Equivalents of the Axiom of Choice II

EQUIVALENTS OFTHE AXIOM OF CHOICE, 11

Herman RUBIN Professor of Mathematics and Statistics Purdue University West Lafayette Indiana U.S.A.

and

Jean E. RUBIN Professor of Mathematics Purdue University West Lafayette Indiana U.S. A.

1985

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

ELSEVIER SCIENCE PUBLISHERS B.V., 1985 All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanicat, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87708 8

Published by:

Elsevier Science Publishers B.V. P.O. Box 1991 1000 6 2 Amsterdam The Netherlands Sole distributors forthe U.S.A. and Canada: Elsevier Science Publishing Company, Inc. 52Vanderbilt Avenue NewY0rk.N.Y. 10017 U.S.A.

Llbrary of Congress Cataloging in Publication Data

Rubin, Herman. Equivalents of the axiom of choice, 11. (Studies in logic and the foundations of mathematics ; V. 116) Bibliography: p. Includes indexes. 1 . Axiom of choice. I. Rubin, Jean E. 11. Title. 111. Series. QA248.R8 1985 511.3'22 84-28692 ISBN 0-444-87706-8

PRINTED IN THE NETHERLANDS

In dedication to o u r children

Arthur Leonard Rubin Leonore Anne Rubin Findsen i n h o p e of t h e future: and t o

Alfred Tarksi i n r e m e m b r a n c e of t h e past.

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PREFACE (1963 Edition)

In 1955 at Stanford University Professor Patrick Suppes gave a course on axiomatic set theory, during which time Professor Herman Rubin was asked to give a series of lectures on the axiom of choice. During the course of these lectures it was noted that,' while there was much in the literature on the axiom of choice, the material was available only in many diverse journals and books. It was suggested that we collect this material to make it more readily available. It seemed like a simple enough project to begin with; then it grew and grew some more and now has blossomed forth into a book. The book consists of a selection of the forms of the axiom of choice which appeared in the literature together with additional forms which were obtained in the process of writing the book. It would have been a hopeless task to try to include all of the forms of the axiom of choice which appeared in the literature, so we chose the forms which in our opinion were either used often in practice, unusual, relatively unknown, or particularly weak or strong. We hope that we have included all of the interesting equivalents of the axiom of choice. We assume a knowledge of logic and elementary set theory (von Neumann-Bernays-Godel set theory), but we do include a list of definitions of set theoretical symbols and terms in the section entitled "Preliminary Definitions and Theorems". In Part I we discuss propositions which are equivalent to the usual form of the axiom of choice. These equivalents will be referred to as set forms. In Part I1 we discuss stronger forms - essentially forms which are obtained from the set forms by changing the word "set" to "class". These latter forms are called class forms. The set forms of the axiom of choice are V

vi

PREFACE (1963 Edition)

the forms which are most often used in practice. In preparing this,monographfor publication we first prepared a draft and sent it to several people for their comments and corrections. We are very grateful to the people who did reply. We believe that the quality and usefulness of the book was greatly improved by their comments. In particular we should like to thank Professor Alfred Tarski for his many useful comments and corrections. Others whom we should like to thank are Professoxs E . W. Beth, A. Levy, D. Scott, and R. Vaught. Our typists, Ann Breen and Barbara Johnson, also deserve credit for bearing with us under strain, and we should like to thank the Mathematics and Statistics departments at Michigan State University for their cooperation. HERMAN RUBIN JEAN E. RUBIN September 1961 East Lansing, Michigan

PREFACE (1985 Edition)

In the twenty years since "Equivalents" (Rubin & Rubin 1963) has been published, there have been so many new developments with respect to the axiom of choice that we thought it was time for a new book to describe these developments. Many new equivalents have been discovered since 1963. The section on algebraic equivalents has been greatly expanded to include statements about vector spaces, groups and other algebras. There are so many forms in logic, analysis and topology that a new section was added to include these forms. Another new section was added for those forms which are equivalent to the axiom of choice only under the axioms of extensionality and foundation, In addition, new forms were added to the other sections. We have also rewritten all the material using modern notation in a style which, we hope, is easier to read. We should especially like to thank the following people for their help: our son Arthur who proof read Part I and made significant contributions to Part 11; Norbert Brunner who, while visiting Purdue during the 1983-84 academic year, made many suggestions for improving the manuscript; Paul Howard who proof read part of the manuscript and made important contributions; Wilfrid Hodges who sent us a list of some of the new algebraic equivalents of the axiom of choice: Andreas Blass who corrected some errors and sent us his unpublished proof that the "vector space theorem'' implies the axiom of choice; and our typist, Judy Snider, who did an admirable job of typing and cutting and pasting. HERMAN RUBIN June 1984 JEAN E. RUBIN West Lafayette, Indiana vii

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TABLE OF CONTENTS

.................. v . . . . . . . . . . . . . . . . . . vii .................... . . . xi . . . . . . . . . . . xv .................... 1 . .......... . . . . . . . . . . . . . . .. .. .. . . . . . . . . . . . . . . 21 . . . . . . . . . . . . . . . . . . 31 . .. . . . . . . . . . . . . . . . . . .. .. .. 7933 .. . . . . . . . . . . . . . . . 137 . . . . . . 163 PART I1 .Class Forms . . . . . . . . . . . . . . . . . . . 185 1 . The Well-Ordering Theorem . . . . . . . . . . . . . 187 2 . The Axiom of Choice . . . . . . . . . . . . . . . . 191 3 . Maximal Principles . . . . . . . . . . . . . . . . . 2 0 3 PREFACE (1963 Edition) PREFACE (1985 Edition) INTRODUCTION PRELIMINARY DEFINITIONS AND THEOREMS PART I .SetForms 1 The Well-Ordering Theorem 2 The Axiom of Choice 3 The Law of the Trichotomy 4 Maximal Principles 5 Forms Equivalent to the Axiom of Choice Under the Axioms of Extensionality and Foundation 6 Algebraic Forms 7 Cardinal Number Forms 8 Forms from Topology, Analysis and Logic

1

i

. . . . . . . . . . . . . . . . . . . 247 . . . . . . . . . . . . . . . . . 271 . . . . . . . 279 . . . . . . . . . . . . . . . . . . . . . . . . 289 . . . . . . . . . . . . . . . . . . . . . 305 . . . . . . . . . . . . . . . . . . . . . 311 . . . . . . . . . . . . . . . . . 315

List of the Set Forms List of the Class Forms List of Forms Related to the Axiom of Choice Bibliography Index of Authors Index of Symbols Index of Technical Terms

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INTRODUCTION

Equivalents of the axiom of choice appear frequently in almost all branches of mathematics in a large variety of different forms. In this monograph we should like to describe some of these equivalents. By the "Axiom of Choice" we mean a statement similar to the following statement given by Russell [1906] and Zermelo [19081.

