have the following form: If A is a recursive subset of Q of order type 5 . q, then A has a H subset B of order type q.

442

16.


We know that if H is IIz,then the statement is correct, whereas if is A, or C , , then the statement is incorrect. But what if H is ll,, Az,or Zz? There are at least the following possibilities. (1) The correct effective version is with H being 112; that is, there is a recursive subset A of Q of order type [ . q that has no &-subset of order type 11. (2) The correct effective version is with being n,;Theorem 16.44 says that no further improvement is possible. (3) The correct effective version is with being A2; that is, there is a recursive subset A of Q of order type [ * v that has no II,-subset and no &subset of order type q. If we do not insist that H be ll, , ll,, or Az, then there are still more possibilities, and the reader is invited to try his hand at determining the best possible conclusion about recursive subsets of Q of order type 5 . q ; the answer can be found in Lerman and Rosenstein [12]. What happens if A is an arbitrary non-scattered recursive subset of Q? The upper bound of llz given for orderings of order type ( . q in $5 no longer works; the lower bound of ll, given by Theorem 16.44, on the other hand, cannot be improved on. That is another challenge for the reader. What happens if A is a dense sum of finite linear orderings? This question is investigated extensively in Watnick [181. This situation is typical. In attempting to find the effective version of a combinatorial statement, we often find that one argument leads to an upper bound and another argument leads to a lower bound; the problem is to close the gap. In the situation of Theorem 16.43, involving automorphisms of C, the gap is closed; in the situation of Theorem 16.44, involving dense subsets of [ . q, it is not. Returning to questions of recursive rigidity, we recall that, although Theorem 16.43 provides a recursively rigid subset A of Q of order type 6, from Exercise 16.4.4 we see that if A is a recursive subset of Q of order type v, then A is not recursively rigid. What if A has order type 2 . q? Although in view of Exercise 16.38 we might expect such orderings to have recursive automorphisms, we will now see that this may not always be the case.

THEOREM 16.45: There is a recursive subset A of Q of order type 2 . q that i s recursively rigid. Proof: The recursive subset A of Q will be constructed in stages. At the beginning of stage k, we assume that Ak consists oft pairs of rational numbers including one pair of rational numbers for each e < k ; that is, Ak contains 2t rational numbers a,, < a , < a2 . . . < alt- divided into t pairs

-=

,,

7.


443

(aZi,ali+I } , such that for each e < k one of these pairs is said to be associated with e, no pair being associated with two different e's. At stage k we first add t + 1 new pairs of rational numbers to A, one pair to the left of a,,, one pair to the right of a 2 t - I , and one pair between each two consecutive pairs of Ak;the first of these pairs is associated with k. To guarantee that 4, is not an automorphism of A, we will compute be(a) and +,(b), where {a, b} is the pair associated with e, and we will make certain that de(a)and $,(b) are not consecutive elements in A. Thus at stage k = ( e , n ) we compute 4:(a) and 4,"(b), and if both are defined, and if 4 / ( a ) < 4,"(b) constitute one of the pairs of Ak, we add to A two elements between them, pairing the leftmost with + / ( a ) and the rightmost with 4,"(b). Then if we avoid putting any elements between a and b into A , our eth requirement will be satisfied. But it is not so easy to avoid putting any elements between a and b into A . For one thing, it is possible that 4,"(a) = a and 4,"(b) = b, in which case it is impossible to satisfy requirement e by considering only a and b. In fact, it is possible that qhe(ai)= ui and $,(bi) = bi for infinitely many pairs {ui,bi) of A without 4e being the identity. If, however, 6Jai) = ai and $,(bi) = bi for a set of pairs { a i , b i )of A that is dense in the set of all pairs of A , then $e would have to be the identity on A. This suggests that at stage k we add ( k + 1) . ( t + 1)new pairs of rational numbers to A , k + 1 of them in each of the t + 1 positions, and associate one of each of the k + 1 pairs with each e I k . Then at stage k = ( e , n ) , we compute and 4,"(b) for each pair {a,b} associated with e, and if for some such pair { a , b } both 4 / ( a ) and 4,"(b) are defined, and if 4;(a) < +,"(b) constitutes one of the pairs of Ak other than the pair {a,b}, then we add to A two elements between them, pairing the leftmost with 4/(a) and the rightmost with $,"(h). Then if we avoid putting any elements between a and b into A , requirement e will be satisfied. Still, it is not easy to avoid putting any elements between a and b into A. For at some later stage k' = ( e ' , n ' ) of the construction we may have an opportunity to satisfy the requirement e'. That is, we may find that for some {a', b'} associated with e', &:(a') and $$(b') are defined and constitute one of the pairs of Ak' other than {a',b'},in which case we would break them up by putting elements between them into A. That would be fine, except that if +:@) = a and &@') = b, then we will have put elements between a and b into A , which is exactly what we want to avoid. Even worse, it is conceivable that every time an opportunity arises to satisfy requirement e', it would have to be done by adding to A elements of an interval that we must avoid in order to satisfy some other requirement; thus requirement e' would never be satisfied and our construction would not yield the desired result.

444

16.


This difficulty is circumvented by assigning higher priority to lowernumbered requirements. That is, if an opportunity arises to satisfy a requirement e’ (which is not already satisfied) but doing so would involve injuring what has already been done to satisfy another requirement e“, then we go ahead and satisfy requirement e’ only if err> e’, that is, only if e’ has higher priority than err.In the construction below we will talk of “requirement e being currently satisfied at stage k,” meaning that at some stage t I k, requirement e was satisfied in the fashion described above, that we agreed to avoid putting elements of a certain interval into A on account of requirement e, and that at no subsequent stage t’, for t I t’ Ik, was requirement e injured. That is, no elements of the interval proscribed on account of e were put into A at stage t’. Now if requirement e is satisfied at some stage k and thereafter is never injured-so we can say that it is permanently satisfied at stage k-then it is clear from the discussion above that 4ecannot be an order-preserving map from A to itself. Note that although, because of the effectiveness of the construction, we can tell whether requirement e is satisfied at stage k, we can never be sure whether it is permanently satisfied at stage k, since it is always possible that it will be injured at some later stage due to some higher-priority requirement. Note also that some requirements may never be satisfied at all, or may never be permanently satisfied-because it is conceivable that no opportunity, or very few opportunities, present themselves for satisfying the requirement. (We hope to show, of course, that such a requirement does not need to be satisfied anyway.) What cannot happen, however, is that a requirement is satisfied and then injured infinitely many times. For it is easy to see, by induction, that each requirement can be injured only finitely many times. Indeed, assuming that each requirement e‘ < e is injured only finitely many times, there is a stage k (which we cannot, in general, find effectively) after which no requirement e’ < e will ever again be injured and another stage k(e) > k such that every requirement e’ < e that will ever be satisfied at some stage t 2 k is satisfied at stage k(e); but after stage k(e) there will be no possible reason ever again to injure requirement e since all requirements that will ever be satisfied after stage k(e) have lower priority than requirement e. (Note that, in our construction, if requirement e is already satisfied at stage k, then we will not, at stage k, take any further steps to satisfy it.) There is yet another problem to consider. Suppose that the pair { c , d } is broken up on account of e and that c is then paired with d’; at some later stage (c,d’} may be broken up on account of e’ and c would then be paired with d “ . If we permitted this to happen infinitely often, c would not be paired with any element of A and hence A would not have order type 2 . q. To guarantee that A has order type 2 . q, we must arrange it so that for any

7.


445

c E A there is a stage k after which l ' is permanently paired with an element d E A . This is done by requiring that we keep track of broken pairs and only permit another breakup if it is caused by a higher-priority requirement. We can now specify how A is constructed. At the beginning of stage k, we assume that Ak consists o f t pairs of rational numbers, to each one of which is assigned a unique u < k. For certain u < k, requirement u is currently satisfied at the beginning of stage k and an interval (a,b) is to be avoided on account of u. Furthermore, for certain u there will be specified pairs that have broken up over u. At stage k we first add ( k + 1 ) . ( t 1) new pairs of rational numbers to A , k + 1 of them in each of the t 1 positions, and associate one of each of the k + 1 pairs with each e I k. Then for k = ( e , n ) , if requirement e is not currently satisfied, we compute 4,"(a) and $I,"@) for each pair {a,b} associated with e; if, for some such pair {a,b}, both $,"(a) and 4,"(b) are defined, and if 4,"(a) < $,"(b) constitute one of the pairs of Ak other than {a,b} and the interval ($/(a), d,"(b))is not being avoided on account of some e' < e and has not been broken up over some e' < e, then we add to A two elements between them, pairing the leftmost with #,"(a) and the rightmost with $,"(b) and associating both pairs with whichever e' the pair { 4e(a),4,(b)} was previously associated; both new pairs are now said to be broken up over e. The interval (a,b)is now to be avoided on account of e, and if ($/(a), $,"(b)) was to be avoided on account of some e' > e, then e' is no longer satisfied and the interval is no longer to be avoided. Finally, if r, is not already in A, we place it in A. To verify that the recursive subset A of Q that we have constructed satisfies the theorem, we must check that each of the following statements is correct :

+ +

(1) Each requirement can be injured only finitely many times. Hence for each e there is a stage k of the construction such that either requirement e is permanently satisfied at stage k or requirement e will never again be satisfied after stage k. Given any c E A, there is a stage k after which c is permanently with an element d E A . Hence A has order type 2 . 1 . For each e there are a finite number of pairs that have been broken up over e. (4) If requirement e is permanently satisfied, then $e cannot define an automorphism of A.

The verification of each of these facts is implicit in the discussion above. Now suppose that, for some e, $e does define an automorphism of A . Choose a stage k , such that, for each e' < e, either requirement e' is permanently satisfied at stage k, or requirement e' is never again satisfied

446

16.


after stage k, . Then, if requirement e is ever satisfied after stage k , , it would be permanently satisfied and hence could not be an automorphism of A , contrary to assumption. Thus requirement e is never satisfied after stage s. At stage s there are altogether a finite number of pairs {c,d 1 that have been broken up over some e' < e or are being avoided on account of some e' < e; no new pairs of this type will arise after stage s. Suppose that {a, b} is a pair that is associated with e, is never again broken up, and {4,(a),4,(b)} is not one of the proscribed pairs { c , d } listed above; then, choosing an n such that ( e , n ) > k, and 4,"(a) and $,"(b) are defined, the only reason for not satisfying requirement at stage ( e , n ) is that &,(a) = a and 4,(b) = b. But the set of such pairs {a, b} is dense in A so that 4, must be the identity on A . Hence the only automorphism of A is the identity and the proof is complete. W This type of diagonal argument is known as a priority argument. It was invented independently by Friedberg and Muchnik to solve Post's problem. The reader should consult Rogers [15] for further discussion of priority arguments. Theorem 16.45 says that if we have a recursive subset A of Q of order type 2.11, the simplest automorphism of A we can expect to find, in general, is a HI-automorphism. On the other hand, it is possible to show that there is always a A,-automorphism of A. To do this, we observe that an automorphism can easily be constructed if we could distinguish first elements of pairs from second elements of pairs. Since the set of first elements is a A,-set, this means that an automorphism can be constructed recursively in a A,-set. Hence, by an argument using a little more recursion theory than what we have developed here, there is a A,-automorphism of A . Thus in this problem we are left with a gap between II, and A2. EXERCISE 16.46: Show that given any infinite recursive subset A of Q there is a A,-map that is an embedding of A into itself but is not the identity. [Hint: Use Theorem 4.6 and the fact that any map constructed recursively in a H,-set is A 3 .] Theorems 16.43 and 16.45 show that there are recursive linear orderings that have many automorphisms but no recursive automorphisms. Our original question dealt not with automorphisms but with embeddings. Given a recursive ordering, is there a recursive map of it into itself? DEFINITION 16.47: Let A be a recursive subset of Q. We say that A is recursively embeddable in itself if there is a recursive function that is an order-preserving map of A into itself other than the identity.

7.


447

Consider for example a recursive linear ordering A of order type 2 .g, such as the one just constructed. Let A = {a,, a,, a 2 ,. . . } be a recursive enumeration of the elements of .4. We will define a recursive function p such that the map a, -ap(,) is an order-preserving map from A to A . To guarantee that the map is not the identity we map a, to a l , Suppose now have been defined and that they are ordered that a,,(,), . . . ,a,,,similarly to a,,a,, . . . ,an- Figure 3 will be helpful in explaining what

,.

aP(iJ

FIGURE

3

we do with a,, : We assume that a, is between ai and a j , where i, j < n. We have to be careful about how we define up(");otherwise, up(,,)might accidentally be the successor of although a,, is not the successor of a i , in which case we will eventually be unable to continue the definition of p. Thus we assume as part of the induction hypothesis that between consecutive images we know of at least one element a, of A ; that is, there is an and mI max(p(i), p ( j ) ) such that a, is between and ap(j).If we have one element between up(i)and then we know that there are infinitely many since A has order type 2 . g. Thus other elements of A between and to define we continue enumerating A until we have found three elements of A between and a p ( j ) and , we choose up(,) to be the middle one. The induction hypothesis continues to hold, and we have thus completed the proof of the following theorem, which should be contrasted with the preceding one. THEOREM 16.48: If A is a recursive subset of Q of order type 2 . g, then A is recursively embeddable in itself. Can we prove that the same is true for every recursive subset of Q of order type The same proof cannot work since it relies heavily on the fact that if there is one element between a and b, then there are infinitely many; this is true in a linear ordering of order type 2 . g but not in a linear ordering of order type 5. As we will see, the statement of Theorem 16.48 is false for (, and, indeed, as we now show, is false for w . This surprising result is due to Louise Hay and the author.

c?

THEOREM 16.49: There is 0 recursive subset A of Q of order type w that is not recursively embeddable in itself.

448

16.


Proof: The basic idea of this construction is similar to that of Theorem 16.45.That is, to guarantee that 4edoes not embed A into itself, we construct A so that if it appears that 4, might be such an embedding, we find x,y E A so that x < y and r $ e ( ~ ) < 4,(y), and we see to it that there are more elements ) 4,(y). If that is done of A between x and y than there are between 4 e ( ~and and never again are elements between 4e(x) and 4,(y) put into A, then requirement e will indeed be satisfied. Can that always be done? Not quite always-since if 4 e ( ~I) x < y < q5e(y),then any numbers between x and y put into A are automatically between @,(x) and +,(y). However, since the set A to be constructed has order type o,the only possible order-preserving maps f satisfy x I f(x)for all x, and all non-identity order-preserving maps satisfy x < f(x) for all but a finite number of x. Thus we can afford to wait for an x and a y satisfying x < y, 4,(x) < 4,(y), and x < 4&x), and once that configuration occurs we can put m 1 elements between x and min{ y, ~ J x ) }into A , where there are m elements already in A between 4e(x)and 4,(y). (Maybe m 1 is more than we need, but there is no reason to be stingy.) Then if we avoid putting any additional elements between $=(x) and 4,(y) into A, requirement e will be satisfied. Why should there be any problem in doing so? Only because there may be an opportunity to satisfy another requirement e' by putting m' 1 elements into A between x' and which would be perfectly fine except if the interval between x' and $e.(~') is wholly within the proscribed interval between +,(x) and &e( y ) described above. Even worse, it is conceivable that every time an opportunity arises to satisfy requirement e', it would have to be done by putting elements into an interval proscribed because of some other requirement, and thus requirement e' would never be satisfied. As in the previous construction, these problems are sidestepped by permitting requirement e to be satisfied only if it can be done without injuring any requirement e' < e with higher priority. One further point about the construction. We will guarantee that A has order type o by making certain that A is infinite and that each element of A has only a finite number of predecessors. At stage k, one new element a(k)will be added to A (so that A will be infinite)and a(k)will be larger than all elements previously put into A . Also, to ensure the general rightward expansion of A, when we search for an x and y with which to satisfy requirement e, we will insist that x be to the right of a(e). [Note that since e < ( e , n ) for all n, the element a(e) will already be in A by the stage that requirement e is first considered.] Now if a E A and a is put into A at stage k, then a Ia(k). But the only elements ever put into A to the left of a(k) after stage k are put there in an attempt to satisfy some requirement e' < k. But, as in Theorem 16.45, each requirement is satisfied only a finite number of times. Thus only finitely

+

+

+

#,.(XI),

7.