AC: If x is a pair-wise disjoint family of non-empty sets, then there is a set C which consists of one and only one element from each set in x. The reluctance that some mathematicians have about using the axiom of choice is due to its non-constructive nature. There is no rule for constructing the choice set, C. Since, in our opinion, it is more desireable to have a constructive proof rather than a non-constructive proof, the axiom of choice and other non-constructive principles should be avoided whenever possible. At times, of course, non-constructive principles are unavoidable and the least the mathematician can do at these times is to declare their use in his/her proof. The set theory in which we prove of the axiom of choice is called NBGO without the axiom of foundation). It Bernays and G6del and it is described entitled "Preliminary Definitions and

most of the equivalences (NBG with atoms but is due to von Neumann, in detail in the section Theorems".

The relative consistency and independence of the axiom of choice with NBG (and NBGO) has now been completely determined. xi

xii

INTRODUCTION

(NBG is consistent if and only if NBGO is consistent). Kurt Godel [1938,1939,1940] has shown that if NBG is consistent so is NBG + AC; Abraham Fraenkel [1922] proved that if NBGO is consistent so is NBGO + 1 AC (see also Fraenkel [19371, Lindenbaum & Mostowski [1938] and Mostowski [1939]); and Paul Cohen [1963,19641 proved that if NBG is consistent so is NBG + 7 AC (see also Cohen [1966] and Shoenfield [1971]) . In Part the axiom of from the set "class". It is a class.

I1 of this monograph we discuss "class" forms of choice, CAC. Generally, these forms are obtained forms in Part I by replacing the word "set" by is clear that CAC implies AC because each set

Using a Fraenkel-Mostowski model in which the class of atoms is a proper class it is easy to see that if NBGO is consistent, so is NBGO + AC + 7 CAC. (This result is credited to E. Specker. See Bernays [1958] p.196.) Later Easton [1964] proved that if NBG is consistent, so is NBG + AC + lCAC. (See Felgner [1976a]). Thus AC does not imply CAC in NBG. However, it has been shown by a number of people that NBG + CAC is a conservative extension of NBG + AC. That is, even though AC does not imply CAC, if a statement about sets, a , is provable in NBG + CAC, Q is also provable in NBG + AC. (See Cohen [19661 ,p.77 , Felgner [1971,1976al, Gaifman [1968], Grishin [1972], Jensen, Kripke and Solovay.) The relative strength of the axiom of choice and other well-known propositions has now been demonstrated using Cohen's forcing techniques. For example, Cohen 119631 has shown that the axiom of choice does not imply the continuum hypothesis in NBG (assuming, of course, that NBG is consistent). It was known much earlier that the generalized continuum hypothesis does imply the axiom of choice in NBGO. (Lindenbaum & Tarski [1926], Sierpinski [1947]). Halpern & Levy [1971] proved that the Boolean prime ideal theorem, BPI, does not imply the axiom of choice in NBG. (Halpern [1961,1964] proved that BPI does not imply AC in NBGO.) Sageev [19751 proved that the 2k = k principle, (that is, for all infinite cardinals k, 2k = k) does not imply AC in NBG. (Halpern & Howard [1974]

INTRODUCTION

xiii

proved that the 2k = k principle does not imply AC in NBGO.) Clearly, AC implies each of BPI and the 2k = k principle. In closing, we'll mention a few of the unsolved problems. It is well-known that the axiom of choice is used in the proof of each of the following propositions: Every subgroup of a free group is free. (NielsonSchreier Theorem.) (Lauchli 119621 proved that its proof is dependent on the axiom of choice and Howard [19831 proved that it implies the axiom of choice for finite sets.) For all sets x

and

Y, F ( x ) < P ( y 1

For all infinite cardinals

2

k, k = k

For each set x, and partition (Partition Principle).

P

of

or 3

P(Y) < $"XI.

. x, P 6 x.

However, it is not known if any of the above propositions imply in NBGO or NBG. Other unsolved problems are mentioned in the text.

AC

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PRELIMINARY DEFINITIONS AND THEOREMS

We assume an elementary knowledge of logic and set theory. We shall use the following logical symbols: 1 A V

-+

(3x1 (VX)

for for for for for for for for

not and or if then if and only if (iff) identity there exists an X for all X.

...

...

The system of axioms for set theory which we adopt is NBGO (von Neumann-Bernays-Godel set theory with atoms and without the axiom of foundation). (See von Neumann [19281, Bernays [19371 and Godel [1940].) Thus, our undefined terms are: class, atom, and the element relation, E . The convention is made that capital letters, XI Y,..., denote variables whose range is all classes and all atoms. AXIOMS Al:

Characterization of Atoms

DEFINITION 0.1 : (a) X is a Set, set(x)

iff

Class(X) xv

A

(3Y)(Class(Y) A X E Y).

xvi

PRELIMINARY DEFINITIONS AND THEOREMS

(b) X

is a proper class, Pr(X)

iff

Class(X)

Set(X).

Axiom of Extensionality Class(X)

A

Class(Y)

[(Vu) (u E X

+

It follows from A2 it is denoted by PI. A4:

7

denote variables whose

Lower case letters, x, y, z , . . . , range is all sets and atoms. A2:

A

that the

X

. u E Y) of

A3

-+

X = Y].

is unique and

Class Construction Schema

If @(u) is a formula in the language of NBGO which has no bound class variables and in which X is not a free variable then (3x1 [Class(XI

(Vu)(u E

A

x

-

Q

(u)1 1

.

(Godel [1940] has shown that in NBGO, A4 can be replaced by a finite number of axioms.) It follows from A2 that the class X of A4 is unique and it is denoted by

x DEFINITION 0.2:

Let

X

= iu: O(u) 1 .

and

Y

be classes.

(a) An element is a set or an atom. (b) Subclass:

X 5 Y

(Vu)(u E X

iff

(c) Proper Subclass: X c Y (d) Power Class: (e) Union: (1) (2)

(f)

ux x u

iff

+

X c Y

p(X) = {u:u _c XI.

= cu: (3v

E X) (u E v) I .

Y = cu:u E

Intersection: (1) nx = {u: (VV E

x

v u E Yl.

x) (U

E

V) I

,

u E Y). A

X # Y.