449

many elements are put into A to the left of a(k) after stage k, and hence a(k) has only finitely many predecessors. The same is therefore true of any element of A . Hence A must have order type w. We are now ready to describe the construction of A . At stage 0 we put ro into A and call it a(0). At stage k = ( e , n ) , if requirement e is not currently satisfied, we look systematically among all ordered pairs (x, y) of elements of Ak until we find one such that u(e) I x < y, #,"(x) and $,"(y) are defined and in Ak, x < #,"(x) < 4,"(y), and the interval between x and 4,"(x) is not contained in any interval that is proscribed on account of some requirement e' < e. If we find such an x and y, we count the number rn of elements in Ak between #e(x) and 4e(y), and we put into A the first m + 1 rational numbers that are not yet in A and are between x and min { y, 4,(x)). The interval between .$e(x) and 4je(y) is now proscribed on account of e, and requirement e is now satisfied. If any of these elements lies in an interval that was proscribed on account of some e' > e, then that requirement e' is injured at stage s and that interval is no longer proscribed on account of e'. If we find no such ordered pair (x,y), then of course we do none of the above. Finally, we find the first rational number that is bigger than all elements of Ak, designate it a(k), and put it into A, and if rk is not already in A, then we put it into A. T h s completes the description of the construction. As in Theorem 16.45, we verify that each requirement can be injured only finitely many times; using that fact, we then verify that each element of A has only a finite number of predecessors, which, as discussed earlier, implies that A has order type w. Also, we can verify that if requirement e is permanently satisfied, then 4, cannot be an embedding of A into itself. Thus we need only show that 4, is not an embedding of A into itself unless it is the identity. Suppose then that 4eis not the identity, yet is an embedding of A into itself. Let k(e) be a stage of the construction after which no requirement e' < e is ever again satisfied; thus if e is ever satisfied after stage k(e),then it would be permanently satisfied, contrary to assumption. Choose xo so that xo 2 a(k(e))and y < #,(y) for all y 2 xo. For some n, 4 / ( x o ) and #,"(yo) will both be defined for some yo > xo. Then at stage k = ( e , n ) we will have an opportunity to satisfy requirement e without injuring any higher-priority requirement (since all proscribed intervals for e' < e lie to the left of a(k(e)).Thus we would satisfy requirement e at stage k and it would be permanently satisfied; thus cPe cannot be an embedding of A into itself. EXERCISE 16.50: Use the result above and its w* version to show that there is a recursive subset of Q of order type 5 that is not recursively embeddable into itself.

450

16.


The following corollary of Theorem 16.49 presents a striking divergence from the classical case. COROLLARY 16.51 : There are two recursive subsets of Q of order type o (or () that are not recursively isomorphic. Proof: Let A be the set of Theorem 16.49 and let B be obtained from A by deleting one of its elements.

Theorem 16.49 is reminiscent of Theorem 9.1. Perhaps more reminiscent of Chapter 9 is the following discussion. Recall that we constructed a dense subset C of the real numbers such that there is no order-preserving map that takes it into itself (except for the identity). If we require instead that C be a recursive dense subset of Q and that no recursive order-preserving map takes C into C (except for the identity), then we will be unable to find such a C by Exercise 16.4.4. It turns out that the proper recursive analogues to dense subsets of reals are lI,-dense subsets of Q. For these sets we get analogues of the theorems of Dushnik and Miller and Sierpinski, and the proofs are constructive versions of their proofs. We will describe the proof of the effective version of Dushnik and Miller’s Theorem, and leave the Sierpinski Theorems to the reader. THEOREM 16.52: There is a subset E E Q such that (1) E is dense in Q, ( 2 ) E is II,, and ( 3 ) every partial recursive function with domain including E that is order-preserving on E is the identity on E . Proof : The proof of Dushnik and Miller’s Theorem consists of enumerating a class of functions { f,I a < c), showing that it suffices to have for each a an x, E E such that fa(xa)4 E , and then actually constructing E so that these requirements are satisfied. In our case, we will use the class { $e 1 e E N } of partial recursive functions, so that to satisfy requirement e it is sufficient to put an x, in E in such a way that q5e(xe)will be kept out of E. That is what we would try to do if we wanted to make E recursively enumerable-but that we already know is impossible. Instead we will enumerate a set E whose complement will be denoted E . We will have an opportunity to satisfy requirement e when for some x, q5&) is defined and x has not yet been put into E, in which case requirement e can be satisfied by putting de(x) into E, and will be permanently satisfied if x is never put into E. Thus when we do this, we say that x is being kept

7.


45 1

out of E for e. The only reason it may be difficult to keep x out of E is if x = 4r(y) for some y which is not in E, so that to satisfy requirement f we would like to put x = g5r(y) into E. As in earlier examples, we see that a priority argument is called for, so we say that requirement f has higher priority than requirement e iff < t', and we satisfy a requirement so long as no higher-priority requirements are thereby injured. At stage k of the construction, where k = (e, n ) , if requirement e is not currently satisfied, we calculate $e"(x) for each x < n until we find an x < n such that x 4 Ek,+ / ( x ) is defined, and 4z(x) is not being kept out of E for any e' < e. If we find such an x , then we put $ / ( x ) into E (if it is not already there), we keep x out of E for e, and we dissatisfy all e' > e such that 4/(x) was being kept out of E for e'. The reader should verify that, as in the previous constructions, each requirement is injured only a finite number of times. The set E constructed is clearly recursively enumerable and hence its complement E is II,.Let + e be a partial recursive function whose domain includes E and that is orderpreserving on E ; we must show that requirement e is permanently satisfied at some stage of the construction, unless 4e is the identity on E . We first observe that if q5e is not the identity on E, then 4e moves infinitely many points; for if 4 e ( ~ # )x for some x, say x < +=(x), then, since 4e is orderpreserving, 4 e ( X ) < 4e(4e(x))< 4 e ( 4 e ( $ e ( X ) ) ) < ' . ' , SO that 4 e moves X , 4 e ( ~$e(4e(x)), ), . . . . Hence if after stage k no e" < e is ever again injured or satisfied, then after stage k there will be an opportunity to satisfy requirement e, and, at the first such opportunity, requirement e will be permanently satisfied. Thus 4e must be the identity on E . Finally, we have taken no steps in the construction above to guarantee that E will be dense. Indeed, it is possible that every element of some interval (a,6 ) is actually put into E. To counteract this, we first let e(a,b) be the index of an algorithm for the function y = x2 [1 - (a b)]x ab whosedomain is the interval (a,b), that is order-preserving on (a,b), and that differs from the identity at each point of (a,b). Then (e(a,b)la < b) = T is a recursive set of indices. We modify the construction above so that if at stage k = (e, n ) we have e E T and e is not currently satisfied at stage k , then we keep computing (more than n steps if necessary) until we actually satisfy requirement e. Since we know that this will be accomplished at some finite time for each e E T, we need not fear that the computations will not terminate. Then, arguing as before, we can show that each e E T is permanently satisfied in the construction; thus for each interval (a, b) there is an x E (a,b) such that x is kept out of E and 4 e ( ~ is put ) in E. Hence both E and E are dense in Q.

+

+

+

A more elaborate version of Theorem 16.52 can be found in Hay, Manaster, and Rosenstein [7]; further results were obtained by Remmel[14].

452

16.


EXERCISE 16.53: Theorems 9.10.

Prove the following effective versions of Sierpinski's

(1) There are Ill-sets { E i ( i 6 N } such that each is dense in Q (with dense complement) and, for each i, E, E Ei+1 , but there is no partial recursive function that maps Ei+ to Eiand is order-preserving on E i . (2) There are lT,-sets I F i ]i E N } such that each is dense in Q (with dense complement) and, for each i, Fi 2 F i + l , but there is no partial recursive function that maps F ito Fi+ and is order-preserving on Fi. ( 3 ) There are Ill-sets (Gili E N } such that each is dense in Q (with dense complement) and, for each i # j , there is no partial recursive function that maps Gi to G j and is order-preserving on Gi . (4) What is the effective version of the fact that there is a partition of 1 into A u B with A 6 A and 1-$B? (See Theorem 9.6 and the discussion of $10.1 about the non-indecomposability of 1.)

As a final example, we will prove the following theorem, due to S. Tennenbaum, which asserts the existence of a recursive subset of Q that has no recursive ascending or descending sequence. On the one hand, this can be understood as providing a counterexample to the effective version of the combinatorial fact that every infinite linear ordering contains either an o-sequence or an w*-sequence. On the other hand, one can regard this theorem as stating that there are infinite effectively finite subsets of Q. Tennenbaum proposed that the class of effectively finite subsets of Q be construed in some way as an extension of arithmetic (as have been the isols), and that an attempt be made to find concrete non-standard models of arithmetic in this context. We first observe that any recursive subset of Q that has no recursive ascending or descending sequence must have both a first element and a last element. Also any element except the last must have an immediate successor and any element except the first must have an immediate predecessor. For if any of these were false, an infinite recursive monotonic sequence could easily be defined. Thus any infinite effectively finite linear ordering must have order type w + 5 . a + o*for some order type a. Now if c( is any countable order type except for q, it is easy to verify that any recursive linear ordering of order type o + C. a + o* has a recursive subset of order type o + o*.Thus we see that every recursive o + o*, subset of Q must contain a recursive subset of order type o,o*, or o + . q o*. Can we improve on this result? The following theorem says that there is a recursive subset of Q of order type o + o* that has no recursively enumerable subset of order type o or w * . Lerman [ll] has now constructed a

+

7.


45 3

recursive subset of Q that has no recursively enumerable subset of order type w , w* or w + w * . THEOREM 16.54: (Tennenbaum) There is a recursive subset A of Q of order type w a* that has no recursive ascending or descending sequence.

+

Proof : We construct a recursive subset A of Q of order type w + w* such that no infinite recursively enumerable subset We of Q is wholly contained in either the w part or the o* part of A . Since the range of any recursive function is a recursively enumerable set, this implies that no recursive function can enumerate an infinite monotonic sequence of elements of A. We will attempt to satisfy requirement e by putting one element of We into the w part A , of A and one element of We into the w* part A , of A . This requires that at any stage k of the construction of A we have clearly specified which elements of Ak are intended to be in A , and which are intended to be in A , . We will speak of a fence at stage k that is located between the largest element intended for A , at stage k and the smallest element intended for A , at stage k . If the fence stays between two fixed elements of We for all stages k 2 k , for some k,, then requirement e is permanently satisfied. But it is possible that an opportunity arises to satisfy another requirement e‘ in that two elements on the same side of the fence turn out to be in We,;if the fence is moved so as to lie between these two elements, then both of the above elements of We may end up on the same side of the fence. Thus we must assign, as in earlier examples, higher priority to lower-numbered requirements, and allow a requirement to be satisfied so long as no higher-priority requirement is thereby injured. Since, as usual, each requirement can only be injured a finite number of times, once requirement e is permanently satisfied, the fence will remain between two fixed elements of We,so if every element added to A at stage k lies near the fence at stage k, all new elements added to A lie between these two elements of We;this implies that A has order type o w * . Formally, at stage 0 we put ro and r l into A and put the fence between r, and r l ; we also put r2 into 2. At stage k = ( e , n) we check to see whether requirement e is currently satisfied (that is, there are elements a, and a2 of We designated as being kept to the left and right, respectively, of the fence because of e), and if not we see if there are elements a, b E Wensuch that a < b and such that if a(e’)< b(e’)are currently satisfying requirement e’ < e, then a < b(e’)and a(e’) < b. (Then there is a place for the fence simultaneously between a and b and each a(e’)and h(e’) for e’ < e.) In that case, requirement e is satisfied by moving the fence so that it is halfway between the largest of {a(e‘)I e‘ < e} u { a } and the next largest element of Ak. All requirements e‘

+

454

16.


for e' > e are declared to be injured. Two elements are added to A, one immediately to the right of the fence, and one immediately to the left of the fence; finally, rk is put into A if it is not already in A . We leave the verification that this construction does what was intended to the reader. We noted above that every infinite effectively finite linear ordering must have order type w + 5 . ci + w* for some order type c1. Tennenbaum showed that there is an infinite effectively finite linear ordering corresponding to a = 0. Are there any others? Watnick showed in [18] that for any constructive order type a-this notion generalizes constructive ordinals-there is an effectively finite linear ordering of order type w i- [ * M + w * ; Watnick subsequently showed [16] that the same is true for any recursive order type CI. EXERCISE 16.55: (1) Verify that the construction above yields the desired result. (2) Show that given any recursive subset A of Q of order type w a*, there is a lT,-function f : N -+ A that defines a &-ascending sequence of elements of A . (3) Let L be a linear ordering and let f : N -P L be a sequence of distinct elements of L ; then the sequence (f(n)(n E N ) has a monotonic subsequence (compare Exercise 7.5.1). What is the effective version of this fact? (4) Show that every infinite recursive subset of Q contains a lT,-subset of order type w or w* (Manaster).

+

It seems appropriate to close with a question that involves concepts from different parts of the book: Does there exist a linear ordering A such that Th(A) has a prime model and a recursive model, but Th(A) has no recursive prime model? REFERENCES Chen, K.-H.. Recursive well-founded orderings, Ann. Math. Logic 13 (1978), 117-147. [ M R We: 030473 [ 2 ] Church, A. and Kleene, S. C . , Formal definitions in the theory of ordinal numbers, Fund. Math. 28 (1937), 11-21, [3] Davis. M.. Contputability and Unsoluability, New York: McGraw-Hill. 1958. [4] Feiner, L., Degrees of non-recursive presentability, Proc. Amer. Math. SOC.38 (1973), 621-624. [ 5 ] Feiner, L., Orderings and Boolean algebras not isomorphic to recursive ones, Thesis. M.I.T., 1967. [6] Fellner, S., Recursiveness and finite axiomatizability of linear orderings, Thesis, Rutgers University, 1976. [I]

REFERENCES

455

[71 Hay, L., Manaster, A., and Rosenstein, J. G., Concerning partial recursive similarity transformations of linearly ordered sets, PaciJic J . Math. 71 (1977). 57-70. [ M R 56, $28061 Hay, L., Manaster, A.. and Rosenstein. I. G., Small recursive ordinals, many-one degrees, and the arithmetical difference hierarchy, Ann. Math. Logic 8 (L975), 297-343. [ M R 53, #7748] Kleene, S. C., Introduction to Metamathematics, Princeton: Van Nostrand, 1952. Kreisel, G.. Shoenfield, J., and Wang, H., Number theoretic concepts and recursive well-orderings, Arch. Math. Logik Grnndlag. 5 (1960), 42-64. [ M R 22, #6709] Lerman, M., On recursive linear orderings, in Proceedings, Logic Year 1979-80: The Uniuersity of’ Connecticut (M. Lerman, J. Schmerl, R. Soare, eds.), Lecture Notes in Mathemarics Vol. 859, Berlin and New York: Springer-Verlag, 1981, pp. 132-142. Lerman, M., and Rosenstein, J. G . , Recursive linear orderings 11, Proceedings, Putras Conference 1980, Amsterdam: North-Holland Publ., 1981. Pinus, A. G., Effective linear orders. Siberian Math. J . 16 (1975), 956-962. [ M R 53,#137] Remmel, J., Realizing partial orderings by classes of co-simple sets, Pacific J . Math. 76 (1978), 169-184. Rogers, H. Jr., Theory of Recursive Functions and Effective ConTputability, New York: McGraw-Hill, 1967. Watnick, R., A generalization of Tennenbaum’s theorem on effectively finite recursive orderings, J . Symbolic Logic. (to be published). Watnick, R., Constructive and recursive scattered order types, in Proceedings, Logic Year 1979-80: The University of(bnnecticut (M. Lerman, J. Schmerl, R. Soare, eds.). Lecture Notes in Marhematics Vol 859, Berlin and New York: Springer-Verlag. 1981, pp. 312-326. Watnick, R., Recursive and constructive linear orderings, Thesis, Rutgers University, 1980.

COMPLETE BIBLIOGRAPHY OF LINEAR ORDERINGS

This bibliography contains all articles that I have been able to find that are related to linear orderings. It includes those items referred to in the text and those not discussed at all. Since I have conducted exhaustive searches of the literature I use the adjective complete with some confidence, even though I am sure that some, perhaps many, items have been omitted. I will compile addenda to this bibliography, based on readers’ additions and corrections, which will be available directly from me on request. I trust that this bibliography, certainly the most extensive to date on the subject, will be helpful to those who have research interests in linear orderings.

Abian, A., A fixed point theorem for nonincreasing mappings, Boll. Un. Mat. Itaf. (4) 2 (1969). 200-201. [ M R 39,#5427] Abian, A,, On the cofinality of ordinal numbers, Math. Ann. 179 (1969). 142-152. [ M R 41, #68] Abian, A., Some special properties of cofinality of ordinal numbers, Math. Ann. 189 (1970). 325-329. [ M R 42, #7517] Abian, A,, The common points of families of normal functions, Canad. J . Math. 25 (1973). 506-510.