PRELIMINARY DEFINITIONS AND THEOREMS

x n

(2)

Y = {u:u E

Class Difference:

X

Y

Complement: - X = {u:u also be denoted by V See 0.4(a).) Disjoint:

X

Y

and

Pair-wise Disjoint: of distinct sets in

4

-

x

A

=

{u:u E X

u E

YI A

X

4

u

Y].

XI. (The complement of X will X where V is the universe.

are disjoint if X

xvii

X

n

Y = 8.

is pair-wise disjoint if each pair are disjoint.

Power Set Axiom Set (x) * Set

)

.

Union Axiom Set(X) * Set(UX). Pairing Axiom (3x1 (Set(x) A (Vu) [u E x It follows from and it is denoted by

A2

x

-

(u = v

V

that the set x

u

=

w) I .

of

A7

is unique

= {v,w).

DEFINITION 0.3 : Unordered Pair: v and w. Unit Set:

{v,w)

{u} = {u,u}

is called the unordered pair of is called a unit set.

Ordered Pair: (u,v) = {{u),{u,v)} pair of u and v. Relation:

is called the ordered

A (binary) relation is a class of ordered pairs.

Function : Func(F) [F is a relationh (Vu,v,w) ((u,v),(u,w)

-

Domain: Range:

EF-+v=w)].

$(X) = {u:(~v)(u,v) E XI. .!#,(X) = {v:(~u)(u,v)

E XI.

R is a relation we frequently write xRy instead of (x,y) E R; and if F is a function, we write F(x) = y or

xviii

PRELIMINARY DEFINITIONS AND THEOREMS

Fx = y A8:

instead of

xFy

or

(x,y) E F.

Axiom of Replacement (Func(F) A Set (3 (F)) ) * Set (2(F)).

A9:

Axiom of Infinity (3x1 [Set(x)

0 E x

A

A

(Vu)(u E x

A

Set(u)

+

u

u

{ul E x)].

Axioms Al-Ag are the axioms of NBGO. A10:

Axiom of Foundation (Class(x)

A

x # 0)

+

(3u E x)(Set(u)

A

u

n x

=

0).

Axioms A7 and A10 imply there donotexist any atoms. Al-AlO are the axioms of NBG. Another form of the axiom of foundation which is sometimes used is the following: A'10:

Axiom of Foundation, alternate form IClass(X)

A

X # 0

A

( V u E X)Set(u)l

+

13u f

x) (u n x = 0)

Of course, A10 i s equivalent to All0 if there do not exist any atoms. As is well-known, the set form of A10, (that is, replace "Class(X)" by "Set(X)" in A101 is equivalent to the class form in NBGO. For suppose the set form of A10 holds and let X be a class. Suppose u E X I u is a set and X n u # fl. Let z = X r l u. Then the transitive closure of z, TC(z), is a set , TC(z) = z U ( U z ) U (UUzIU

... .

Let w = TC(z) n X. Then w # 0 is a set so by the set form of the axiom of foundation, there is a set v E w such that v n w = 0. Clearly v E X. If v n X # 0 then there exists an a E v n X. By definition of TC(z), a E v E TC(z) implies a E TC(z). Thus a E v n w, which is a contradiction. All proofs, unless otherwise stated, are in are independent of the existence of atoms.

NBGO

and

PRELIMINARY DEFINITIONS AND THEOREMS

xix

DEFINITION 0.4:

v

(a) Universe:

=

iu:u = ul [(vu) (u E v)].

(b) Unordered n-tuple: (xltx2r * * txn-1) u {xn}-

-

(cI

Ordered n-tuple:

=

...rxn) =

(xlrx2r

(d) If F i s a function and the language of NBGO, (F(X1t...rXn):~(~lr...rX

(x1r(x2r...rxn)).

@(X~,...~X n)

i s a formula in

n ) I = Iu:(3Xl...x n )(U = F(X1r.a-rX n ) O(Xlr...,x n 1 ) ) .

A

(e) Cartesian or Direct Product: Class(X) (f) Image:

A

Class(Y)

-t

x

(h) Inverse:

RI\X = ((urv):u

R-l = {(x,y):(y,x)

( i ) Indexed Classes:

Suppose

u xi

=

t~

{xi:

i E I).

r) xi

=

n {xi:

i E I}.

iE1

Y

{(u,v):u

=

E

x

A

v E Y}.

R"X = {v: (3u E X) ((ulv) E K) 1.

(9) Restriction:

iE1

x

(vi E I) (xi = Y)

-t

X xi

iE1

E X

A

(u,v) E R}.

E R).

(Vi E I)Set(Xi).

= Y '

.

We may also take the union, intersection, and Cartesian product of proper classes by using the following device: Let R be a relation such that for each a set. (R"Xi could be a proper class.)

i E 1, Xi

is

PRELIMINARY DEFINITIONS AND THEOREMS

xx

x

R"Xi = {f:Func(f)

A

$(f) = I

A

(vi t I) (f(i) E R"Xi)I.

A

$(F) = X

A

%(P) _C Y.

if1

(j) Functional Notation: F:X

+

Y

iff

Func(F)

is 1-1 or an injection iff E F u =

F

(VUrVrW) ((UrW)r(VrW)

F:X < Y

X < Y F:X

F:X X

iff iff

Func(F) (3F)(F:X

Onto> Y iff surjection. )

2 and 1 < k < n then AC 16(nfk) WO 4(n-k). -+

PROOF: Suppose x is an infinite set, and n and k satisfy the hypothesis. (There is no l o s s of generality in proving WO 4(n-k) for an infinite set x.) Let y be a well-ordered set such that y n x = and y w where

w

=

{u _c x: u

sz

n

- kl.

(For example, let y = I'(P(x)), where r is Hartogs' function; see 0.11.) Apply AC 16(n,k) to x U y. Let tn be a set of n-tuples of elements of x U y such that each k-tuple of elements of x U y is a subset of exactly one element of tn. For each u E x, let s

U

=

{V

E v

E tn: u

A

k

-

1 < v

n y}.

Then there is a v E su such that k 4 v y. For suppose not. Let a be any (k-1)-element subset of y. Then a U {u} is a subset of exactly one element in su. Moreover, each element of su contains exactly one subset a U {u} where a is a (k-1)-element subset of y. Consequently, sU

{ a _CY: a

F=

k

- 13

= K.

Since y can be well-ordered, K FS y. We claim K 4 w. function F such that for each a E K, F(a) = va (a U where va is the unique element of su with a U { u } 5 is a 1-1 function. This contradicts the definition of for y 4 w. Therefore, there is a v f su such that k j,

A

=

D

or

J,

(5)

ui D u!

3'

54.

MAXIMAL P R I N C I P L E S

for

i # j.

69

The set {vi:i E In) is both a 5-maximal subset of y with the property J (by (2) and (4)) and also a 5-maximal subset of y with the property D (by (2) and ( 3 ) ) . Therefore, the hypotheses of both M 15(3:D) and M 1 5 ( D : J ) are satisfied. M 1 5 ( 5 : D ) implies that y has a 5-maximal subset z1 with the property D. It follows from (2) that at most one vi can be an element of zl. If no vi is an element of z1 then {u:ui E zl} is a 5-maximal subset of x with the property D for each i E In (by (1)). I f vi E z, then for each j > i, { u : u j E z,} is a 5-maximal subset of x with the property D (by (1) and ( 3 ) ) . This proves part (a).

To that y follows vi E z2, then i

prove part (b), we use the fact that M 1 5 ( 5 : J ) implies has a 5-maximal subset z 2 with the property J. It from ( 2 ) , ( 3 ) and ( 5 ) that there is exactly one and it also follows from (5) that if ui, u!J 22 = j . Thus, there is a j E In, j s i such that if then k = j . Then by (1) { u : u . E 2 , ) is a Uk '2 3 c-maximal subset of x with the property J, q.e.d. This completes the discussion of M 1 5 ( A : B ) and M 16(A:B) where A # B and A , B = D, D l J, 5, K, or We shall show in section 5 that in NBG M 8 ( K ) AC.

E.

-f

The next form to be considered in this section is an obvious extension of the idea of constructing maximal subsets with respect to "natural" relations. The relation to be considered is the €-relation. M 17: Every set x contains a 5-maximal subset y such that for every s , t E y, s # t, either s E t or t E s.

It is clear that M 7 + M 17 since M 17 deals with a property of finite character. We shall prove that M 17 + WO 1. THEOREM 4 . 4 4 :

M 17

-f

WO 1.

PROOF: Let x be an arbitrary set and let {W: (3y)[ y _c x and W is a well-ordering of define a relation R on @ as follows:

&

=

y] 1 .

We

70

PART I, SET FORMS If

W, W' E 2)

then

is an extension of W. For all function f on %r as follows:

W'

(i) f(W) = f(U) (ii) f(W) E f(U)

We claim:

W E

we define a

.+W

-

= U and W R U.

It is clear that if W = U then f(W) = f(U). converse it is sufficient to prove that (iii) for all

W ' E %r

{{WI}

.

n {f(W'):W' R W and W'

We leave the proof of

To prove the

E

%r)

=

fl

(iii) as an exercise.

To prove (ii) we note that if W R U then clearly f(W) E f(U). Suppose f(W) E f(U); then either f(W) = {U) or W R U, but the first alternative is impossible. M 17 implies that % ( f ) contains a maximal subset Q such that for every S, T E Q, if S # T then either S E T or T f S. Let 2 = {U:f(U) E Q ) . Since f is a 1-1 mapping of U onto $ ( f ) preserving order (by (i) and (ii)) it follows that ! ,? is a maximal subset of @ such that for all U, W E 21, either U R W or W R U.

We claim that U z is a well-ordering on x. Since Ug is a union of R-comparable well-orderings of subsets of x, u i l is clearly a well-ordering on a subset of x. Suppose there exists an s E x such that s p a ( U 2 ) . Then for all

u

E

il,

s fi

i(u). Let

v

=

u 2L u ( B ( U 2 1 )

x

CS})

u

t(S,S)I,

A*

= 21 U {V}. Since for each U E Z*, U R V, and the maximality of 2 is contradicted. Hence, Ug is a well-ordering on x .

a n d let

:d c d * ,

M 18, the next maximal principle in this section, is put in primarily as a curiosity. Blair and Tomber [1960] discuss

71

MAXIMAL PRINCIPLES

94.

two forms of the axiom of choice for finite sets. It was pointed out to them by H. Rubin that if a slight change was made in the wording, one would obtain forms of the general axiom of choice. M 18 is a combination of these two forms. M 18: Suppose x is partially ordered by t E x, let + t = { s E x : t R S A t # s ~ .

Suppose every R-linearly ordered subset of bound. Then there is a t E x such that element.

x

R.

For each

has an R-upper has no R-first

We shall show M 18 is equivalent to M 1. THEOREM 4.45:

M 1

-+

M 18.

PROOF: If x satisfies the hypothesis of M 18. Then it also satisfies the hypothesis of M 1. Consequently, x has -+ -+ an R-maximal element, t. Since t = 8 , t has no R-minimal element. THEOREM 4 . 4 6 :

M 18

+

M 1.

PROOF: Suppose M 1 is false. Then there is a set x a partial ordering relation R such that every R-linearly ordered subset of x has an R-upper bound, but x has no R-maximal element. Define a relation S on x x w as follows: If (u,m), (vln) E x x w then (u,m)S(v,n)

iff

(u # v

A

u R v) v (u = v

A

m

5

and

n).

satisfies the hypothesis of M 18. For suppose Then (x x w,S) u is an S-linearly ordered subset of x x w. Then is an R-linearly ordered subset of x. Every R-linearly ordered subset of x has a strict R-upper bound. If not,then an R-upper bound would be an R-maximal element of x. Let b E x be a strict R-upper bound of a ( u ) . Then (b,O) is an S-upper bound of u.

a(~)

-

Suppose (ulm) E x x w. Then we claim (u,m+l) is an R-first element of (u,m), because (u,m)S(u,m+l) and if (u,m)S(v,n) and (u,m) # (v,n), then (u,m+l)S(v,n). This contradicts M 18, q.e.d.

PART I, SET FORMS

72

We end this section with fix-point theorems which are equivalent to maximal principles. Abian [1983] gave M 19(P,W) in an abstract. The other forms and all the proofs are our own. M 19(Q,U): Every set x such that x is Q-ordered by R and every U-ordered subset has an R-upper bound,has the property that every function f from x to x such that for all u E x, u R f(u), has a fixed point.

We shall show M 19(Q,U) * M 1 (Q,U). It is clear that M l(Q,U) -+ M 19(Q,U) because an R-maximal element must be a fixed point. THEOREM 4.47:

M 19(Q,U)

+

M l(Q,U).

PROOF: Suppose x is a set and R a relation which satisfy the hypotheses of M l(Q,U). Suppose x has no and define a relation R-maximal element. Let y = x x w on y so that (u,m)S(v,n)

iff

(u # v A u R v)

V

(u = v A m

s

S

n).