Abian, A., and Deever, D.. On the bounds of the minimal length of sequences representing simply ordered sets. Arch. Math. Logik Grundlag. 10 (1967), 3-5. [ M R 35,#6587] Abian, A,, and Deever, D., On the minimal length of sequences representing simply ordered sets, Z . Math. Loyik Grundlag. Math. 13 (1967), 21-23. [ M R 36, #4978] Abian, A., and Deever, D., Representation of simply ordered sets and the generalized continuum hypothesis, Prace Mat. 11 (1967). 183-186. [Also Notices Amer. Math. Soc. 12 (1965), 718.1 [ M R 36, #4998] Aczel, P.. Describing ordinals using functionals of finite type, J. Symbolic Logic 37 (1972). 35-47. [ M R 48,#72] Adams, M. E., Weakly homogeneous order types, Canad. Math. Bull. 18 (1975), 159-161. [ M R 52, #7976] Aigner, A , , Der multiplikative Aufbau beliebiger unendlicher Ordnungszahlen, Monatsh. Moth. 55 (1951) 297-299. [ M R 13, p. 5421 Aigner, A,, Der multiplikative aufbau der polynome in der unendlichen Ordnungszahl w , Monatsh. Math. 55 (1951), 157-160. [ M R 13, p. 1201 Alling, N. L., On the existence of real-closed fields that are q.-sets of power &, Trans. Amer. Math. Soc. 103 (1962), 341-352. [ M R 26, #3615] Alo, R. A., A proof of the complete normality of chains, Acta Math. Acaa. Sci. Hungar. 22 (1971/72). 393-395. [ M R 4S, #I8121 Alo, R.A,, and Frink, O., Topologies of chains, Math. Ann. 171 (1967), 239-246. [ M R 35,#99] 456


457

Amerbaev, V. M., and Kasimov, Ju. F., Formula expressibility of order relations in algebraic systems, Vestnik Akad. Nauk. Kazak. SSR 1978,55559. [ M R 58, #21582] Arnit, R., and Shelah, S., The complete finitely axiomatized theories of order are dense, Israel J . Math. 23 (1976), 200-208. [ M R 58. #5162] Anderson, J. A. H., The maximum sum of a family of ordinals, in Cambridge Summer School in Mathematical Logic (Cambridge, 1977), Lecture Notes in Mathematics, Vol. 337. Berlin and New York: Springer, 1973, pp. 419-438. [ M R 49, #2395] Anderson, J. A. H., The minimum sum of an arbitrary family of ordinals, J . London Math. SOC. (2) 7 (1974), 429-434. [ M R 48, #I08121 Aronszajn, N.. Characterization of types of order satisfying a, + a, = a1 + a,, Fund. Math. 39 (1952), 65-96. [ M R 14, p. 8541 Ash, C. J., Dense, uniform and densely subuniform chains, J.Australian Math. Soc. 23 (1977), 1-8. Ash, C . J., A supply of Hopfian chains (unpublished). Assous, M., Caracterisation du type d’ordre des barrikres de Nash-Williams, Pubf. D6p. Math. (Lyon) 11 (1974), 89-106. [ M R 51, //?004] Assous, M., Une caracterisation du on-meilleurordre, C . R . Acad. Sci. Paris 285 (1977), A597A599. [ M R 56, #5298] Avraham, U., and Shelah, S., Martin’s Axiom does not imply that every two N,-dense sets of reals are isomorphic, IsraelJ. Math. 38 (1981), 161-176. Bachmann, H.. Die Normalfunktionen und das Problem des ausgezeichneten Folgen von Ordnungszahlen, Vierfeljschr. Naturforsch. Ges. Zurich 95 (1950), 115-147. [ M R 12, p. 1651 Bachmann, H., Vergleich und Kombination zweier Methoden von Veblen und Finder zur Losung des Problems der ausgezeichneten Folgen von Ordnungszahlen, Comment. Math. Helu. 26 (1952), 55-67. [ M R 13, p. 7281 Bachrnann, H., Normalfunktionen und Hauptfolgen, Comment. Math. Helv. 28 (1954), 9-16. [ M R 16, p. 201 Bachmann, H., Transfinite Zahlen, Ergebnisse der Mathematik und ihre Grenzgebiete, new series Vol. 1, Berlin and New York: Springer-Verlag, 1955. [ M R 17, p. 134; 36, #2506] Bagemihl, F., and Gillman, L., Generalized dissimilarity of ordered sets, Fund. Math. 42 (1955), 141-165. [ M R 17, p. 2431 Bagemihl, F., and Gillman, L., Some cofinality theorems on ordered sets, Fund. Math. 43 (1956), 178-184. [ M R 18, p. 5511 Banaschewski, B.. Orderable spaces, Fund. Math. 50 (1961). 21-34. [ M R 25, #2007] Baumgartner, J., All K,-dense sets of reals can be isomorphic, Fund. Math. 79 (1973), 101-106. [ M R 47, #6483] Baumgartner, J., Improvement of a partition theorem of Erdos and Rado, J . Combin. Theory Ser. A 17 (1974), 134-137. [ M R 49, $88661 Baurngartner, J., Canonical partition relations, J . Symbolic Logic 40 (1975), 541-554. [ M R 53, #2688] Baumgartner, J., Partition relations for uncountable ordinals, Israel J . Math. 21 (1975), 296-307. [ M R 53, #2687] Baumgartner, J., Anewclassofordertypes, Ann. Math. Logic9(1976), 187-222. [MR54, #4988] Baumgartner, J., Order types of real numbers and other uncountable linear orderings, in Proceedings qf Symposium on Ordered Seis (Banff, 1981). Berlin and New York: Springer, to be published. Baumgartner, J., and Hajnal, A,, A proof (Involving Martin’s axiom) of a partition relation, Fund. Math. 78 (1973), 193-203. [MR 47, #8310]

458


Baumgartner, J., Calvin, F., Laver, R., and McKenzie, R., Game-theoretic versions of partition relations, in Infinite and Finite Sets, Colloq. Math. SOC.Janos Bolyai Vol. 10. (Colloquiwn, Keszthely, 1973: Dedicated to P. Erdos on h b 60th Birthday), Amsterdam: North-Holland Publ., 1975, Vol. I , 131-135. [ M R 53, #12956] Benda, M., Somepropertiesof mirrored orders, Math. Scand. 37(1975), 5-12. [ M R 5 6 , #15411] Bennett, A. A,, Some arithmetic operations with transfinite ordinals, Amer. Math. Monthly 28 (1921),427-430. Benos, A., Resultats sur la comparaison des types d’ordres, Bull. Soc. Math. Grece (N.S.) 14 (1973), 148-151. [ M R 52, 8104321 Bloch, G., Sur les ensembles stationnaires de nombres ordinaux et les suites distinguees de fonctions regressives. C. R. Acad. Sci. Paris 236 (1953), 265-268. [ M R 14, p. 7331 Bonnet, R., Chaines de Ramsey, C. R. Acad. Sci. Paris 274 (1972), A605-A606. [ M R 45. # 17671 Bonnet, R., Sur les algebres de Boole rigides, Thesis, Universitt Claude-Bernard, Lyon I, 1978. Bonnet, R., Corominas, E., and Pouzet, M., Simplification pour la multiplication ordinale. C. R. Acad. Sci. Paris 276 (1973), A221LA224, A339-342. [ M R 47, #4799, #4800] Bosch, J., Fixed points of transfinite ordinal operators (Spanish), Univ. Nac. La Plata Publ. Far. Cienc. Fisicomat. Ser. Segunda Rev. 5 (1965), 201-214. [ M R 19, p. 10311 Bridge, J., A simplification of the Bachmann method for generating large countable ordinals, J. Symbolic Logic 40 (1975), 171-185. [ M R 53, #2644] Buchholz, W., Normalfunktionen und Konstruktive Systeme von Ordinalzahlen, in Proof Theory Symposium, Kid, 1974, Lecture Notes in Mathematics, Vol. 500, Berlin and New York: Springer, 1975, pp. 4-25. [ M R 54, #72] Biichi, J. R.,Weak second order arithmetic and finite automata, Z. Math. Logik Grundlag. Math. 6 (1960).66-92. Biichi, J . R., On a decision method in restricted second order arithmetic, Proc. 1960 Int. Cong. for Logic, Methodology, and Philosophy of Science, Stanford: Stanford Univ. Press, 1962, pp. 1-1 1. [ M R 32, #1116] Buchi, J. R.,Decision methods in the theory of ordinals, Bull. Amer. Math. Soc. 71 (1965), 767-770. [ M R 32, #7413] Biichi, J. R., Transfinite automata recursions and weak second order theory of ordinals, in Logic, Methodology, and Philosophy of Science (Proc. 1964 Int. Cong.), Amsterdam: North-Holland Publ.. 1965, pp. 3-23. [ M R 35, #1480.] Biichi, J. R., The monadic second-order theory of wl, in Decidable Theories I1 (eds. G. H . Miiller and D. Siefkes). Lecture Notes in Maihematics, Vol. 328. Berlin and New York: Springer, 1973, pp. 1-127. [ M R 57, #I60331 Biichi, J. R., The monadic second-order theory ofw,, in Lecture Notes in Mathematics. Vol. 328. Berlin and New York: Springer, 1973, pp. 1-126. Biichi, J. R., and Siefkes, D.. Axiomatization of the monadic second order theory of w, , in Decidable Theories II (eds. G. H. Muller and D. Siefkes), Lecture Notes in Mathematics, Vol. 328, Berlin and New York: Springer, 1973, pp. 129-217. [ M R 58, #198] Cantor, G., Beitrage zur Begrundung der transfiniten Mengenlehre, Math. Ann. 46 (1895), 481-512; 49(1897), 207-246. Carruth, P. W., Arithmetic of ordinals with applications to the theory of ordered abelian groups. Bull. Amer. Math. SOC.48 (1942), 262-271. [ M R 3, p. 2251 Carruth, P. W., Roots and factors of ordinals, Proc. Amer. Math. Soc. 1 (1950), 470-480. [ M R 12, p. 1661 Case, J., Sortability and extensibility of the graphs of recursively enumerable partial and total orders, Z. Math. Logik Grundlag. Math. 22 (1976), 1-18. [ M R 53, #2654] Chajoth, Z., Beitrag zur Theorie der geordneter Mengen, Fund. Math. 16 (1930), 132-133.


459

Chang. C. C . , A partition theorem for the complete graph on ww,J . Combin. Theory Ser. A 12 (1972), 396-452. [ M R 48, ,#I9301 Chang, C. C., and Ehrenfeucht, A,, A characterization of Abelian groups of automorphisms of a simple ordering relation, Fund. Math. 51 (1962), 141-147. [ M R 26, #I981 Chen, K. H., Recursive well-founded orderings, Ann. Math. Logic 13 (1978), 117-147. [ M R 80e: 030471 Church, A,, and Kleene, S. C., Formal definitions in the theory of ordinal numbers, Fund. Math. 28(1937), 11-21. Cohn, P. M., Groups or order automorphisms of ordered sets, Mathematika 4 (1957), 41-50. [ M R 19, p. 9401 Combebiac, G., Sur les elements de la thtorie des ensembles ordonnes, Enseign. Math. 8 (1906), 201-203. Corominas, E., see Bonnet, R., Corominas, E., and Pouzet, M. Crossley, J. N., Constructive Order Types. Amsterdam: North-Holland Publ., 1969. [ M R 41, #5214] Cuesta, N., Construction of an ordered dense set which is not continuous and whose cardinal is K, (Spanish), Rev. Mat. Hisp.-Amer. ( 4 )3 (1943), 38-40. [ M R 4, p. 2121 Cuesta, N., Continuous permutations with real numbers (Spanish), Rev.Mat. Hisp.-Amer. (4) 5 (1945), 191-203. [ M R 7, p. 2771 Cuesta, N., Dense perfectly ranked ordering (Spanish), Rev. Mat. Hisp.-Amer. ( 4 ) 8 (1948), 57-71. [ M R 10, p. 231 Cuesta, N., Ascending sequences of ordinal numbers (Spanish), Rev. Mat. Hisp.-Amer. ( 4 ) 9 (1949), 83-96, 168. [ M R 11, p. 6461 Cuesta, N.. Ordinal algebra (Spanish), Reu. Acad. Cienc. Madrid48 (1954), 103-145. [ M R 16, p. 10911 Cuesta, N., Ordinal arrangement (Spanish). Rev. Mat. Hisp.-Amer. ( 4 ) 14 (1954), 237-268. [ M R 16, p. 10911 Davis, A. C., Cancellation theorems for products of order types, Bull. Amer. Math. SOC.58 (1952),63. Davis, A. C., On order types whose squares are equal, Bull. Amer. Math. Soc. 58 (1952), 382. Davis, A. C., Sur l’equation 5” = u pour les types d’ordre, C. R . Acad. Sci. Paris 235 (1952). 924-926. [ M R 14, p. 3611 Davis, A. C., see also Morel, A. C. Davis, A. C., and Sierpinski, W., Sur les types dordre distincts dont les c a d s sont egaux, C. R . Acad. Sci. Paris 235 (1952), 850-852. [ M R 14, p. 3611 Day, G. W., Maximal chains in atomic Boolean algebras, Fund. Math. 67 (19701, 293-296. [ M R 41, #3344] Deever, D., see Abian, A., and Deever, D. Denjoy, A., L’enumeration transfinie, I . Ln notion de rang. Paris: Gauthier-Villars, 1946. [ M R 8, p. 2541 Denjoy, A,, Quelques proprietes des ensembles ranges, Ann. Soc. Polon. Math. 21 (1948) 187-195. [ M R 11, p. 161 Denjoy, A,, L’enumeration transjinie, I I . L’arithmitisation du transfini. Paris: Gauthier-Villars, 1952. [ M R 15, p. 4081 Denjoy, A,, L’ordination des ensembles, c‘. R. Acad. Sci. Paris236 (1953), 1393-1396. [ M R 14, p. 10691 Denjoy, A., Sur les ensembles disperses, C . R Acad. Sci. Paris 265 (1967), A529-A533. [ M R 37, #6188] Delvin, K. J . , Order types, trees, and a problem of Erdos and Hajnal, Period. Marh. Hungar. 5 (1974), 153-160. [ M R 51, #I571

460

COhQ'LETE

BIBLIOGRAPHY OF LINEAR ORDERINGS

Delvin, K. J., A note on a problem of Erdos and Hajnal, Discrete Math. 11 (1975), 9-22. [M R 50, #6842] Doets, H. C., A generalization in the theory of normal functions, Z . Math. Logik Grundlag. Math. 16(1970), 389-392. [ M R 4 5 , #1768] Doner, J . , Definability in the extended arithmetic of ordinal numbers, Dissertationes Math. (Rozprawy Mat.) 96 (1972), 1-50. [ M R 49, #2396] Doner, J., Mostowski, A,, and Tarski, A,, The elementary theory of well-ordering-a metamathematical study, Logic Colloquium '77 (Proc. Conf., Wroclaw, 1977), 1-54, Studies in Logic and Foundations of Math. 96, Amsterdam: North-Holland Publ.. 1978. [ M R 80d:03027a] Doner, J., and Tarski, A,, An extended arithmetic of ordinal numbers, Fund. Math. 65 (1969). 95-127. [ M R 39, #5374] Dushnik, B., A note on transfinite ordinals, Bull. Amer. Math. Soc. 37 (1931), 860-862. Dushnik. B., Maximal sums o f ordinals, Trans. Amer. Math. Soc. 62 (1947), 240-247. [ M R 9, p. 1771 Dushnik, B., Upper and lower bounds of order types, Michigan Math. J . 2 (1954), 27-31. [ M R 16, p. 191 Dushnik, B., and Miller, E. W., Concerning similarity transformations of linearly ordered sets, Bull. Amer. Math. Soc. 46 (1940), 322-326. [ M R I, p. 3181 Ehrenfeucht, A,, An application of games to the completeness problem for formalized theories. Fund. Math. 49 (1961). 129-141. [ M R 23, #3666] Ehrenfeucht, A,, Decidability of the theory of the linear order relation, Notices Amer. Math. SOC.6 (1959), 556-38. Ehrenfeucht, A,, Polynomial functions with exponentiation are well-ordered, Algebra Universalis 3 (1973). 261-262. Ehrenfeucht, A., see also Chang, C. C., and Ehrenfeucht, A. Eilenberg, S., Ordered topological spaces. Amer. J . Math. 63 (1941), 39-45. [ M R 2, p. 1791 Eisenbud, D., Groups of order automorphisms of certain homogeneous ordered sets, Michigan Math. J . 16 (1969), 59-63. [ M R 39, #1373] Erdos, P., Some remarks on set theory, Proc. Amer. Math. Soc. 1 (1950), 127-141. [ M R 12. P. 141 Erdos. P., and Hajnal, A., On a property of families of sets (Russian summary), Acta Math. Acad. Sci. Hungar. 12 (1961), 87-123. [ M R 27, #50]. Erdos, P., and Hajnal, A,. On a classification of denumerable order types and an application to the partition calculus, Fund. Math. 51 (1962), 117-129. [ M R 25, #5000] Erdos. P., and Hajnal, A,, Ordinary partition relations for ordinal numbers, Period. Math. Hung. 1 (1971), 171-185. [ M R 45, #8531] Erdos. P., and Hajnal, A., Unsolved problems in set theory, in Axiomatic Set Theor??(Proceedings ojsjlmposia in Pure Mathematics. Yol. XIII. Part I ) , Providence. Rhode Island : Amer. Math. SOC.Publ., 1971, pp. 17-48. [ M R 43, #6101] Erdos, P., and Milner, E. C., A theorem in the partition calculus, Canad. Math. Bull. 15 (1972). 501-505 [Correction, Canad. Math. Bull. 17 (1974), 305.1 [ M R 50, #12734; 48, #lO819] Erdos, P., and Rado, R., A problem on ordered sets, J . London Math. Soc. 28 (1953), 426-438. [ M R 15, p. 4101 Erdos, P., and Rado, R., A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956). 427-489. [ M R 18, p. 4583 Erdos, P., and Rado, R., Partition relations and transitivity domains of binary relations, J . London Math. Soc. 42 (1967), 624-633. [ M R 36,#I3351 Erdos, P., Hajnal A., and Milner E., A problem on well-ordered sets, Acta Math. Acad. Sci Hungar. 20 (1969), 329. [ M R 41, #5222]


46 1

Erdos. P., Hajnal A,, and Milner E., Set mappings and polarized partition relations, in Combinatorial Theory and its Application.\, I (Proc. Colloq. Balatonfured, 1969), Amsterdam: North-Holland Publ., 1970, pp. 327-363 [ M R 45, # S S S S ] ErdBs, P., Hajnal A,, and Milner E., Polarized partition relations for ordinal numbers, in Studies in Purr Mathematics (Rado Volume),pp. 63-87. New York: Academic Press, 1971. [ M R 43. fi3123J Erdos. P.. Hajnal A , , and Milner E., Partition relations for q, and &-saturated models, in Theory of Sets and Topology (in Honour q f F . Hausdorff), pp, 95-108. Berlin: VEB Deutsch. Verlag Wissensch., 1972. [ M R 49, #7143] Erdos, P., Milner, E. C., and Rado, R., Partition relations for ?,-sets, J . London Math. SOC.(2) 3(1971), 193-204. [MR43,#60] Ershov, Y. L., Restricted theories of totally ordered sets, Algebra and Logic 7 (1968), 153-159. [ M R 41,#44] Feferman, S., Some recent work of Ehrenteucht and Fraisse, in Summer Institute for Symbolic Logic, Cornell University, 1957, Summaries, pp. 201 -209. Communications Research Division, Institute for Defense Analysis. Princeton, New Jersey 1957. Feferman, S., Systems of predicative analysis 11. Representations of ordinals, J . Symbolic Logic 33 (1968), 193-220. [ M R 41, #5213] Feferman, S., Hereditarily replete functionals over the ordinals, in Intuitionism andproof Theory (ed. J . Myhill), Amsterdam: North-Holland Publ., 1970, pp. 289-301. [ M R 44, #3863] Fefferman, C., Cardinally maximal sets of non-equivalent order types, Z . Math. Logik Grundlug. Matb. 13 (1967). 205-212. [ M R 36, 413131 Feiner, L., Orderings and Boolean algebras not isomorphic to recursive ones, Thesis, Massachusetts Institute of Technology, 1967. Fellner, S., Recursiveness and finite axiomatizability of linear orderings, Thesis, Rutgers University, 1976. Finder, P., Eine transfinite Folge arithmetischer Operationen, Comment. Math. Helu. 25 (195l), 75-90. [MR 13, p. 1201 Fleischer, I . , Embedding linearly ordered sets in real lexicographic products, Fund. Math. 49 (1960/61), 147-150. [ M R 23, #A1557; 27, #58] Fleischer, I., A characterization of lexicographically ordered ?,-sets, Proc. Nut. Acad. Sci. USA 50 (1963). 1107-1 108. Reviewed in ZcrirralbIutt,fur Math. 137 (1967), p. 21. Fodor, G . , Generalization of a theorem of Alexandroff and Urysohn, Acta Sci. Matb. 16 (1955), 204-206. [ M R 17, p. 8311 Fodor, G.. Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. 17 (1956), 139-142. [ M R 18, p. 5511 Fodor. G.. On regressing functions, Z . Matk. Logik Grundlay. Math. 4 (1958), 148-156. Fodor, G., Uber transfiniteFunktionen, Acta Sci. Muth.21(1960), 343-345; 22(1961),289-295; 22 (1961). 296-300. [ M R 24, #A48; 27, 1/44] Fodor, G., On stationary sets and regressive functions, Acta Sci. Math. 27 (1966), 105-110. [ M R 34,#66] Fodor, G., Principal sequences and stationary sets, Acta Sci. Math. 34 (1973), 81-84. [ M R 47, #6491] Foldes, I., Familles w,-commutatives de nombers ordinaux, C. R. Acad. Sci. Paris 276 (1973), A1539-Al540. CMR48, #I08131 Foldes, I., Fermetures sur les chaines, C. R. Acad. Sci. Paris276 (1973),A140-Al406. [ M R 48, #8315] Foldes, I., Sur la limite des suites compostes de nombres ordinaux, C. R. Acad. Sci. Paris 280 (1975), A65-A67. [ M R 51, #2919] Folley, K. W., Simply ordered sets, Trans. Royal SOC.Canada 22 (1928), 225.

462


Fraisse, R., Sur la comparaison des types d'ordres, C . R . Acad. Sci. Paris 226 (1948), 1330-1331. [ M R 10, p. 5171 Fraisse, R., Sur quelques classifications des systtmes de relations, Publ. Sci. Uniu. Alger. S i r . A , 1 (1954). 35 -182. [Abstract: Applications scientifques de la logique rnathimatique. Paris and Louvain, 1954, p. 85.1 Frai'sse, R., Abritement entre relations et specialement entres chaines, in Symposia Mathematica. Vol. V, pp. 203-251. New York: Academic Press, 1971. [ M R 43, #IlO] Fraisse, R.. Course of Mathematical Logic, Dordrecht: Reidel, 1973. [Translation of Cours de logique mathlmatique, 2 Vols.] [ M R 37, #3902] Frasnay. C.. Relations invariantes dans un ensemble totalement ordonnt, C. R . Acad. Sci. Paris 225 (1962), A2878-A2879. [ M R 27,11571 Frasnay, C., Quelques theorems combinatoires faisant intervenir un ordre total, C. R . Acad. Sri. Paris 257 (1963), A1825-Al828. [ M R 28, #1141] Friedman, H., Onclosed sets of ordinals, Proc. Amer. Math. SOC.43(1974), 190-192. [ M R 48, #5863] Frink. 0.. see Alo, R. A,, and Frink, 0. Froda, A., Sur les reunions ordonnees d'ensembles, (Romanian; Russian and French summaries), Acad. R . P . Rom6w Stud. Cerc. Mat. 7 (1956), 7-35. [ M R 18, p. 2741 Galvin, F., see also Baumgartner, J., Galvin. F., Laver, R., and McKenzie, R. Galvin, F., and Larson, J., Pinning countable ordinals, Fund. Math. 82 (1974), 357-361. [ M R 50, #I27301 Garland, S. J., Second-order cardinal characterizability, in Axiomatic Set Theory (Proc. Syrnp. Pure Math., Vol. XIII, Part I I , Unir. Calif., Los Angeles, Calif., 1967). pp. 127-146. [M R 54, #4982] Gerber, H., An extension of Schutte's Klammersymbols, Marh. Ann. 174 (1967), 203-216. [M R 36, 1162951 Gillman, L.. On intervals of ordered sets, A n n . of Math. 56 (1952), 440-459. [ M R 14, p. 5431 Gillman, L., Remarque sur les ensembles q%.C . R . Acad. Sci. Paris 241 (1955). 12-13. [ M R 17. p. 1351 Gillman. L., On a theorem of Mahlo concerning anti-homogeneous sets, Michigan Math. J. 3 (1955/56), 173-177. [ M R 18, p. 4951 Gillman, L., Some remarks on qz-sets, Fund. Math. 43 (1956), 77-82. [ M R 18, p. 241 Gillman. L., A continuous exact set. Pro(,. Amer. Math. Soc. 9 (1958), 412-418. [ M R 20, 1116371 Gillman, L., see also Bagemihl, F., and Gillman, L. Ginsburg, S . ,On thedistinct sums of I-type transfinite series obtained by permuting theelements of afixed i-type series. Fund. Math. 39(1952), 131-132. [ M R 14, p. 10691 Ginsburg, S., Some remarks on order types and decompositions of sets, Trans. Amer. Math. SOC.74 (1953),514-535. [ M R 14, p. 8531 Ginsburg, S.. Fixed points of products and sums of simply ordered sets, Proc. Amer. Math. Soc. 5 (1954), 554-565. [ M R 16, p. 211 Ginsburg, S., Further results on order types and decompositions of sets, Trans. Amer. M o f h . Soc. 77 ( 1 954). 120- 150. [M R 16, p. 201 Ginsburg. S., Order types and similarity transformations, Trans. Amer. Math. Soc. 79 (1 955). 341-361. [ M R 17, p. 201 Ginsburg. S.. Uniqueness in the left division of order types, Pror. Amer. Math. SOC.6 (1955). 120-123. [ M R 16, p. 6821 Glass, A., Ordered Permutation Groups. Bowling Green, Ohio: Bowling Green University, 1976. [ M R 54, #10097] Glass, A.. Gurevich. Y., Holland. W. C., and Jambu-Giraudet, M., Elementary theory ofautomorphism groups of doubly homogeneous chains, Proceedings, Logic Year 1979-80 :


463

The University of Connecticut (M. Lerman, J. Schmerl, R. Soare, Eds.) Lecture Notes in Marhematics Vol. 859, Berlin and New York: Springer, 1981, 67-82. Glass, A., Gurevich, Y., Holland, W. C . , and Shelah, S., Rigid homogenous chains, to appear in Cambridge Philosophical Socieiy, Mathematical Proceedings. Gleyzal, A,, Order types and structure of orders, Trans. Amer. Math. Soc. 48 (194Q),451-466. [ M R 2, p. 1291 Goffman, C., The group of similarity transformations of a simply ordered set, Research Prob60 (1954), 289. lem 10, Bull. Amer. Math. SOC. Grossman, A., Sur une propriete des ensembles ordonnes, Hroatsko Privodosloono DrrrStro. Glasnik Mat.-Fiz. Astr. Ser. II 8 (1953), 24-26. [MR 14, p. 10691 Gurevich, Y . , Monadic theory of order and topology I, Israel J . Math. 27 (1977), 299-319. [MR 56, #5259] Gurevich, Y., Modest theory of short chains I, J . Symbolic Logic, 44 (1979), 481-490. [ M R 81a:03038a] Gurevich, Y., Monadic theory of order and topology 11, Israel J . Math. 34 (1979), 45-71. [ M R 81f:03049] Gurevich, Y., Crumbly spaces, to appear in Logic, Philosophy, and Methodology of Science, Proc. Hannover Conference (1979). Gurevich, Y., and Holland, W. C., Recognizing the real line, Trans. Amer. Math. Soc. (to be published). Gurevich, Y., Magidor, M., and Shelah, S., The monadic theory of w 2 ,to appear in J. Symbolic Logic. Gurevich, Y., and Shelah, S., Modest theory of short chains 11, J . Symbolic Logic 44 (1979), 491 -502. [ M R 81a:03038b] Gurevich, Y., and Shelah, S.. Monadic theory of order and topology in ZFC, to appear in Annals of Math. Logic. Gurevich, Y., see also Glass, A., Gurevich, Y., Holland, W. C., and Jambu, M.; Glass, A,, Gurevich, Y., Holland, W. C., and Shelah, S. Haddad, L., and Sabbagh, G., Calcul de certain nombres de Ramsey genkralisis, C. R. Acad. Sci. Paris 268 (1969), 1233-1234. [MR 40,#I2851 Haddad, L., and Sabbagh, G., Nouveaux resultats sur les nombres de Ramsey gtneralisCs, C. R. Acad. Sci. Paris 268 (1969), 1516-1518. [MR40, #I2861 Haddad, L., and Sabbagh, G., Sur une extension des nombres de Ramsey aux ordinaux, C. R. Acad. Sci. Paris 268 (1969), A1 165-,41167. [ M R 40, #47] Hagendorf, J . G., Extensions immediates de chaines et de relations, C. R. Acad. Sci. Paris 274 (1972). A607-A609. [MR 48, #5927] Hagendorf, J. G., Extensions immediates et respectueuses de chaines et de relations, C. R. Acad. Sci. Paris 275 (1972). A949-A950. [MR 47, #4797] Hagendorf, J. G., Extensions immediates respectueses de chaines, C . R. Acad. Sci. Paris 275 (1972), A1273-Al275. [MR 47. #4798] Hagendorf, J. G., Sur certaines extensions de chaines, C. R. Acad. Sci. Paris 276(1973), A1589A 1592. [ M R 49, #46] Hagendorf, J. G:, Chaines incassables, C. R. Acad. Sci. Paris 278 (1974), A1475-Al478. [MR 50, #I 8851 Hagendorf, J. G., Extensions de chaines, Thesis, Universite d’Aix-Marseilles I, 1975. Hagendorf, J. G., Sur distributivite du plongement entre chaines, C. R. Acad. Sci. Paris 282 (1976), A1391LA1394. [MR 57, #5756] Hagendorf, J. G., Extension de chaines, Thesis de Doctoral d’Etat, UniversitC d’Aix-Marseilles I, 1977. Hagendorf, J. G., Extensions respectueuses de chaines, Z . Math. Logik Grundlag. Math. 25 (1979),423-444. [MR 81c: 1060021

464


Hajnal, A,, Some results and problems in set theory, Actu Math. Acad. Sci. Hungar. 11 (1960). 277-298. [ M R 27, #47] Hajnal, A., see also Baumgartner, J., and Hajnal, A,; Erdos, P., and Hajnal, A,; Erdos, P., Hajnal, A., and Milner, E. C. Hajnal, A., and Milner, E. C., Some theorems for scattered ordered sets, Period. Math. Hungar. 1, (1971). 81-92. [ M R 44, #3930] Hanazawa, M., A remark on ordered structures with unary predicates, J . Math. SOC.Japan 27 (1975), 345-349. [ M R 52, #I091 Hartogs, F., Uber das Problem der Wohlordnung, Math. Ann. 76 (1914), 438-443. Harzheim, E., Uber Permutationen von totalgeordnete Mengen, Math. Ann. 147 (1962), I20125. [ M R 25,#2008] Harzheim, E., Dualzerlegungen in totalgeordneten Mengen, Fund. Math. 53 (1963), 81-91. [ M R 27, #5700] Harzheim, E., Beitrage zur Theorie der Ordnungstypen, insbesondere der q,-Mengen, Math. Ann. 154 (1964), 116-134. [ M R 28, #SO111 Harzheim, E., Bemerkungen zu den Satzen von Hausdorff-Urysohn und Padmavally, Z . Math. Logik Grundlag. Math. 10 (1964), 17-21. [ M R 28, #2055] Harzheim, E., Einbettungssatze fur totalgeordneten Mengen, Math. Ann. 158 ( I 965), 90-108. [MR 30,#4695] Harzheim, E., Einbettung totalgeordneten Mengen in lexikographische Produkte, Math. Ann. 170 (1967), 245-252. [ M R 35, #2744] Harzheim, E., Uber universal geordnete Mengen, Math. Nachr. 36 (1968). 195-213. [ M R 39. #I011 Hausdorff, F., Untersuchungen uber Ordnungstypen, Ber. Verhand. konig. Sachsischrn Gesellsch. Wiss. Leipzig, Math.-Phys. Klasse 58 (1906). Hausdorff, F..Grundzugeeiner Theorieder geordneten Mengen, Math. Ann. 65(1908),435-505. Hausdorff, F., Grundziige der Mengenlehre, Leipzig 1914. Reprinted Chelsea Publishing Co., New York 1949. Hay, L., Manaster, A., and Rosenstein, J. G . , Small recursive ordinals, many-one degrees, and the arithmetical difference hierarchy, Ann. Math. Logic 8 (1975), 297-343. [ M R 53, #7748] Hay, L., Manaster, A,, and Rosenstein, J. G . , Concerning partial recursive similarity transformations of linearly ordered sets, Pacific J . Math. 71 (1977), 57-70. [ M R 56, #2806] Herre, H., and Wolter, H., Entscheidbarkeit der Theorie der linearen Ordnung in L,, , Z . Math. Logik Grundlag. Math. 23 (1977), 273-282. [ M R 56, #5240] Herre, H., and Wolter, H., Entscheidbarkeit der Theorie der linearen Ordnung in LQKfur regulares K , 2. Math. Logik Grundlug. Math. 24 (1978), 73-78. [ M R 57, #16034] Herre, H., and Wolter, H., Decidability of the theory of linear orderings with cardinality quantifier Q,. Notices Amer. Math. SOC.26 (1979).A-16. Hessenberg, G., Grundbegriffe akr Mengenlehre, Abhandlungen der Fries’schen Schule, New Series I,4, Gottingen, 1960. Hessenberg, G., Potenzen transfiniter Ordnungszahlen, Zber. Deutsch. Math. Verein. 16 (19071, 130-137. Hickman, J. L., Aproblem on series of ordinals, Fund. Math. 81 (1973).49-56. [ M R 4 8 , #I08141 Hickman, J. L., Concerning the number of sums obtainable from a countable series of ordinals by permutations that preserve the order-type, 1.London Mach. Soc. 9 (1974/75), 239-244. [ M R 51, #I651 Hickman, J . L., General-well-ordered sets. J . Australian Math. Soc. 19 (1975), 7-20. [ M R 52, 1133931 Hickman, J . L., Reducing series of ordinals, PacificJ. Math. 59 (1975),461-473. [ M R 54, #98]