It is clear that when one considers the possible values for Q and U given on pp. 37-8, that if R is a Q-ordering on x then S is a Q-ordering on y. Also, since x has no R-maximal element, if each U-ordered subset of x has an R-upper bound then each U-ordered subset of y has an S-upper bound. Let f be a function defined on y such that for all (ulm) E y, f(u,m) = (u,m+l). Then for all (u,m) E y, (u,m)Sf(u,m), but f has no fixed point. This contradicts M 19(Q,U). Thus, x must have an R-maximal element, q.e.d. COROLLARY 4.48: M 19(Q,U) is equivalent to the axiom of choice if Q = TR, P I RA or F and U = C, AS & C, TR & C, L, D, or W and also if Q = D and U = W. PROOF: Theorem 4.47 and preceding remarks, and Theorems 4.9 and 4.10.

85.

FORMS EQUIVALENT TO THE AXIOM OF CHOICE UNDER THE

AXIOMS O F EXTENSIONALITY AND FOUNDATION

As we state in the Introduction, all proofs in this book, unless otherwise specified, are in NBGO. Thus, the proofs are valid whether or not there are atoms and whether or not the axiom of foundation holds. H. Rubin [19601 discovered two forms of the Axiom choice, PW and LW below, in which he assumed there were no atoms (or the class of atoms was a set and could be well-ordered) and the axiom of foundation to prove they implied the axiom of choice. Also, since both PW and LW are true in the Fraenkel-Mostowski model given for example in Halpern [1962] and the axiom of choice is false in most Fraenkel-Mostowski models, it was clear the proof of equivalence did not hold in NBGO. Later Felgner & Jech 119731, Howard & Rubin I19771 and others discovered other statements in the same category. This section is devoted to such statements. Our proofs below which require the stronger assumptions will be prefaced by "NBG IThus, we shall assume in these proofs that there are no atoms and that the axiom of foundation holds. (In most instances, the conditions on atoms can be weakened.)

".

PW: The power set (the set of all subsets) of a well-ordered set can be well-ordered. LW:

Every linearly ordered set can be well-ordered.

INJ: If x < A.

x

is a non-empty set and

13

A

is a proper class then

PART I, SET FORMS

74

PROJ: If x is a non-empty set and A can be mapped onto x.

A

is a proper class then

MC 1: If s is a set of non-empty sets then there is a function f such that for each x E s, f(x) is a non-empty finite subset of x. MC 2: sets.

Every set is the union of a well-ordered set of finite

If s is a set of infinite sets, then there is a function f such that for each x E s, f(x) is a partition of x into pairwise disjoint finite sets, each of which has at least 2 elements.

MC 3:

DEFINITION 5.1: If R is a partial ordering relation on X then Y 5 X is called an R-antichain iff for all u, v E Y, if u # v then 1 uRv A vRu. For example, a set with the property can t icha in -

.

A:

-

K

(see 4.20) is a

Every partially ordered set contains a 5-maximal antichain.

MC 1 is usually called the multiple choice axiom and A. Levy [1962] proved that it does not imply the axiom of choice in NBGO. MC 2 and MC 3 are variations of MC 1. (See WO 4 ( m ) and AC 10(m).) INJ and PROJ are forms similar to a form given by von Neumann [1928]. (They are called the injection and projection principles in Felgner & Jech 119731.) The statement A appears in Kurepa (19521. If R is taken (As is well known, there is no to be 5, then A is M S ( i ? ) . loss of generality in assuming that a partial order is 5, because if R is any partial ordering relation on a set x c c and for each u E x, u = {v E x: vRu1, then if y = {u:u E x), (XtR)

(Yis).) Felgner

&

INJ

Jech [1973] prove the following implications: +

PROJ

+

AC

+

MC 1 * A

-t

LW

+

PW.

(They also proved that none of the arrows are reversible in NBGO). H. Rubin [1960] proved PW implies AC in NBG and

55,

Rubin

&

EQUIVALENCES UNDER EXTENSIONALITY AND FOUNDATION Rubin [ 1 9 6 3 ] prove that

AC

-+

INJ

15

in NBG.

The following implications are clear: INJ PROJ, AC 1 MC 1, MC 2 MC 1. The proofs that MC 1 MC 2 and MC 1 -+ MC 3 are similar to the proof of 2.9. The proof that MC 1 is similar to the proof of 2 . 1 2 . MC 3 +

-+

-+

-+

-+

THEOREM 5.2:

PROJ

WO 3.

-t

PROOF: Let x be a non-empty set. PROJ implies that there is a function F mapping On, the class of ordinal - 1 { s j is a non-empty numbers, onto x. For each s E x, F class of ordinal numbers. Let as be the first element of F-l" {s]. Then x w { a s : s E XI. 11

THEOREM 5.3:

MC 1

A.

-+

PROOF: Suppose x is partially ordered by R. MC 1 implies there is a function f such that for each non-empty subset u 5 x, f(u) is a non-empty finite subset of u. Define a function g such that €or all u such that BZU_cX, g(u) = {a: a

is an R-minimal element of

f (u)1.

Then g(u) is finite, non-empty and an R-antichain. Now, using transfinite induction, we construct a 5-maximal antichain, A _c x. A = (J A, , where aEOn Aa = g(1a E x: a is R-ipcomparable with all b E U A B ] ) . B are linearly ordered by 5 . ) We shall prove that x contains a 5-maximal linearly ordered subset. Let F be the field of rational numbers and let F[x] be the abelian polynomial ring with elements u E x as indeterminates. (That is, F 5 F[x] and each element of F[x] F is a sum of elements of the form aulu2...um where a E F, ui E x and the ui are not necessarily distinct.) F[x] is a unique factorization domain and for y C_ x if yF[x] is the set of all finite sums from {uv:u E y A v E F[xl] then yF[x] is a prime ideal in F[x]

-

.

x

Let L and let

be the set of all 5-linearly ordered subsets of

u

S = F[xl

yF[x].

YEL S

is a subalgebra of

F[x]

R = S-'F[x]

.

Let

= {u/s:

u E F[x]

A

s E S).

If addition and multiplication are defined in the natural way, R is a unique factorization domain. u/s E R is a unit (invertible) iff u E S. If, for each w E x, f w = {v E x:v s; w} then u/s E R is a unit iff (Vv E x)u g fF[x]

.

AL 15 implies that M. Let c = u/s E M I factors. Then

R

contains a 5-maximal proper ideal

so that u

and

s

have no common

114

PART I , SET FORMS

u = altl

+

a Z t Z +...+

a t

j j f

ai E F and ti i s a monomial i n n o t a u n i t , f o r i f it w e r e w e would have

where

if

u # 0

ti,

monomials,

x.

of

U

u E

u f! S , so

then

yF[x].