465

Hickman, J. L., Analysis of an exponential equation with ordinal variables, Proc. Amer. Math. SOC.61 (1976), 105-111. [ M R 56, #836X] Hickman, J. L., Critical points of normal functions I, Notre Dame J . Formal Logic 18 (1977), 527-534. CMR58, #5233] Hickman, J. L., Regressive order types, A’olre Dame J . Formal Luyic 18(1977), 169-174. [ M R 56,1128791 Hickman, .I. L., Rigidity in order-types, J . Australian Math. SOC.24 (1977), 203-215. [ M R 58, #3971 Hickman, J. L., Some results on series of ordinals, 2. Math. Logik Grundlag. Math. 23 (1977), 1-18. [ M R 58.1152321 Hickman, J. L., Commutativity of generalized ordinals, Notre Dame J . Formal Logic 19 (1978), 702-704. [ M R 80c:04005] Hickman, J. L., Cofinalities of doubly transitive sets, J . Reine Angew. Math. 299/300 (1978), 7-15. [ M R 58, #398] Hickman, J. L., Critical points of normal functions 11, Notre Dame J . Formal Logic 19 (1978), 20-24. [ M R 58, #27495] Hickman, J. L., Doubly transitive sets. Notre Dame J . Formal Logic 19 (1978), 386-394. [ M R 58, #399] Hickman, J. L.. Polynomials in a single ordinal variable, Marh. Logik Grundlagen Math. 25 (1979), 173-178. [ M R 80d:04003] Hickman, J. L., Semi-monotone series of ordinals, Notre Dame J . Formal Logic 20 (1979), 196-200. [ M R 80d:04004] Hickman, J. L., Conrnutativity in series of ordinals: a study of invariants, Trans. Amer. Math. SOC. 248(1979),411-434. [MR80h:04002] Hickman, J. L., A class of “near-finite” order types, Z . Math. Logik Grundlag. Math. 25 (1979), 79-92. [ M R 80h:04003] Hoborski, A,, Une rernarque sur la limitedes nombres ordinaux, Fund. Math. 2 ( 1921), 193-198. Holland, W. C., The lattice ordered group of automorphisms of an ordered set, Michigan Math. J . 10 (1963), 399-408. [ M R 28, #I2371 Holland, W. C., Transitive lattice ordered permutation groups, Math. 2. 87 (1965), 420-433. [ M R 31, #2310] Holland, W. C., A class of simple lattice ordered groups, Proc. Amer.. Math. Soc. 16 (1965), 326-329. [ M R 30,#3927] Holland, W. C., Outer automorphisms of ordered permutation groups, Proc. Edinburgh Math. SOC.19 (19751, 331-344. [ M R 52, #7995] Holland, W. C . , Equitablepartitionsof thecontinuum, Fund. Math. 92(1976), 131-133. [ M R 5 4 , #5076] Holland. W. C., see also Glass, A , , Gurevich, Y . , Holland, W. C., and Jambu, M.; Glass, A., Gurevich, Y., Holland, W. C., and Shelah, S . ; Gurevich, Y., and Holland, W. C. Holland, W. C., and McCleary, S. H., Wreath products of ordered permutation groups, Pacific 1. Math. 31 (1969), 703-716. [ M R 41, #3350] Howard, P. E., and Rubin, J. E., The axiom of choice and linearly ordered sets, Fund. Math. 97 (1977). 111-122. [MR57,#115] Huntington. E. V., The Continuum and Other Types of Serial Order. With an Introduction to Canror’s Transfinite Numbers, 2nd ed., Cambridge, Massachusetts: Harvard Univ. Press, 1917. [Reprinted by Dover Publications, New York, 1955.1 [ M R 16, p. 8041 Isles, D., Natural well-orderings, J . Symbolic Logic 36 (1971), 288-300. [ M R 46, #8522] Isles. D., Regular ordinals and normal forms, in Intuitionism and Proof Theory (ed. J. Myhill), Amsterdam: North-Holland Publ., 1970, pp. 339-361. [ M R 43, #I8421 Jacobsthal, E., Vertauschbarkeit transfiniten ordnungszahlen, Math. Ann. 65 (1908), 160.

466


Jacobsthal, E.,Uber den Aufbau der transfiniten Arithmetik, Math, Ann. 66 (1909), 145194. Jacobsthal, E., Zur Arithmetik der transfiniten Zahlen, Math. Ann. 67 (1909), 130-144. Jambu-Giraudet, M., Theorie des moddes de groups d’automorphismes d’ensembles totalement ordones, Thtse 3Cme cycle, Universitt de Paris. [MR, 81f:O3046] Jambu-Giraudet, M., see also Glass, A., Gurevich, Y., Holland, W. C., and Jambu-Giraudet, M. Jullien, P.,Sur quelques problemes de la theorie des chaines permutes, C . R. Acad. Scz. Paris 262 (1966), A205-A208. [ M R 33, #2576] Jullien, P., Sur la comparaison des types d’ordres disperses, C . R . Acad. Sci. Paris 264 (1967), A594-A595. [ M R 39, #2642a] Jullien, P., Sur une nouvelle classification des types d‘ordres disperses denombrables, C . R . “ad. Sci. Paris 266 (1968), A327-A329. [ M R 39, #2642b] Jullien, P., Sur l‘insecabiliti finie des types d’ordres disperses, C. R. Acad. Sci. Paris 266 (1968), A389-A391. [ M R 39, #2642~] Jullien, P., Contribution a I’etude des types d’ordres disperses, Thesis, Marseilles, 1969. Kaluza, T., Zu einer Wachstumfrage bei Zuordnungen zwischen Ordinalzahlen, Math. Ann. 122 (1950), 323-325. [ M R 12, p. 5961 Kaluza, T., Zur Rolle der Epsilonzahlen bei der Polynomdarstellung von Ordinalzahien, Math. Ann. I22 (1950), 321-322. [ M R 12. p. 6261 Karp, C., Finite quantifier equivalence, in Proceedings of the Symposium on Ihe Theory of Models, Berkeley, 1963, Amsterdam: North-Holland. 1965, pp. 407-412. [ M R 35, k36] Kasimov, Ju. F., see Amerbaev, V. M., and Kasimov, Ju. F. Katriiiak, T., Note sur les ensembles ordonnes, (Czech, French, and Russian summaries), Acta Fac. Nut. Univ. Comenian 4 (1959), 291-294. [ M R 24, #A591 Kino, A., On definability of ordinals in logic with infinitely long expressions, J . Symbolic Logic 31 (1966), 365-375. [ M R 34,#4113]. Kita, T., A theorem on limit ordinals, Math. Japon. 3 (1954), 62. [ M R 16, p. 10071 Kleene. S. C . , see Church, A,, and Kleene, S. C . Kleinberg, E. M., and Seiferas, J. I., Infinite exponent partition relations and well-ordered choice, J . Symbolic Logic 38 (1973), 299-308. [ M R 49, #4782] Knight, P. R. S., Algebraic equivalence of ordinal numbers, Fund. Math. 73 (1972), 235-247. [MR45,#1769] Komjath, P.. Rearranging transfinite series of ordinals, Bull. Australian Math. SOC.16 (1977), 321-323. [ M R 57, #I22321 Koppelberg, S., Groups cannot be Souslin ordered, Arch. Math. (Basef)29 (1977), 315-317. [ M R 57, #5741] Kotlarski, H., On the existence of well-ordered models (Russian summary), Bull. Acad. Pofon. Sci. Ser. Sci. Math. Astronom. Phys. 22 (1974),459-462. [ M R 50, #6825] Kowalski, D., and PondiliEek, B., On the characters of chains, &sopis P2st. Mat. 91 (1966), 1-3, [ M R 32, #7459] Kreisel, G., Shoenfield, J., and Wang, H., Number theoretic concepts and recursive wellorderings, Arch. Math. Logik Grundlag. 5 (1960), 42-64. [ M R 22, #6709] Kruse, A. H., A note on the partition calculus of P. Erdos and R. Rado, J . London Math. Sac. 40 (1965), 137-148. [ M R 35, #5326] Kulunkov. P. A., Questions of definability in ordered sets (English trans.), M o s c o ~ ,Uniu. Math. Bull. 35 (1980), 51-55. [ M R 81g:O3038] Kuratowski, C . , Sur la notion de I’ordre dans la theorie des ensembles, Fund. Math. 2 (1921). 161-171. Kuratowski, C., Les types d’ordre definissables et les ensembles boreliens, Fund. Mafh. 29 (l937),97-100.


467

Kuratowski, C., Sur la geometrisation des types d'ordre denombrable, Fund. Math. 28 (1937), 167-1 85. Kuratowski, C., and Mostowski, A,, Set Theory. Amsterdam: North-Holland Publ.. 1968. Kurepa, D., Ensembles ordonnis et ramifies, These, Paris 1935. [Published in Math. Belgrade 4(1935), 1-138.1 Kurepa, C., Sur les ensembles ordonnes denombrables, Hrvatsko Prirodoslovno DruStro. Glasnik Mat.-Fiz. Astr. Ser. 113 (1948). 145-151. [ M R 10, p. 4371 Ladner. R. E., Application of model theoretic games to discrete linear orders and finite automata, Information and Control 33 (1977), 281-303. [ M R 58, #lo3871 Lal, R. N., A complete extension of ordinal numbers, Amer. Math. Monthly 70 (1963), 501-505. [ M R 27, #2437] Landraitis, C., Infinitary properties of linear orderings, Thesis, Dartmouth College, 1974. Landraitis, C., Definability in well-quasi-ordered sets of structures, J . Symbolic Logic 42 (1977), 288-291, Landraitis, C., A combinatorial property of the homomorphism relation between countable order types, J . Symbofic Logic 44 (1979), 403-41 1. [ M R 80h:04001] Landraitis, C., L , ,,-equivalence between countable and uncountable linear orderings, Fund. Math. 107 (1980), 99-112. Larson, J., A short proof of a partition theorem for the ordinal wo, Ann. Math. Logic 6 (1973), 129-145. [ M R 49, #2401] Larson, J., On some arrow relations, Thesis, Dartmouth College, 1972. Larson, J., A solution for scattered order types of a problem of Hagendorf, Pacific J . Math. 74 (1978), 373-379. [ M R 58, #5422] Larson, J., An independence result for pinning for ordinals, J . London Math. Soc. 19 (1979), 1-6. [ M R 80e: 030591 Larson, J., Partition theorems for certain ordinal products, in Infinite and Finite Sets (Colloq., Kesztheiy, 1973; Dedicated to P . Erd6s on his 60th Birthday), Colloq. Math. SOC.Janos Bolyai, Vol. 10, Vol. 11, pp. 1017 -1024. Amsterdam: North-Holland Publ., 1975. [ M R 51, #I25321 Larson, J., A counterexample in the partition calculus for an uncountable ordinal, Notices Amer. Math. Soc. Abstract #774-E7 (1980), 334. Larson, J., Pinning for pairs of countable ordinals, Fund. Math. 108 (1980), 7-21. [ M R 81i :040071 Larson, J., see also Calvin, F., and Larson, J. Lauchli, H., Mischsummen von Ordnungszahlen, Arch. Math. 10 (1959), 356-359. [ M R 22, #1517] Lauchli, H., A decision procedure for the weak second order theory of linear order, in Contributions 10 Mathematical Logic (Colloquium, Hannover, 1966)Amsterdam: North-Holland, 1968, pp. 189-197. [ M R 39, #5343] Lauchli, H., and Leonard, J., On the elementary theory of linear order, Fund. Math. 59 (1966), 109-1 16. [ M R 33, #7258] Laver, R., On Fraissss order type conjecture, Ann. Math. 93 (1971), 89-1 11. [ M R 43, #4731] Laver, R., An order type decomposition theorem, Ann. of Math. 98 (1973), 96-119. [ M R 47, #8361] Laver, R., Better-quasi-orderings and a class of trees, in Studies in Foundations and Combinatorics: Advances in Mathematics, Supplementary Series 1, pp. 39-48. New York: Academic Press, 1978. Laver, R., see also Baumgartner, J . . Calvin, F., Laver, R., and McKenzie, R. Lemonnier, B., Deviation ordinale des ensembles ordonnts et groupes abtliens totalement ordonnes, C . R . Acad. Sci. Paris 273 (1971), A1013-A1016. [ M R 47, #8388]

468


Leonard, J., see Lauchli, H., and Leonard, J. Lerman. M.. On recursive linear orderings, in Proceedings. Logic Year 1979-80: The Uniuersiry of’ Connecticut (H. Lerman. J. Schmerl. R. Soare, Eds.), Lecture Notes in Mathematics. Vol. 859, Berlin and New York: Springer-Verlag, 1981, pp. 132-142. Lerman, M., and Rosenstein, J. G., Recursive linear orderings 11, to appear in Proceedings, Patras Conference 1980, Amsterdam: North-Holland Publ., 1981. Lmman, M., and Schmerl, J., Theories with recursive models, J. Symbolic Logic 44 (1979). 59 -76. Levitz, H., Uber die Finslerschen hoheren arithmetischen Operationen, Comment. Math. Helu. 41 (1966/67), 273-286. [ M R 35, #4102] Levitz, H., On the Finsler and Doner-Tarski arithmetical hierarchies, Comment. Math. Helr;. 44 (1969), 89-92. [ M R 39, #5375] Levitz, H., An ordered set of arithmetic functions representing the least &-number,2. Math. Logik 21 (1975), 115-120. Lindenbaum, A,, and Tarski, A,, Communication sur les recherches de la theorie des ensembles, C. R. SOC.Sci. Lettres de Varsouie, CI. 111 19 (1926), 299-330. Litman, A., On the monadic theory of o1without A. C., Israel J. Math. 23 (1976), 251-266. [ M R 54, #I25261 Liu, S.-C., Four types of general recursive well-orderings, Notre Dame J. Formal Logic 3 (1962). 75-78. [ M R 29, #I21 Liu, S.-C., Recursive linear orderings and hyperarithmetical functions, Notre Dame J. Formal Logic 3 (1962), 129-132. [ M R 29, #I31 Lloyd, J. T., Lattice ordered groups and o-permutation groups, Ph.D. Thesis, Tulane University, 1964. Longyear, J. Q., Patterns: The structure of linear homogeneous sets, Notices Amer. Math. Soc. 21 (19741, A-293. Longyear, J. Q., The structure of linear homogeneous sets, J . Reine Angew. Math. 266 (1974), 132-135. [ M R 49. #4876] Lopez-Escohar, E. G. K., On defining well-orderings, Fund. Math. 59 (1966), 13-21. [See also Fund. Math. 59 (1966), 299-300.1 [ M R 34,#7354; 34, #30] Lopez-Escobar, E. G. K., Well-orderings and finite quantifiers, J. Math. SOC.Japan 20 (1968), 477-489. [ M R 37, #6169] Magidor, M.. see Gurevich. Y., Magidor, M., and Shelah, S. Manaster, A,, see Hay, L., Manaster, A., and Rosenstein, J. G. Marek, W., Consistance d’une hypothese de Frai’sse sur la definissabilitt dans un langage du second ordre, C. R . Acad. Sri. Paris 276 (1973). A1147-A1150, A1169-AI172. [ M R 52, #2879] Markwald, W., Zur Theorie der konstruktiven Wohlordnungen, Math. Ann. 127 (1954). 135 149. [ M R 15, p. 7711 Matsuzaka, K., On the definition of the product of ordinal numbers (Japanese), Scgaku 8 (1956/57), 95-96. [ M R 22, #10915] Mayer, R. D., and Pierce, R. S., Boolean algebras with ordered bases, Pacijic J . of Math. 10 (1960), 925-942. [ M R 25, #A6961 McBeth, R., Fundamental sequences for initial ordinals smaller than a certain O,, 2. Math. Logik Grundlag. Math. 22 (1976), 97-104. [ M R 56, #2829] McCleary. S . H., see Holland, W. C. and McCleary, S. H. McKenzie, R.,see Baumgartner, J., Calvin, F., Laver, R., and McKenzie, R. Mendelson, E., On aclass of universal ordered sets, Proc. Amer. Math. Soc. 9 (1958), 712-713. [ M R 20, #3075] Metcalf. F. T., and Payne, T. H., On the existence of fixed points in a totally ordered set, Pror. Amer. Math. Sor. 31 (1972). 441 -444.[ M R el, #3931]


469

Meyer. A. R., The inherent computational complexity of theories of ordered sets, Proc. Int. Cony. of Math. Vancouver 1974, pp. 477-482. Miller, E. W., see Dushnik, B., and Miller, E. W. Milner, E. C., On complementary initial and final sections of simply ordered sets, J. London Math. SOC.42 (1967), 269-280. [ M R 35, #2745] Milner, E. C . , Well-quasi-ordering of sequences of ordinal numbers, J. London Math. SOC.43 (1968), 291-296. [ M R 37, #78]. Milner, E. C., Partition relations for ordinal numbers, Canad. J . Math. 21 (1969), 317-334. [ M R 40,#2548] Milner, E. C., Remark on a theorem of Rado, J. London Math. Soc. 44 (1969),222-224. [ M R 38, #4324] Milner, E. C., A finite algorithm for the partition calculus, Proc. Canad. Math. Congress 1971, pp. 117-128. [MR48, #I08341 Milner, E. C., see also Erdos, P., Hajnal, A , , and Milner, E. C.; Erdos, P., and Milner, E. C.; Erdos, P., Milner, E. C., and Rado, R.; Hajnal, A,, and Milner, E. C . Milner, E. C., and Rado, R., The pigeon-hole principle for ordinal numbers, Proc. London Math. SOC.15 (1965), 750-768. [ M R 32, #7419] MiSik, L., On one ordered continuum, Czechoslovak Math. J . 76 (1951), 81-86. [ M R 14, p. 1461 Morel, A. C., On the arithmetic of order types, Trans. Amer. Math. Soc. 92 (1959), 48-71. [ M R 21, #4927] Morel, A. C., Ordering relations admitting automorphisms, Fund. Math. 54 (I964), 279-284. [ M R 30,#lo611 Morel, A. C., A class of relation types isomorphic to the ordinals, Michigan Math. J . 12 (1965), 203-215. [ M R 31, #2153] Morel, A. C., Structure and order structure in Abelian groups, Colloq. Math. 19 (l968), 199-209. [ M R 38, # I 1661 Morel, A. C., On groups admitting a scattered ordering, Ann. Uniu. Sci. Budapest Eiituiis Sect. Math. 14 (1971), 67-76. [ M R 47, #99] Morel, A. C., see also Davis, A. C. Mostowski, A,, see also Doner, J., Mostowski, A., and Tarski, A,; Kuratowski, C . ,and Mostowski, A. Mostowski, A., and Tarski, A,, Boolesche Ringe mit geordnete Basis, Fund. Math. 32 (1939), 69-86. Mostowski, A., and Tarski, A., Arithmetically definable classes and types of well-ordered systems, Bull. Amer. Math. SOC.55 (1949), 65, 1192. Motohashi, N., Some proof-theoretic properties of dense linear orderings and countable wellorderings, Proc. Japan Acad. Ser, A 51 (1975), 301-303. [ M R 51, #7836] Mycielski, J . , and Sierpinski, W., Sur une proprikte des ensembles lineaires, Fund. Math. 58 (1966). 143-147. [ M R 34, #4140] Myers, D., The Boolean algebra of the theory of linear orders, Isruel J . Math. 35 (1980),234-256. [ M R 81i:03038] Myers, D., The Boolean algebras of abelian groups and well-orders, J. Symbolic Logic 39 (1974), 452-458. [ M R 51, #138] Nadel, M., and Stavi, J., La,-equivalence, isomorphism, and potential isomorphism, Trans. Amer. Math. Soc. 236 (1978), 51-74. [ M R 57, #2907] Nagai, S., La solvabilite de certains equations sur les nombres ordinaux transfinis, Proc. Japan Acad. 37 (1961), 121-126, 175-178,276-281,331-335. [ M R 24, #A688; 25, #I9981 von Neumann, J., Zur Einfiihrung der transfiniten Zahlen, Acta Lift. Scient. Uniu. Szeged 1 (1923), 199-208. Neumer, W., Einige Eigenschaften und Anwendungen der 6- und E-Zahlen, Math. Z . 53 (1950), 419-449. [Correction, 54 (1951), 338.1 [ M R 13, p. 1201

470


Neumer, W., Uber den Aufbau der Ordnungszahlen, Math. Z. 53 (1951), 59-69. [ M R 12, p. 3231 Neumer, W., Verallgemeinerungeines Satzes von Alexandroff und Urysohn, Math. Z. 54(1951), 254-261. [ M R 13, p. 3311 Neumer, W., Zum Beweis eines Satzes uber die Polynomdarstellung der Ordnungszahlen, Math. 2. 55 (1952), 399-400. [ M R 13, p. 9231 Neumer, W., Zur Konstruktion von Ordnungszahlen, Math. 2 . 5 8 (1953), 391-413; 59 (1954), 434-454; 60 (1954), 1-16; 61 (1954). 47-69; 64 (1956), 435-456. [ M R 15, pp. 512, 689; 16, pp. 19, 343; 18, p. 1391 Neumer, W., Uber Mischsummenvon Ordnungszahlen, Arch. Math. 5(1954), 244-248. [ M R 16, P. 191 Neumer, W., Uber Folgen von Ordnungszahlen, 2. Math. Logik Grundlag. Math. 1(1955), 109-126. [ M R 17, D. 9521 Neumer, W., Algorithmen fur Ordnungszahlen und Normalfunktionen, Z. Math. Logik Grundlug. Math. 3 (1957), 108-150; 6(1960), 1-65,16 (1970), 1-112. [ M R 20, #1631; 23, #A2314; 43, #7335] Neumer, W., Kritische Zahlen und bestimmt divergente transfinite Funktionen, Math. Z. 70 (1958), 190-192. [ M R 20, #5742] Nosal, E., On a partition relation for ordinal numbers, J. London Math. Sac. (2) 8 (1974), 306-310. [ M R 50, #I27311 Nosal, E., Partition relations for denumerable ordinals, J . Combin. Theory Ser. B 27 (1979). 190-197. [ M R 80h:040051 Novak, J., A paradoxical theorem, Fund. Math. 37 (1950), 77-83. [MR 13, p. 3301 2' containing a dense set of power K,, Novak, J., On some ordered continuum of power O Czechoslovak Math. J. 76 (1951), 63-79. [ M R 14, p. 1461 Novak, J., On partition of an ordered continuum, Fund. Math. 39 (1952),53-64. [MR 15, p. 171 Novak, J., On some characteristics of an ordered continuum (Russian, English summary), Czechoslovak Math. J. 77 (1952), 369-386. [ M R 15, p. 9431 Novak, V., On the lexicographic dimension of linearly ordered sets, Fund. Math. 56 (1964), 9-20. [MR 34, #7412] Novak, V., Uber Realisierungen geordneter Mengen, Arch. Math. (Brno) 11 (1975), 241 -245. [ M R 53, #lo6631 Novotny, M., Construction de certains continus ordonnes de puissance 2'", Czechoslovak Math. J . 76 (1951), 87-95. [ M R 14, p. 1461 Novotny, M., Sur la representation des ensembles ordonnes, Fund. Math. 39 (1952), 97-102. [MR 15, p. 171 Novotny, M., Sur une caracteristique du continu ordonne (Russian and French summary), Czechoslovak Math. J . 78 (1953), 75-82. [ M R 15, p. 9431 Ohkuma, T., On discrete homogeneous chams, Kodai Math. Sem. Rep. (1952), 23-30. [ M R 13, p. 8281 Ohkuma, T., Sur quelques ensembles ordonnes lineairement, Proc. Japan Acad. 30 (1954), 805-808. [ M R 17, p. 201 Ohkuma, T., Sur quelques ensembles ordonnes lineairement, Fund. Math. 43 (1956), 326-337. [MR 18,p. 8681 Padmavally, K., Generalization of rational numbers, Rev. Mat. Hisp.Amer. 12 (1952), 249-265. [ M R 15, p. 18) Padmavally, K., Generalisation of the order type of rational numbers, Rev. Mat. Hisp. Amer. 14 (1954), 50-73. [ M R 16, p. 119) Padmavally, K., A remark on order-types, Fund. Math. 42 (1955), 312-318. [ M R 17, p. 9521 Parikh, R. J., Some generalizations of the notion of well-ordering, Z. Math. Logik Grundiag. Math. 12 (1966), 333-340. [ M R 34, #4131]


47 1

Payne, T. H., see Metcalf, F. T., and Payne, T. H. Peretyatkin, M. C., Every recursively enumerable extension of a theory of linear order has a constructible model, AIgebra i Logrka 12 (1973), 21 1-219. Pfeiffer, H.. Ausgezeichnete Folgenfir Getrisse abschnitte der zweifen und weiterer Zahlenklassen, Technischen Hochschule Hannover. Stuttgart : Karl Mayer, 1964. [ M R 31, #4730] Pfeiffer, H., Ein Bezeichnungssystem fur Ordinalzahlen, Arch. Math. Logik Grundlag. 12 (1969), 12-17. [MR40, #5419] Pfeiffer, H., Ein Bezeichnungssystem fur Ordinalzahlen, Arch. Math. Logik Grundlag. 13(1970), 74-90. [MR 44,#60] Pfeiffe;, H., Uber zwei Bezeichnungssysteme fur Ordinalzahlen, Arch. Math. Logik Grundlag. 16 (1974), 23-36. [ M R 50, #I8861 Pierce, R. S . , see Mayer, R. D., and Pierce, R. S. Pinus, A. G., On the theory of convex subsets, Siberian Math. J . 13 (1972), 157-161. Pinus, A. G., The imbedding of totally ordered sets, Math. Notes 11 (1972), 54-57. [See also 12 (1972), 733.1 [ M R 45, #3256; 47, #6560] Pinus, A. G., Lexicographic powers of totally ordered sets, Siberian Math. J . 14(1973), 478-482. [ M R 4 8 , #I08171 Pinus, A. G., The number of pairwise incornparable order types, Siberian Math. J . 14 (1973), 164-167. [MR 48, #I08161 Pinus, A. G.. Countable indecomposable scattered order types, Math. Notes 13(1973), 67-70. [ M R 47, #6561] Pinus, A . G., Effective linear orders, Sibcriczrz Math. J . 16 (1975), 956-962. [ M R 53, #I371 Rnus, A. G., Collections of pairwise incomparable linear orders of a given degree of dispersal (Russian), I m . Vysi. UEebn. Zaved. Mathemarika 1979 no. 7.62-65. TMR 80m:06001] Polakova, N., Note on characteristics of ordered sets (Czech, Russian, and English summaries), Casopis Pest. Mat. 88 (1963), 387-390. [ M R 31, #4741] PondEliEek, B., see Kowalski, D.. and PondEliCek, B. Pouzet, M., Sur les conjectures de Fraisse et les premeilleurs ordres, C. R. Acad. Sci. Paris 270 (1970), 1-3. [ M R 41, #6728] Pouzet, M., Les chaines completes sont simplifiable a droite pour la multiplication ordinale, Portugal. Math. 35 (1976), 127-135. [MR 57, #5841] Pouzet. M., Ensemble ordonne universe1 recouvert par deux chaines, J . Combin. Theory Ser. B 25 (1978). 1-25. [MR 58, 8164341 Pouzet, M., see also Bonnet, R., Corominas, E., and Pouzet, M. Prikry, K., On a set-theoretic partition relation, Duke Math. J . 39 (1972), 77-83. [ M R 45. #I7721 Rabin, M. O., Decidability of second order theories and automata on infinite trees, Trans. Amer. Math. Soc. 149 (1969), 1-35. [ M R 38, #44] Rado, R., The minimal sum of a series of ordinal numbers, J . London Math. Soc. 29 (1954), 218-232. [MR 16, p. 191 Rado, R., The partition calculus, in Recerrt Progress in Combinatorics (Proc. Third Waterloo Conf. Combinatorics. May 1968). pp. 151-159. New York: Academic Press 1969. [ M R 41, #76] Rado, R., see also Erdos, P., Milner, E. C., and Rado, R.; Erdos, P., and Rado, R.; Milner, E. C., and Rado, R. Rautenberg, W., Die elementare Theorie der diskret geordneten Mengen, Wiss. Z. HumboldtUniv., Berlin Math.-Natur. Reihe 15 (1966). 677-680. [ M R 36, #I3151 Rautenberg, W . , Unterscheidbarkeit endlicher geordneter Mengen mit gegebener Anzahl von Quantoren, Z . Math. Logik Grundlug. Math. 14 (1968), 267-272. [ M R 37, #2595] Remmel. J., Realizing partial orderings by classes of co-simple sets, PaciJic J . Math. 76 (1978), 169-184

472


Remmel, J., Recursively categorical linear orderings, to appear. Ricabarra, R., Partitions in sets of ordinal numbers(Spanish), Rev. Mat. Cuyana 2(!956), 1-27. [ M R 22, #1518] Rice. H. G., Recursive and recursively enumerable orders, Trans. Amer. Math. Soc. 83 (1956). 277-300. [ M R 18, p. 7121 Rosenstein, J. G., KO-categoricity of linear orderings, Fund. Math. 64 (1969), 1-5. [ M R 39, #3982] Rosenstein, J. G., see also Hay. L., Manaster, A,, and Rosenstein, J. G . ; Lerman, M., and Rosenstein, J. G. Norre Dame J . Formal Logic 15 (1974), 122-132. Rosenthal, J. W., Models of Th((w", i)), [MR 49, #8853] Rotman, B., Principal sequences and regressive functions, J. London Math. SOC.38 (1963). 501-504. [ M R 28, #SO061 Rotman, B., A note on principal sequences, Proc. Glasgow Math. Assoc. 6 (1964), 133-135. [ M R 29,1557351 Rotman, B., On the comparison of order types, Acta Math. Acad. Sci. Hungar. 19 (1968). 31 1-327. [ M R 38, #86] Rotman. B.. On countable indecomposable order types, J . London Math. Soc. (2) 2 (1970). 33-39. [ M R 40, #5504] Rotman, B., A mapping theorem for countable well-ordered sets, J. London Math. Soc. (2) 2 (1970), 509-512. [ M R 42, #4405] Rotman, B., Note on the decomposition of ordered sets, J . London Math. Soc. (2) 3 (1971), 561-562. [ M R 44,#2618] Rotman, B., Remark on countable indecomposable order types, J. London Marh. SOC.(2) 4 (1972), 725-728. [ M R 45, #6702] Rotman, B., Boolean algebras with ordered bases, Fund. Math. 75 (1972), 187-197. [ M R 46, #1671] Rousseau, G., Remark on rigid order types (loose Russian summary), Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 15 (1967), 599-602. [ M R 37, #1253] Roy. D. K., Preliminary results on r.e. presented linear orders, J. Symbolic Logic. (to be published). Rubin, A. L., and Rubin, J. E., Extended operations and relations on the class of ordinal numbers, Fund. Math. 65 (1969), 227-242. [ M R 39, #6756] Rubin, A. L., and Rubin, J. E., Accumulation functions on the ordinals, Fund. Math. 70 (1971). 205-220. [ M R 43, #7338] Rubin, J. E., Several relations on the class of ordinal numbers, Z. Math. Logik Grundlag. Murh. 9 (1963), 351-357. [ M R 27, #4761] Rubin, I. E., see also Howard, P. E., and Rubin, J. E.; Rubin, A. L., and Rubin, J. E. Rubin, M., Theories of linear order, Israel J . Math. 17 (1974), 392-443. [ M R 50, #1871] Rubin, M., Vaught'sconjecture for linear orderings, Notices Amer. Math. SOC.24(1977), A-390. Rudin, M. E., A subset of the countable ordinals, Amer. Math. Monthly 64 (1957). 351. [ M R 19, p. 41 Ruziewicz, S., and Sierpinski, W., Sur un ensemble parfait qui a avec toute sa translation au plus un point commun, Fund. Math. 19 (1932), 17-21. Saarnio, U., Von den Rechenoperationen hoherer ordnung bei der Darstellung der transfiniten Ordnungszahlen, Marh. Ann. 146 (1962), 217-225. [ M R 25, #I91 Saarnio. U ., Die kritischen Zahlen hoherer ordnung inverhalb der zweiten Cantorschen Zahlenklasse, Math. Ann. 178 (1968). 173-183. [ M R 38, #I0041 Sabbagh, G., see Haddad, L., and Sabbagh, G. Schmerl, J., see Lerman, M., and Schmerl, J.


473

Schmidt, D., Bounds for the closure ordinals of replete monotonic increasing functions, J . Symbolic Logic 40 (1975), 305-316. [ M R 52, #7865] Schmidt, D., Associative ordinal functions, well partial orderings and a problem of Skolem, Z . Math. Logik and Grundlag. Math. 24 (1978), 297-302. Schmidt, D., A partition theorem for ordinals, J . Combin. Theory Ser. A 27 (1979), 382-384. [M R 81d :04004) Schoenflies, A,. Entwickelung der Mengenlehre und Ihrer Anwendungen, 2nd ed. Leipzig : Teubner, 1913. Schutte, K., Kennzeichnung von Ordnungszahlen durch rekursiv erklarte Funktionen, Math. Ann. 127 (1954), 15-32. [ M R 15, p. 6891 Schutte, K., Predicative well-orderings, in Formal Systems and Recursive Functions (eds. J. N. Crossley and M. A. E. Dummett), Amsterdam: North-Holland Publ., 1965, pp. 280-303, [ M R 33, #5466] Schutte, K., Ein konstruktives System von Ordinalzahlen, Arch. Math. Logik Grundlag. 11 (1968), 126-137; 12 (1969), 3-11. [ M R 4 0 , #2534, #5418]. Schutte. K., Primitiv-rekursive Ordinalzahl-funktionen, Buyer. Akad. Wiss. Math.-Natur. K1. S.-B. (1975), 143-153. [ M R 54, #12503] Schwartz, N., qa-structuren, Math. Z . 158 (1978), 147-155. [ M R 57, #9523] Seiferas, J. I., see Kleinberg, E. M., and Seiferas, J. I. Shelah, S., The monadic theory of order. Ann. of Math. (2) 102 (1979, 379-419. [ M R 58, +/ 103901 Shelah, S., see also Amit, R., and Shelah. S ; Avraham, U. and Shelah, S.; Glass, A., Gurevich Y., Holland, W. C.. and Shelah. S . ; Gurevich, Y., Magidor, M., and Shelah, S . ; Gurevich, Y.. and Shelah, S. Shepherdson, J. C., Well-ordered subseries of general series, Proc. London Math. Soc. ( 3 ) 1 (1951), 291-307. [ M R 13, p. 3301 Sherman, S., Some new properties of transfinite ordinals, Bull. Amer. Math. Sue. 47 (1941). 111-1 16. [ M R 2, p. 2551 Shoenfield, J . , see Kreisel, G . , Shoenfield, J., and Wang, H. Sieczka, F., Sur l’unicite de la decomposition de nombres ordinaux en facteurs irreducibles, Fund. Math. 5(1924). 172-176. Siefkes, D., see Biichi. J . R., and Siefkes. D. Sierpinski, W., Une remarque sur la notion de l’ordre, Fund. Math. 2 (1921), 199-200. Sierpinski, W., Sur un probleme concernant les sous-ensembles croissants du continu, Fund. Mafh. 3(1922), 109-112. Sierpinski. W., Lecons sur les nombres transfinis, Paris: Gauthier-Villars, 1928. Sierpinski, W.. A property of ordinal numbers, Bull. Calcutta Math. Sue. 20 (1930), 21 -22. Sierpinski, W., Generalisation d’un theorime de Cantor concernant les ensembles ordonnks denombrables, Fund. Math. 18 (1932), 280-284. Sierpinski, W., Sur les translations des ensembles lineaires, Fund. Math. 19 (1932), 22-28. Sierpinski, W., Un theoreme concernant les transformations des ensembles lineaires, Fund. Math. 19 (1932), 205-210. Sierpinski, W., Sur un probleme de la theorie des relations, Ann. R . Scuola Norm. Sup. Pisa, Ser. 2 2 (1933), 285-287. Sierpinski, W., Sur l’existence d’une base dtnombrable d’ensembles lineaires denombrables, Fund. Math. 31 (1938), 259-261. Sierpinski, W., Remarque sur les ensembles des nombres ordinaux de classes I et 11, Rev. Cien. (Lima) 41 (1939), 289-296. [ M R 1. p. 2061 Sierpinski, W., Sur les types ordinaux dont tous les vrais restes sont egaux, Fund. Math. 39 (1952), 1-7. [ M R 14, p. 10691