YE L

c is Thus,

x. Moreover, cc-l = 1 E M.

This implies t h e

have f a c t o r s i n some l i n e a r l y o r d e r e d s u b s e t

Therefore, i f

f i n i t e non-empty

c # 0,

t h e r e must b e a t l e a s t one

s - l i n e a r l y ordered set

A

c_ x

such t h a t

( i )e a c h monomial, t i , h a s a f a c t o r i n A, and ( i i ) e a c h A o c c u r s a s a f a c t o r o f some ti. The s e t A

element of

is

n o t u n i q u e , b u t t h e r e a r e a t most a f i n i t e number o f them, Let

A1,A 2 , . . . , A k .

E ( c ) = { v : ( ~ ~ E( w 1 )s i s k A v i s t h e < - l a r g e s t e l e m e n t of A i ) ] .

v E E(c)

If

then

c E fR.

c = 0

If

then

E(c) =

8.

We

define D = {v

E x : ( V c E M) ( c # 0

(3w E E(c)) ( v

5;

w

V

w < v)}.

i s a 5-maximal l i n e a r l y o r d e r e d s u b s e t The p r o o f depends on t h e f o l l o w i n g t h r e e lemmas:

W e s h a l l show t h a t

of

.+

x.

(1) D (2)

D

(3)

M

D

- M, i s a l i n e a r l y o r d e r e d i n i t i a l segment of

x,

DR.

By (1) and ( 3 1 , M = DR. I f D is n o t a c_maximal l i n e a r l y o r d e r e d s u b s e t o f x , t h e n , by ( 2 ) , t h e r e i s a which is < - l a r g e r t h a n e v e r y c o n t r a d i c t s t h e m a x i m a l i t y of

u E D.

Thus,

DR C GR,

v E x

which

M.

W e s h a l l s k e t c h t h e p r o o f s of

(l), ( 2 ) and ( 3 ) .

PROOF OF (1): D _C M. Suppose

is a

u E R

v E D and a

and

c E M

v

M.

Since

such t h a t

M

i s maximal t h e r e

u - v + c = 1.

Moreover,

ALGEBRAIC FORMS

96.

115

v S because {v} E L and v E {v)F[x]. Therefore, v is not a unit in R (u-v # l), so c # 0. By the definition of D, there is a w E E(c) which is comparable with u and c c E GR. If w s v, then c, v E vR, which implies the contradiction that 1 = U - v + c E GR. If v i w, then c, v E wk so again we have the contradiction 1 = u'v + c E GR. PROOF OF (2):

of

D

is a linearly ordered initial segment

x.

Suppose u , v E D _c M. Then u + v E M I so u + v is not a unit element. Therefore, there is a w E x such that u + v E GF[x]. This implies u , v E w' so u and v are comparable. Suppose u E D, v s u and v E x. If c is a element of M then there is a w E E(c) such that are comparable. If u 2 w then v s w so v E D. f then v, w E u , so v and w are comparable. This implies that v E D. PROOF OF ( 3 ) :

non-zero u and w If w s u, again

M _C DR.

-

Suppose c E M DR and suppose c # 0. No element u E D is in E(c); for otherwise c E f R _C DR by (2). Let v1,v2, ,vk be the distinct elements of E(c). Since the vi t D , there is a non-zero element wi E M such that for all u E E(wi) , u is incomparable with vi. Since F is an infinite field, we may choose scalars al,a2,...,ak E F such that in

...

d = c + a w

+ a2w2 +...+

akwk E

no monomial which occurs with a non-zero coefficient in c or some wi, vanishes in R. Suppose w E E(d) Then there is some vi such that vi 5 w. Also, there is a v E E(wi) f such that v i w. Therefore, v, vi E w, which implies that v and vi are comparable. This contradicts the choice of the vi, q.e.d.

.

It follows as a simple corollary that if in AL 15, "unique factorization domain" is replaced by either "commutative

116

PART I, SET FORMS

ring with multiplicative identity" or "integral domain with multiplicative identity" the resulting statement remains equivalent to the axiom of choice. We continue with a form which was given originally by Klimovsky [1962] and later extended and generalized by Felgner [1976]. AL

16:

Every group contains a 5-maximal abelian subgroup.

The proof that M 3 + AL 16 is similar to the proof of Alternatively, it is easy to show that M 7 + AL 16 because being abelian is a property of finite character and a c-maxima1 abelian subset of a group is a subgroup. 4.3.

DEFINITION 6.28: The group (G,{*,e,-1 3 ) is said to be free if there is a non-empty X 5 G such that:

(1) e t x I (2)

If x E X, then

x-l E X I

(3) If y E G, then either y = e or there is a unique f 0 ) and a unique sequence (yl,...,yn) such that n E w each yi E X U X - l , yi*yi+l-# e for all i, and = {x-I:x E X I . y = y1*y2*...BY, , where X

-

The set X is said to generate G and the representation specified in (3) is called the reduced form of y. Given any non-empty set X there is a free group which is generated by X I and any two such groups are isomorphic, with the isomorphism being the identity on X. Hence, we call such a group the free group generated by X. THEOREM 6.29:

AL 16

-+

AC 13(2).

PROOF: Let x be a set of non-empty sets. For each u E x, let FU be the free group generated by u. Let lU be the identity element of FU. Let G be the weak direct product of the FU1s. G =

If

E

x FU:{u

UEX

E x: f(u)

# lU}

is finite).

56.

ALGEBRAIC FORMS

117

If f, g E G, f*g(u) = f(u).g(u) and f - l ( u = (f(u))-l for all u E x. The identity element 1 is the function f such that for all u E x, f(u) = lU. Then ( G , { * 1,-1l ) is a Let A be a 5-maximal abelian subgroup of G. For group each u E x, let AU be the set of uth coordinates of elements Each AU is an abelian subgroup of Fu and no of A - l U l . For -if AU = {luJ and a E u we could extend AU AU to All - U [a], where [a] is the cyclic group generated by a, (all powers of a and a-1 ) thereby contradicting the maximality of A.

-

{lUl and w is in reduced form then w If w E AU must be a power, positive or negative, of a product of elements of u U u-’. Otherwise, AU would not be abelian. Let f(w) be the first factor of w, %(w) be the last factor of w and - ~ ~ . for all C(w) = { f ( w ) , ~ ( ~ ) , ( f ( w ) ) - ~ , ( ~ ( w ) ) Then w1,w2 E AU {lul, C(w,) = C(w2) and 1 2 (C(w,) n uI s 2. Therefore, for each u E x, define g(u) = C(w) n u where w is any element of AU {lull then g is the required choice function. DEFINITION 6.30: Suppose (G,{.,e, -1 1) is a group.

-

-

(a) (Va,b E G) (a,b) = a-lb-lab a and b.

of H

is called the commutator

(b) If H is the group generated by {(a,b):a,b E GI, is called the commutator subgroup of G.

*,...

( (alIa2,.. . fan-l),an) and ...,aklak+l) = e for all

(c) If (al,a ,an-l,an) = G has the property that (al,a2, ai E G I then G is called nil-k. all

(d) G is called metabelian if a,b,c,d E G.

(Nil-1 is abelian.) ((a,b),(c,d)) = e

for

(e) If k t 1 is a natural number and G I is the commutator subgroup of G I define = GI, and G (k+l) = (G(k)) G‘O) = G, G We say the solvability degree of G is k if G(k = {el and G(k-l) # { e l . (Solvability of degree 1 is abe ian and solvability of degree 5 2 is metabelian.)

PART I, SET FORMS

118

Felgner [1976] extended Klimovsky's results and showed that each of the following are equivalent to the axiom of choice. Let

k

2

1 be a natural number.

AL 17(k):

Every group contains a 5-maximal nil-k subgroup.

AL 18(k): Every group contains a 5-maximal subgroup with solvability degree s k. The proof that each implies the axiom of choice is similar to the proof of 6.29. Our next forms are statements about vector spaces.

We shall use the notation av shall use the same symbols + and

instead of Oa(v), and we 0 for V as for F.

The notions of independent, basis, etc. which were defined earlier (6.12) for algebras in general can be defined more simply for vector spaces. DEFINITION 6 . 3 2 : (V,{+,O] U {Oa:a E F))

Suppose F is a field and is a vector space over F.

(a) W _c V

is called a subspace if

(b) A _c V

is said to be linearly dependent if

(3n E

...,vn

w ) (3v1,

W

is .a subalgebra.

E A ) (3all...lan E F) O = alvl

and for some i, ai # 0. linearly independent.

Otherwise, A

+...+

is said to be

anvn'

ALGEBRAIC FORMS

56.

119

said

v = alul (V

+...+anun .

is said to be a linear combination of the

ui's.)

(d) B _c V is said to be a basis of V if B is linearly independent and generates V. ( A basis is a c-maximal linearly independent subset of V . ) (e)

T:V

+

W,

If W is also a vector space over T is called linear iff

(i) T(vl

+

(ii) T(av) Let

B

v2) = T ( v l )

=

aT(v)

+

T(v2)

for all

for a l l

v E V

F

and

vl, v2 E V.

and for all

a E F.

be the following statement.

B:

Every vector space has a basis.

AL 19: For every field over F, every subset of V basis. Let

F

F and every vector space V which generates V contains a

be a field:

AL 20(F): Every vector space V over F has the property that for all A _ c V , if A is linearly independent then A can be extended to a basis of V. AL 21(F): Every vector space V over F has the property that if S _c V is a subspace of V then there is a subspace S ' 5 V such that S n S ' = {O} and S U S ' generates V. It is clear that the axiom of choice implies B, for example AL 12 + B, but it was unknown up to the present time whether B implies the axiom of choice. However, Andreas Blass sent us a proof that B implies the multiple choice axiom. (Blass credits Dr. R. Harting for asking the question

120

PART I, SET FORMS

which Theorem 6 . 3 3 answers and for calling attention to the construction of the ground field, which was a key step in the It proof.) We shall include his proof here (Theorem 6 . 3 3 ) . is still unknown whether B implies the axiom of choice in NBGO. Bleicher [1964] proved that AL 20(F) and AL 21(F) each imply the multiple choice axiom and had some partial results involving AL 19. Halpern [1966] proved that AL 19 implies the axiom of choice. Armbrust 119721 proved that if F is a field of characteristic zero, then the multiple choice axiom implies AL 21(F). AL 19 B and AL'12 AL 20(F) -+ B. We Clearly AL 13 AC 2 , AL 20(F) -+ AL 21(F) MC 1, shall show that AL 19 B + MC 1, and if F is a field of characteristic zero then MC 1 AL 21(F). -f

-f

-+

-f

-f

-f

First, it is easy to see that if F is a field, AL 21(F). For suppose V is a vector space over AL 20(F) -f

F and S 5 V is a subspace. AL 20(F) implies that S has a basis B' and that B' can be extended to a basis B of F. Take S' to be the subspace generated by the elements in B B'. Then S n S ' = t o ] and S U S ' generates V.

-

THEOREM 6 . 3 3 :

B

-f

MC 1.

PROOF: Let x = {ui:i E I) be a non-empty set of pairwise disjoint non-empty sets. Let R be any field such that R (Ux) = pI. Let F be the field obtained from R by adding U x as a set of indeterminates. That is, each f E F is a rational function of the "variables" in Ux. For each i E I, we define the i-degree of a monomial to be the sum of the exponents of elements of ui in that monomial. A rational function f E F is called i-homogeneous of degree d if it is the quotient of two polynomials such that all monomials in the numerator have the same i-degree, n, while all monomials in the denominator have the same i-degree, n - d. The rational functions that are i-homogeneous of degree 0 for all i E I form a subfield K _c F. Thus, F is a vector space over the field K and we let V be the subspace which is generated by

Ux.

ALGEBRAIC FORMS

56.

121

By hypotheses V has a basis, B. For each i E I and t E ui we can express t as a finite linear combination of elements of B with coefficients in K:

where B(t) is a finite subset of B and for each b E B(t) , kb(t) is a non-zero element of K. If s is another element of the same ui as t then s =

c

b€B ( s )

kb(s)*b

If we multiply (1) by the element

.

s/t E K

we obtain

(3)

Since B is a basis each element in V is uniquely expressible as a linear combination of elements of B. Thus, it follows from ( 2 ) and ( 3 ) that B(s) = B(t) and kb(s) = (s/t)kb(t). This means that the finite subset B(t) _c B and the elements kb(t)/t depend only on i, not on the particular t 6 ui; we therefore call them Bi and kbi, respectively. Since kb(t) € K, kbi is i-homogeneous of degree -1 (and j-homogeneous of degree 0 if j # i). This means that when kbi is written as a quotient of polynomials in reduced form, some variables from ui must occur in the denominator. We define a multiple choice function g on x such that for each i E I, g(ui) = the set of those members of ui which occur in the denominator of kbi (in reduced form) for some b E Bi. Then g(ui) is a non-empty finite subset of ui, q.e.d. If a vector space V has a basis and all bases of V have the same cardinal number K, then K is called the dimension of V. Let B' be the following statement: B':

Every vector space has a dimension.