474


Sierpinski, W., Sur une propnett des ensembles ordonnes, Pontijicia Acad. Sci. 4 (1940). 207208. [ M R 2, p. 2561 Sierpinski, W., Sur la division des types ordinaux, Fund. Math. 35 (1948), 1-12. [ M R 10, p. 3581 Sierpinski, W., Sur les translations des ensembles Iineaires, Fund. Math. 35 (1948), 159-164. [ M R 10, p. 2871 Sierpinski, W., Sur les ensembles lineaires denombrables non equivalents par decomposition finie, Fund. Math. 36 (1949). 1-6. [ M R 11, p. 1651 Sierpinski, W., Sur les series infinies de nombres ordinaux, Fund. Math. 36 (1949), 248-253. [ M R 12, p. 141 Sierpinski, W., Sur une propritte des ensembles ordonnes, Fund. Math. 36 (1949), 56-67. [ M R 11, p. 1651 Sierpinski, W., Le dernier theoreme de Fermat pour les nombres ordinaux, Fund. Math. 37 (1950), 201 -205. [ M R 12, p. 6831 Sierpinski, W., Solution de ]’equation o5= s“ pour les nombres ordinaux, Acta Sci. Math. 12B (1950), 45-60. [ M R 11, p. 6461 Sierpinski, W . ,Sur les produits infinis de nombres ordinaux, C . R . Soc. Sci. Lift. Varsouie I l l 43 (1950), 201 -205. [ M R 14, p. 10683 Sierpinski, W., Sur les types d’ordre des ensembles lineaires, Fund. Math. 37 (1950), 253-264. [ M R 13, p. 191 Sierpinski, W., Sur les types ordinaux des ensembles lineaires, Atti Accad. Naz. Lincei Rend. CI. Sci. Fix Mat. Nut. (8)8 (1950), 427-428. [ M R 13, p. 191. Sierpinski, W., Sur I’extension d’un theoreme de M. D. Pompeiu aux nombres transfinis, C. R . SOC.Sci. Litt. Varsouie I l l 43 (l950), 1-3. [ M R 14, p. 10681. Sierpinski, W., Sur un type ordinal denombrable qui a une infinite indknombrable de diviseurs gauches, Fund. Math. 37 (1950), 206-208. [ M R 12, p. 6831 Sierpinski, W., Dernieres recherches et problemes de la theorie des ensembles, Rend. Mat., Ser. VlO(1951). 1-11. Sierpinski, W., Sur les fonctions continues d’une variable ordinale, Fund. Math. 38 (1951). 204-208. [ M R 13, p. 8281 Sierpinski, W., Sur les diviseurs de types ordinaux, C. R. Premier Cong. Math. Budapest (1952). 397-399. [ M R 14, p. 10691 Sierpinski, W., Sur I’equation 5’ = q3 1 pour les nombres ordinaux transfinis, Fund. Math. 43(1956), 1-2. [ M R 17, p. 11901 Sierpinski, W., Sur quelques problemes arithmetiques de la theorie des nombres ordinaux, Czechoslovak Math. J. 6 (1956), 161 -163. [ M R 18, p. 71 I] Sierpinski, W., Sur une propriete des nombres ordinaux, Fund. Math. 43 (1956), 139-140. [ M R 17, p. 11901 Sierpinski, W., CardinalandOrdinal Numbers, Warsaw: PWN, 1958 (2nd rev. ed. 1965).[MRZO. #2288; 33, #2549] Sierpinski, W., see also Davis, A. C., and Sierpinski, W.; Mycielski, J., and Sierpinski, W . ; Ruziewicz, S., and Sierpinski, W. Skolem, Th., Logisch-Kombinatorische Untersuchungen iiber die Enjiillbarkeit order Beweisharkzeit mathematische Sitze nebst eirenz Theoreme iiber dichte Mengen, Skrifter utget av Videnskapsselskapet i Kristiania, no. 4, I classe, 1920. Skolem, Th., An ordered set of arithmetic functions representing the least s-number, Det Kon. Norske V. Selskabs Forhardlinger 29 (1956), 54-59. Slater, M., On a class of order-types generalizing ordinals, Fund. Math. 54 (1964), 259-277. [ M R 29, #5736] Slomson, A. B., Generalized quantifiers and well-orderings, Arch. Math. Logik Grundlug. 15 (1972), 57-73. [ M R 4 8 , #3698].

+


475

Slupecki, J., Sur la multiplication des types ordinaux, Colloq. Math. 3 (1954), 41 -43. [ M R 15, p. 9421 Sonenberg, E. A., On the elementary theory of inductive order, Arch. Math. Loqik Grundlag. 19 (1978/9), 1-2, 13-22. [ M R 80d:03027b] Sonenberg. E. A,, Nonstandard models of ordinal arithmetics, Z . Math. Loqik Grundlag. Math. 25 (1979), 5-27. [ M R 80h:03051]. Specker, E., Teilmengen yon Mengen mit Relationen. Comment. Math. Helv. 31 (1957), 302-314. Spector, C., Recursive well-orderings, J . Symbolic Logic20 (1955), 151-163. [ M R 17, p. 5701 Stavi, J., see Nadel, M., and Stavi, J. Steckel, S., Remarque sur une classe d’ensetnbles ordonnes, Fund. Math. 11 (1928), 285-287. Sudan, G., Zur Jacobsthalschen transfiniten Arithmetik, Math. Ann. 105 (1931), 40-51. Sudan, G., Sur un theoreme de Hessenberg, Fund. Math. 18 (1932), 293-297. Sudan, G., Sur certains nombres principaux, Bull. Math. SOC.Roum. 35 (1933),237. Sudan, G., Sur les singularites des fonctions transfinis, Disquisitiones Math. Phys. I (1941), 315-320. [ M R 8, p. 5051. Sudan, G . , Sur les nombres delta, Acad. Roum. Bull. Sect. Sci.26 (1946), 212-223. [ M R 10, p. 2381 Sudan, G., Sur la “forme normale” de Cantor et la definition de Hessenberg pour la puissance des nombres ordinaux, Acad. Roum. Bull. Sect. Sci. 27 (1947), 108-117. [ M R 10, p. 2391 Sudan, G., Sur une propritte des nombres epsilon, Acad. Roum. Bull. Sect. Sci. 27 (1947), 258-264. [ M R 10, p. 2391. Sunyer i Balaguer, F., Sur les types d‘ordre distinct dont les n-itmes puissances sont tquivalentes, Fund. Math. 46 (1958), 221-224. [ M R 20, #6983] Swierczkowski, S., On some equation in transfinite ordinals, Fund. Math. 45 (1958), 213-216. [ M R 20, #3076] Szpilrajn, E., Sur l’extension de I’ordre partial, Fund. Math. 16 (1930), 386-389. Tarski, A,, Sur les principes de l’axiomatique des nombres ordinaux, Ann. SOC.Polon. Math. 3 (1924), 148. Tarski, A., Sur les classes d’ensembles closes par rapport a certaines operations elementaires, Fund. Math. 16 (1930), 181-304. Tarski, A,, On well-ordered subsets of any set, Fund. Math. 32 (1939) 176-183. Tarski, A,. Ordinal Algebras, Amsterdam: North-Holland Publ. 1956. [ M R 18, p. 6321 Tarski, A,, see also Doner, J., and Tarski, A. ; Doner, J., Mostowski, A,, and Tarski, A , ; Lindenbaum, A., and Tarski, A. ; Mostowski, A,, and Tarski, A. Thomas, W., On the bounded monadic theory of well-ordered structures, J. Symbolic Logic 45 (1980),334-13R. [ M R 81h:03080] Toulmin, G . H., Shuffling ordinals and transfinite dimension, Proc. Lond. Math. SOC.(3) 4 (1954), 177-195.[MR 16, p. 5021 Truss, J., The well-ordered and well-orderable subsets of a set, 2. Math. Loqik Grundlaq. Math. 19 (1973), 211-214.[MR 47, #8308] Tuschik, H. P., On the decidability of the theory of linear orderings in the language L ( Q l ) , in Lecture Notes in Math. Vol. 619, Berlin and New York: Springer-Verlag, 1977. pp. 291-304. [ M R 57, #I60371 Tuschik, H. P., On the decidability of the theory of linear orderings with generalized quantifiers, Fund. Math. 107 (1980), 21-32. Urysohn, P., Un t h e o r h e sur lapuissance desensembles ordonnes, Fund. Math. 5(1923),14-19. [See also 6 (1924), 278.1 Vaquer Timoner, J., On the ordinal product of two iixed sets (Spanish), Rev. Acad. Cien. Madrid 54 (1960), 189-192.[MR 23, #A155Y] Vaughan, H. E., Well-ordered subsets and maximal members of ordered sets, PaciJic J . Math. 2 (1952), 407-412.[MR 14, p. 3621

476


Veblen, O., Definition in terms of order alone in the linear continuum and in well-ordered sets. Trans. Amer. Math. Soc. 6 (1905), 165-171. Veblen, O., Continuous increasing functions of finite and transfinite ordinals, Trans. Amer. Math. SOC.9 (1908), 280-292. Venkataraman, R.,Symmetric ordered sets, Math. Z . 79 (1960), 10-20. [ M R 25. #I 1151 Venkataraman, R., Natural sums on ordinal numbers, Indian J . Math. 7 (1965), 47-59.[MR 36. #5000] Wachs. E. A., Models of well-orderings, Ph.D. Thesis, Monash Univ., Clayton, Victoria. Australia, 1976. Wakulicz, A., Sur les sommes de quatre nombres ordinaux, SOC.Sci. Lettres Varsooie. C . R. C1. I l l Sci. Math. Phys. 42 (1949), 23-28. [ M R 13, p. 9231 Wakulicz, A,, Sur les sommes d’un nombre fini de nombres ordinaux, Fund. Math. 36 (1949). 254-266. [Correction, Fund. Math. 38 (1951), 239.1 [MR 12, p. 141 Wang, H., see Kreisel, G., Shoenfield, J., and Wang, H. Wang, K., see Wang, S. and Wang. K. Wang, S., and Wang. K . , On some equations of ordinal numbers (Chinese, English summary). Adu. in Math. 3 (1957),646-649. [ M R 20, #3791] Watnick, R., Recursive and constructive linear orderings, Thesis, Rutgers University, 1980. Watnick, R., Constructive and recursive scattered order types, Proceedings. Logic Year 1979-80: The University of Connecticuf (M. Lerman, J. Schmerl, R. Soare, Eds.), Lecture Notes in Mathematics Vol. 859, Berlin and New York: Springer-Verlag, 312-326. Watnick, R., A generalization of Tennenbaum’s theorem on effectively finite recursive orderings, J . Symbolic Logic. (to be published). Weyhrauch, R., Relations between some hierarchies of ordinal functions and functionals, Ph.D. Thesis, Stanford University, 1976. Williams, H.P., A formalization of the arithmetic of the ordinals less than ww, Notre Dame J . Formal Logic 10 (1969), 77-89. [ M R 45, ti721 Wolter, H., see Herre, H., and Wolter, H. Zakon. E.. Left side distributive law of the multiplication of transfinite numbers (Hebrew, English summary), Riueon Lematematika 6 (1953), 28-32. [MR 14, p. 7331 Zakon, E.. On fractions of ordinal numbers, Technion. Israel Inst. Tech. Sci. Publ. 6 (1954/55). 94-103. [ M R 17, p. 3511 Zakon, E., On the relation of “similarity” between transfinite numbers (Hebrew, English summary), Riueon Lematematika 7 (1954), 44-49. [ M R 15, p. 4091 Zakon, E.. On common multiples of ordinal numbers, Cunad. Math. Bull. 3(1960), 31-33.[MR 22, #2551] Zuckerman. M.. Ordinal sum-sets, Proc. Amer. Math. SOC.35 (1972). 242-248. [MR47,#3177] Zuckerman, M . , Natural sums of ordinals, Fund. Math. 77 (1973),289-294. [ M R 4 8 , #I 101 Zuckerman. M., Sums of at least 9 ordinals. Notre Dame J . Formal Logic 14 (1973), 263-268. [ M R 49.1147801 Zuckerman, M . . Sums ofat most 8 ordinals, Z . Math. Logik Grundlag. Math. 19(1973),435-446. [ M R 49. #2398] Zuckerman, M., Arithmetic operations on ordinals, Notre Dame J , Formal Logic 16 (197% 578-582. Zvengrowski, P., Perfect transfinite numbers, Fund. Math. 52 (1963), 123-128. [ M R 26, #4923]

LIST OF NOTATION

8 9 9 10 10-1 1 I1 I1 11 13 14 16, 17 18 19 21 21 38 49 50 51 52 56 57 58 60, 132 61, 69, 91. 127 65 65 65 70 71 72 74 76 82 477

Page 84-85 86 96 97 99 108 111 113 116 118 133 137 148 163 183 183 183 184 185 186,402 188 189 190 190 208 -209 224 232 233 233 233 235 237 239 247 249 250,344 250 274 276

478

LIST OF NOTATION Page 219 304 312 318 318 3 50 353 355 361

Page

371 396 402 403 403 404 415 41 7,430 42 1

Author Index

This index refers only to those authors cited in the text itself. A complete list of those mentioned in the book can be obtained by adding to this index the names of the authors in the Complete Bibliography of Linear Ordermps (pp. 456-476) and in the Bibliography to Chapter 11. Numbers in italics refer to the pages on which the complete references are listed. Erdos, P., 76, 92, 169,172, 205,206,207, 211, 212, 214,215,216,217

A

Adams, M. E., 155. 171 Amit, R., 256, 325, 334, 341,342 Ash, C. J., 155, 162, 163, 171, 172

F Feferman, S., 96, 109 Feiner, L., 430, 454 Fellner, S., 337, 342, 427, 454 Fleischer, I., 170, 172 Fraisst, R.,84,92,96, 108,109, 145,172, 177, 179, 196,203

B Bachman, H., 59,66 Baldwin, J . , 279, 342 Barwise, J . , 246, 355, 358,369 Baumgartner, J., 154, 159,172, 216 Blass, A,, 279, 342 Bonnet, T., 158, 172 Biichi, J. R., 397, 399, 400

G Galvin, F., 194,203, 215, 216, 217 Garland, S. J., 399, 400 Gillman, L., 169, 172 Ginsburg, S., 145, 158, 159, I72 Glass, A., 136, 143, 279, 342 Goffman, C., 133, 143 Gurevich, Y.,136, 143, 397, 398, 399,400

C

Cantor, G., 26,28,61,66, 145 Chang,C. C., 133, 143,215,216 Chen, K.-H., 421,454 Church, A,, 413, 454 Cohn, P. M., 134,143

H

D Davis, M., 246, 405, 454 Doner, J., 269, 342 Dushnik, B., 71, 73, 75, 145, 147, 155, /72

E Ehrenfeucht, A., 96, 108, 109, 133, 143, 169, 271,282,342, 370, 376, 311, 381,400 Enderton, H., 229, 246,282 Engeler, E., 297, 342

Haddad, L., 212,214,217 Hagendorf, J. G., 179, 180, 181, 183,202,203 Hajnal,A.,76,92,209,211,2l4,215,216,217, 218 Harzheim, E., 167, 169, 171, 172 Hausdorff, F., 74, 75, 76, 84, 86, 87, 92, 163, 164, 165, 166, 169, 172 Hay, L., 421,447,451,455 Higman, G., 187,203 Holland, W. C., 135, 136, 143

479

AUTHOR INDEX

480 J

Jambu, M., 136, 143 Jenkyns, T. A., 187,203 Jullien, P., 176, 179, 203

K Karp, C., 349,357,369 Keisler, H. J., 237, 240, 246, 367, 369 Kleene, S. C., 242,246,405,413,418,454,455 Kochen, S., 169,172 Kreisel, G., 421, 455 Kruskal, J. B., 187,203 Kueker, D., 279,342,358,369 Kuratowski, C., 163, 169, 172

Nosal, E., 215,219 Novak, J., 59,66 Novak, V., 171, 172 0 Ohkuma, T., 135,143, 155, 172

P Padmavally, K., 167, 172 Parsons, T. D., 113,120 Passow, E., 134, 144 Pinus, A. G., 413,455 Prikry, K., 194,203,217

L Landraitis, C., 201, 203, 346, 358, 361, 363, 367,368,369 Larson, J., 176, 203, 215, 217,218 Lauchli, H., 113, 115, 119,120,256,271,325, 342,383,396,397,400 Laver, R., 178, 179, 185, 188, 190, 191, 196, 197,200,201,202,203,216 Leonard, J., 113, 115,119, 120,256,271,325, 342 Lerman, M., 427,442,452,455 Litman, A., 397, 398, 399,400 Lloyd, J. T., 135,143 Longyear, J. Q., 136,143 Lopez-Escobar, E. G. K., 358,369