Then we have AC +. B' implies AC in NBGO

B MC 1. It is not known if B or if B' implies AC in NBGO.

+

+.

PART I, SET FORMS

122

(The statement B '

was suggested by Norbert Brunner.)

THEOREM 6.34: AL 19

AC 2 .

+

PROOF: Let x be a set of pairwise disjoint non-empty and let sets. Let R be any field such that R n (Ux) = F be the field obtained from R by adding U x as a set of indeterminates. (The same construction which was used in

Theorem 6 . 3 3 . ) =

Let V

{f: f

is a function, f:x

-+

{u E x: f(u) # 0)

is finite)

be the vector space of

For each

s E Ux

and

-s(u) =

F

and

.

F L X 1 over the field

u E x,

F.

define

s

if

s E u,

0

otherwise,

Is:

and let A = s E Ux}. Then it is easy to see that A is a linear combination generates V because each f E of s ' s . Thus, AL 19 implies that there is a B 5 A such - that B is a basis of V. Suppose s l , s 2 E B and there is a u E x such that s 1 (u) = s1 E u and s2(u) = s2 E u. Then (sil)sl ( s -1 2 ) s 2 = 0, contradicting the fact that B is linearly independent. Thus, for each u E x there is at most one E B such that s ( u ) # 0. On the other hand,since B generates V, for each u E x there is at least one f B such that s(u) # 0. Thus, { s : E Bl is a choice set for x.

-

-

s

s

THEOREM 6.35:

If

F

is a field, AL 21(F)

-t

MC 1.

PROOF: Let x be a set of pairwise disjoint non-empty sets. Let F be any field. Let L(u,F) = {alsl +...+

ansn: n E w,

ai E F

and

si E u3

L(u,F) is the set of all finite formal sums with indeterminates from u and coefficients in F. It can be considered to be a vector space over F, if we assume the usual conventions about addition and scalar multiplication. The set u is then a basis of L(u,F). Let LO(u,F) be the subspace of L(u,F)

56.

ALGEBRAIC FORMS

123

in which the sum of the coefficients of each finite sum is zero. That is, LO(u,F)

=

{alsl

+...+

ansn E L(u,F):al +

..+ an

= 01

.

Let V be the weak direct product of the L(u F ) for u E x and Vo be the weak direct product of the Lo u,F) for u E x.

v

=

X

{f E

is finite}.

L(u,F): {u E x: f(u) # 0 1

UEX

v0= {f E X L~(u,F):{u E x: f(u) #

01

is finite}.

UEX

Then Vo is a subspace of V, and is a subspace V ' c V such that

AL 21(F) implies that there

0 -

(i) V o

n

V;,

=

and (ii) V o U V;,

{O)

generates V.

In what follows we shall identify each function fs E V such that s if s E u , f (u) = 0 otherwise

s E Ux

with the

.

For each v E w' E V b such s1 = v1 + vi vi, v; E V b .

V there is a unique w E Vo and a unique that v = w + w'. Suppose sl, s2 E u E x, and s2 = v2 + v;, where vl, v2 E Vo and Then

s1

-

s2

=

(vl - v2)

+

'vi

Thus, by the definition of V o , we have so vi v; E V o n V b = {O}. Therefore, Consequently, for each u E x there is a that for each s E u there is a vo E Vo For each u E x, let g(u) be the finite u which generate the uth coordinate of multiple choice function on x.

-

If AL 21(F).

THEOREM 6 . 3 6 :

then

MC 1

-+

PROOF: Let F a vector space over

F

- "1,.

s1 - S 2 I v1 - v2 E V O l vi = vi. unique vu E V b such with s = vo + vu. set of elements in vu. Then 9 is a

is a field of characteristic zero

be a field of characteristic zero and V F. Let S be a subspace of V. We use

124

PART I, SET FORMS

MC 1 to prove that there is a linear mapping T from V to S such that TrS is the identity function. Then, if S ' = {v E V:T(v) = O}, the kernel of T, it is easy to prove that S ' is the required subspace. ( S ' is clearly a subspace. If u E S then T(u) = u and if u E S ' then T(u) = 0 so S n S ' = {O}. For any v E V, T(v) = s E S and T(s) = s. Consequently v = s + (v-s), where s = T(v) E S and v - s E S'.) It remains to be shown that such a linear mapping exists. Let f be a multiple choice function on P(V) { @ I . Let L be the set of all linear mappings from a subspace of V into S and let g be a multiple choice function on f?(L) {fl]. Define f(0) = g(pI) = 0. For each ordinal a we define a linear function ha and a set B a as follows:

-

-

ho = identity map on If

a

S,

Bo

= S.

is a limit ordinal, ha =

U h B and B 0) (Vp

2

KO)

( ~ * = p K

+

K*).

+

p).

(Tarski 1 9 2 4 ; 1 3 9 )

(Tarski 1 9 2 4 ; 1 3 9 )

LIST OF SET FORMS

262

CN 1F:

CN 2:

CN'4:

x is finite iff x has at most one element or there exists a non-empty set u disjoint from x such that u u x < u x x. (Tarski 1938a; 140)

A set

( V ~ , p ~ zv N 0 )

(VK

No)

2

(K

s p

[K

+

=

K*

+

u

-+

[K

s p

V K i

v)].

(Tarski;

.

(Tarski; 139)

&),

]

.

(Tarski;

(Lesniewski, Sierpinski 1947;

K*).

139) CN 4F:

CN 5:

set x is finite iff x has at most one element or there exist disjoint sets u and v such that ]u( < 1x1 and I v / < 1x1, and u u v = x. (Tarski 1938a; 139)

A

(VK) (vp

2 No) ( K * V = K

V K-U = p).

(Lesniewski,

Sierpinski 1947; 140) CN'5:

%) (K*K*

(VK z

(Lesniewski, Sierpinski 1947;

= K*).

140) CN 6: CN 6F:

(VK

2

L

t Z O ) ( ~= K ) .

(Tarski 1924; 140)

A set

x is finite iff x has at most one element or x < x x x. (Tarski 1938a;140)

CN 7:

( Y K t p ) (K2 = p2

-+

K

= p).

(Tarski 1924; 140)

CN 8:

( V K , ~ )( K * = U* *

K

= p).

(Sobocinski 1961b; 140)

CN 9:

(VK

2

KO)

( 3 ~ ( )K *