M Makkai, M., 346,361, 363,369 Manaster, A. B., 229,246,421,451,455 Matt, A., 205, 217 Mendelson, E., 170, 172, 229, 242,246 Miller, E. W., 71, 73, 75, 145, 147, 155, 172 Milner, E. C., 214,219 Morel, A. C., 121, 123, 124, 133, 143 Morley, M.,89, 92, 285, 296,343 Mostowski, A., 108, 109, 163, 169, 172, 269, 342,343 Myers, D., 256, 325, 342,343 N Nadel, M., 346, 368, 369 Nash-Williams,C. St. J. A., 178,187,190,194, 203,204

R Rabin, M. O., 397,400 Rado, R., 188, 190, 191, 204, 205, 206, 207, 212, 215, 217,219 Ramsey, F. P., 11 1, 120,219 Remmel, J., 451,455 Robinson, A., 274,343 Rogers, H., 405,413,417,418,446,455 Rosenstein, J. G., 137,144,286,297,299,316, 343,421,427,442,447,451,455 Rosenthal, J., 270,343 Rotman, B., 145, 158, 171,172, 176, 177,204 Rubin, M., 264,265,266,277,286,287,296, 307,311,312, 313, 325,343,368,369 Ryll-Nardzewski, C., 297,343

S

Sabbagh, G., 212,214,217 Scott, D., 359, 369 Shelah, S., 136, 143, 217, 256, 334, 341, 342, 397,398,399,400 Shepherdson, J. C., 84,92 Shoenfield, J., 421, 455 Siefkes, D., 397, 400 Sierpinski, W., 22, 59,66, 112, 120, 129,144, 145, 152, 153, 154, 155, 156, 158, 160, 163, 165, 168, 169,173,174,204 Skolem, Th., 116, 120 Specker, E., 212,219 Stavi, J., 368, 369 Svenonius, L., 297,343

48 1

AUTHOR INDEX

T

V

Tarski, A., 108, 109, 169, 172, 217, 253. 269, 270, 214,342, 343

Vaught, R., 274,279,282, 292,295,296,343 Venkataraman, R., 123, 124, 144

W U

Urysohn, P., 164, 166,173

Wang, H., 421,455 Watnick, R., 413,442,454, 455

Subject Index

A

C

Additively indecomposable order type, I74 hereditarily additively indecomposable, I96 KO-categoricallinear ordering, 297 Algorithms and decision procedures, 241 -246 Almost transitive linear ordering, 136 almost n-tuply transitive linear ordering, 137 Anti-chain, 183 Anti-lexicographic product, 130 Arithmetical hierarchy, 4l4ff Arithmetical set, 415 Aronszajn trees, 398 Atomic formula, 229 Automorphism, 8 of linear orderings. 132- 136 Axiomatizable, 234, 267 finitely axiomatizable, 253 Axiomatization, 234, 253

B Back and forth property, 352 6-regressive back and forth property. 353 Backwards linear ordering, 14 backwards order type, 15 Barrier, 190 sub-barrier, 192 Better-quasi-ordering (bqo), I88ff Block, 190 Boolean combination. 231 Bound greatest lower bound, 13 least upper bound, 13 lower bound, 13 upper bound, 13 Bounded, 13 above, 13 below, 13 482

Cantor Normal Form, 61 Cardinal number, 50-52, 56-58 arithmetic of, 64-66 cardinality, 50 regular, 57, 165 of a set, 50 strongly inaccessible, 166 Chain, 4 Characterizable linear ordering completely characterized, 238, 363ff Church’s Thesis, 242,401 Closed interval, I 1 Closed subset of o , , 58 Cofinal, 13 Cofinality of an ordinal, 56 Coinitial, 13 Compactness Theorem, 236 Complete diagram, 276 Complete linear ordering, 33 Complete theory, 233 Completeness Theorem, 236 Completion of a linear ordering. 37 Condensations of linear orderings, 69-75 cD, 304 cE, 318 cFr71 C G , 119 cs, 74 cw, 72 finite condensation, 76 is condensed to, 70 iterated condensations, 79ff label condensation c,, 138 shuffle condensation cs, 138 Conjunctive normal form, 232 Consistent, 236 Continuous functions of ordinals, 62-64 Continuum, 65-66

483

SUBJECT INDEX

Continuum hypothesis, 65 generalized continuum hypothesis, 65 Convex subset, 10 Correctness Theorem, 235 Countable linear ordering, 23 D Decidable, 241 Decidable theories of linear orderings, 267ff. 383ff Decision procedure, 241 Dedekind complete linear ordering, 34 Dedekind cut, 34 Deduction, 235 Definable, 261 Degree of an ordinal, 62 &equivalence, 353 Ak order type, 430 A&-predicate,417 ®ressive back and forth property, 353 Dense linear orderings, 25-40 K,-dense linear ordering, 75, 154 c-dense linear ordering, 154 dense in a linear ordering, 36 uncountable dense linear orderings, 145173 Densely subuniform linear ordering, I62 Diagonal arguments, 145ff, 43% Disjunctive normal form, 232 Domain, 224 Doubly transitive linear ordering, 31, 155

E Ehrenfeucht-Frai’ssk game, 96ff. 247ff, 349ff. 370ff Elementarily equivalent, 232 Elementary class, 234 Elementary embedding, 274 elementarily embeddable, 274 elementary substructure, 274 Elimination of quantifiers method, 293 Embeddable, 10 q,-orderings, 163ff Exponentiation exponentiable linear ordering, 132 of cardinal numbers, 65 of linear orderings, 130- 132

of ordinals, 60 of Z , 90-92

F False, 23 I Fieldable linear ordering, 126 Finitary structure, 254 Finite isomorphism, 352 Finitely axiomatizable, 253 linear orderings, 256ff First element, I 1 First €-number,64 First-order class, 234 First-order definable, 224 First-order equivalent, 232 First-order language, 223, 229-236 First-order property, 224 First-order theory, 233 First-order theory of linear orderings, 233 First-order equivalent, 232 Fixed point of a continuous function, 63 of an order type, I59 FraissC’s Conjecture, l77ff, 196ff Frank, 82 G G-equivalent, 99 G,-equivalent, 99 G,-equivalent, 351 Gap in a linear ordering, 34 of an order type, 38 Generalized continuum hypothesis, 65 Generalized sum of linear orderings, 19 General recursive function, 242 Generate a type, 280 Godel numbering, 243 Greatest element, 11 Greatest lower bound, 13 Groupable order type, 125

H Hausdorff’s Theorem, 84-87 H,-equivalent, 373 H-equivalent, 373

484

SUBJECT INDEX

Hereditarily additively indecomposable linear orderings. I968 Homogeneous for a kdbeling, 138 for a partition, 110, 1 1 I , 207 Homogeneous linear ordering, 30, 31 Homomorphic image, 70 Homorphism, 69

Isomorphism into, 7 onto, 8, 232 K Karp game, 349ff Kleene Hierarchy Theorem, 41 8 k-tuply transitive linear ordering, 30

I Immediate predecessor, I 1 Immediate successor, I I of a linear ordering. 179 Incomparable order types, 10. I83 Indecomposable order types, 174- 177 additively indecomposable, 174 hereditarily additively indecomposable, 196 hereditarily indecomposable, 196 left indecomposable, 175 right indecomposable, 175 Indecomposable sequence, 188 Induction definition by, 55 mathematical, 52-53 principle of, for formulas, 230 transfinite, 53-55 lnfinite descent, method of, 42 Infinitary languages, 223, 236-240 lnfinitary theoriesof linear orderings, 344-369 Initial formula, 318 Initial interval, I0 Initial segment, 43 proper, 43 Interpretation of L, 230 Interval, 10 closed, 1 i initial, 10 open. 1 1 proper, 10 terminal. 10 Interval formula, 318 Interval type limit, 319 non-principal, 31 9 principal, 3 19 of a theory, 3 18 Isolated n-type, 281 Isomorphic, 8, 232

L Labeling, 138 homogeneous for, I38 label condensation, 133 Last element, 11 Least element, 1 I Least upper bound, I3 Left indecomposable order type, 175 Lexicographic product, 127 anti-lexicographic product, I30 Limit interval type, 3 19 Limit n-type, 281 Limit order type, I56 Limit ordinal. 48 ath, 55 /3-Iimit point, 78 P-limit ordinal, 77 Linear ordering, 6 &,-categorical, 297 almost transitive, 137 a-sum of, 20 automorphisms of, 132-6 characterizable, 238 complete, 33 countable, 23 Dedekind complete, 34 dense, 25 densely subuniform, 166 doubly transitive, 31 exponentiable, 132 exponentiation of, 130-2 fieldable, 126 generalized sum of, 19 groupable, 125 homogeneous, 30 k-tuply transitive, 30 k-universal, 170 perfectly symmetric, 123

485

SUBJECT INDEX

primitive, 135 product of, 21 recursive, 409 recursively rigid, 438 rigid, 133 scattered, 32 self-additive, 301 separable, 36 of a set, 3 short, 88 sum of, 16 symmetric, 58 transitive, 30, 121ff uncountable, 23 uniform, 166 uniquely transitive, 135 universal, 26 very discrete, 84 well-ordering, 41 Linear sets, 145, 159 Logical consequence, 233 Logically equivalent, 101, 231, 232, 235 Logically implies, 233, 235 L,,, . 237 L,,,-equivalent, 239 Lm1,-substructure. 361 Lowenheim-Skolem Theorem. 103, 104. 2 36, 239, 382, 398 Lower bound. 13

M Method of infinite descent, 42 Model. 224,233 Monadic second-order language, 227 Monadic second-order theory of linear orderings, 396-399 N

Non-decreasing function, 149 Non-principal interval type, 31 9 Non-principal n-type. 281 n-type, 279 generate an n-type, 280 isolated, 281 limit, 281 non-principal, 28 I omit an n-type, 280

principal, 28 1 realize an n-type, 280

0 Omit a type, 280 Open interval, 1 1 Orbit, 136 Ordered Abelian group, 124 Order-preserving map, 7 Order type, 9 Ordinal, 46-50 arithmetic of, 59-62 p-limit, 77 and cardinal numbers, 50-52 degree of, 62 and induction, 52-59 limit, 48 successor, 48 tail of, 62

P Partial recursive function, 403,406 Partitioned ordering, 327 k-partitioned ordering, 327 Partitiun theorems, 205ff Perfectly symmetric IInear ordering, 123 n, order type, 430 II,-predicate, 41 7 Positive formula, 368 Power set, 65 Predecessor, 11 immediate, 11 Prenex normal form. 294 Prime model, 278 Primitive linear ordering, 135 Principal interval type, 3 I9 Principal n-type, 281 Principle of Induction for formulas, 230 Priority argument, 446ff Product of linear orderings, 2 I Proper initial segment, 43 Proper interval, 10 Pseudo-elementary class, 297

Q Quantifier depth, 250, 344 Quasi-ordering, 183

486

SUBJECT INDEX

R Ramsey cardinal, 207 Ramsey ordering, 205 Ramsey’s Theorem, I I 1ff Rank F-rank r F . 82 5’-ran k r v , 86 C’D-rank rvD, 85 2-rank r z , 92 Realize a type, 280 Recursive functions, 242, 401. 406 general recursive functions, 242 partial recursive functions, 403, 406 set. 243, 401 Recursive linear orderings. 409ff recursive order type, 410 recursive ordinal, 41 2 Recursively embeddable, 446 Recursively enumerable, 244,40 I linear ordering. 410 order type, 410 Recursively isomorphic, 406 Recursively rigid linear ordering. 438 Regular cardinal number, 57. 165 Relativization, 259 Representation of a set in order of magnitude, 424 T-representation, 424 Right indecomposable order type, 175 strictly right indecomposable, 175 Rigid linear ordering. I33 recursively rigid, 438 S

Satisfaction. 231 Saturated, 290 weakly saturated, 292 Scattered linear ordering, 32 Scott sentence, 359 Second-order definable, 226 Second-order language, 223, 240-241 monadic, 228 weak, 228 Second-order property, 226 Segment. 10 initial, 43 proper initial, 43 Selecting formula. 267

Self-additive linear ordering, 301 Separable linear ordering, 36 Short linear ordering, 88 Shuffle operation, 116ff Shufflecondensation, 138 Xk order type, 430 &-predicate, 417 Simple ordering, 4 Splitting of a linear ordering, 69 Standard partition, 205 Statement, 230 Strongly inaccessible cardinal number. 166 Subordering, 6 generated by a subset, 7 Substructure, 274 elementary substructure, 274 La,,-substructure, 36 1 Successor, 1 1 immediate, I I Successor ordinal, 48 Sum of linear orderings, I6 a-sum, 20 generalized sum, 19 Symmetric linear ordering. 58 perfectly symmetric, 123

T Tail of an ordinal, 62 Tail of a sequence, 188 Tarski-Kuratowski algorithm. 41 7 Terminal formula, 31 8 Terminal interval, 10 Testing formula, 265 Theorem-checking algorithm, 268 Theory, first-order, 233 complete theory, 233 Total ordering, 4 Transfinite induction, 53-55 Transitive linear ordering, 30, l2lff almost transitive linear ordering, 136ff almost n-tuply transitive linear ordering, 137 doubly transitive linear ordering, 3 I , I55 uniquely transitive linear ordering, 135, 155 True, 231 Type of a structure, 230 U

Uncountable linear ordering, 23

487

SUBJECT INDEX

Undecidable, 267 Uniform linear ordering, 162 Uniquely transitive linear ordering, 135, I55 Universal formula, 367 Universal linear ordering, 26 K-universal linear ordering, I70 Universally valid, 232 Universe, 224 Upper bound, 13 V

Variables bound, 230 free, 230 b’D-rank, 85

Very discrete linear ordering, 84 V-rank, 86 W

Wait-for-a-proof algorithm, 268 Weakly saturated, 292 Weak second-order language, 228 weak second-order theories of linear orderings, 378-396 Well-ordering, 41 -46 Well-quasi-ordering (wqo), 183ff Winning strategy, 96, 248, 350, 372, 378 2

Z-rank, 92

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors

Samuel Eilenberg and Hyman Bass

Columbia University, N e w York

RECENT TITLES

CARLL. DEVITO.Functional Analysis MICHIELHAZEWINKEL. Formal Groups and Applications SIGURDUR HELCASON. Differential Geometry, Lie Groups, and Symmetric Spaces ROBERTB. BURCKEL. An Introduction to Classical Complex Analysis : Volume 1 JOSEPH J. ROTMAN. An Introduction to Homological Algebra C. TRUESDELL A N D R. G. MUNCASTER. Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas : Treated as a Branch of Rational Mechanics BARRY SIMON. Functional Integration and Quantum Physics A N D ARTOSALOMAA. The Mathematical Theory of L Systems. GRZEC~RZ ROZENBERC DAVIDKINDERLEHRER and GUIDOSTAMPACCHIA. An Introduction to Variational Inequalities and Their Applications. H. SEIFERTAND W. THRELFALL. A Textbook of Topology; H. SEIFERT. Topology of 3-Dimensional Fibered Spaces LOUISHALLEROWEN. Polynominal Identities in Ring Theory DONALD W. KAHN.Introduction to Global Analysis DRAGOS M. CVETKOVIC, MICHAEL DOOB,A N D HORST SACHS.Spectra of Graphs M. YOUNG.An Introduction to Nonharmonic Fourier Series ROBERT MICHAELC. IRWIN. Smooth Dynamical Systems JOHN B. GARNETT. Bounded Analytic Functions EDUARD PRUWVEEKI. Quantum Mechanics in Hilbert Space, Second Edition M.SCOTTOSBORNE A N D GARTHWARNER. The Theory of Eisenstein Systems JEAN DIEUDONN~. A Panorama of Pure Mathematics ; Translated by I. G. Macdonald JOSEPH G. ROSENSTEIN. Linear Orderings

IN PREPARATION

K. A. ZHEVLAKOV, A. M. SLIN’KO,I. P. SHESTAKOV, A N D A. I. SHIRSHOV. Translated by HARRY SMITFI.Rings That Are Nearly Associative ROBERT B. BURCKEL. An Introduction to Classical Complex Analysis : Volume 2 HOWARD OSBORN. Vector Bundles : Volume 1, Foundations and Stiefel- Whitney Classes RICHARD V. KADISON A N D JOHN R RINGROSE. Fundamentals of the Theory of Operator Algebras AVRAHAM FEINTUCH A N D RICHARD SAEKS. System Theory : A Hilbert Space Approach BARRETT O’NEILL. Semi-Riemannian Geometry : With Applications to Relativity ULFGRENANDER. Mathematical Experiments on the Computer

Linear Orderings

Linear orderings

Linear orderings

Convex functions, partial orderings, and statistical applications

Spaces of Orderings and Abstract Real Spectra

Linear spaces and linear operators

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Algebra and Linear Models

Linear Operators and Linear Systems)

Linear spaces and linear operators

Linear Operators and Linear Systems)

Linear Algebras

Linear Algebra

Linear Algebra

Linear Algebra

Linear Algebra

Linear Accelerators

Linear Programming

Linear Algebra

Linear Programming

Linear Algebra

Linear algebra

Linear estimation

Linear Systems

Linear Estimation

Linear geometry

Linear algebra

Linear Orderings