THE AXIOM OF CHOICE FOR COLLECTIONS OF FINITE SETS ,.. ._. by Robert James Gauntt Dissertation submitted to the Faculty of the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1969 C', f·' [ APPROVAL SHEET Title of Thesis: The Axiom of Choice for Collections of Finite Sets Name of Candidate: Robert James Gauntt Doctor of Philosophy, 1969 Thesis and Abstract Approved: ~ ~ ~ Carol R. Karp'f2-: Professor Department of Mathematics ABSTRACT Title . of Thesis: The Axiom of Choice for Collections of Finite Sets Robert James Gauntt, Doctor of Philosophy, 1969 Thesis directed by: Professor Carol Karp Some implications among finite versions of the Axiom of Choice are considered. In the first of two .. chapters some theorems are proven concerning the dependence or independence of these implications on the theory ZFU, the modification of ZF which permits the existence of atoms. The second chapter outlines proofs of corresponding theorems with 11 ZFU 11 replaced by "ZF" .. The independence proofs involve Mostowski type permuta­ tion models in the first chapter and Cohen forcing in the second ·chapter. n The finite axioms considered_ are C , "Every collection of n-element sets has a choice function"; n W , "Every well-orderable collect ion of n-element sets n has a choice function"; D , "Every denumerable collection 0 9f n-element sets has a choice function"; and A (x), "Every collection Y of n-element sets, with~~ X, has a choic e function". The conjunction cnl & ••• & cnk is denoted by CZ where Z = (n , ••• ,nk). Corresponding 1 conjunctions of other finite axioms are denoted similarly by W, D and A (X). z z z Theorem: The following are provable in ZFU: wk1n1+ ••• +krnr wnl . nr ➔ V • • • '7 w · I / okJ)'l.1+ •.. +krnr onr ➔ Dn1 V ••• V , and ck1n1+ ••• +krnr cn1 v wn2 nr ~ v •.. v w . The principal result involves the condition Tn,z= = For every subgroup G of Sn without fixed points, f\•,, there is a finite sequen~e (H1 , •.• ,Hm) of proper subgroups of G such that I G I . + ••. + I G I e: z. I H1I I HJ Theorem T: If ZF is consistent, then T n,Z is necessary and sufficient for (I) lz'Fu (Dz ➔ Dn) and suff~cient for (II) lzFu (Cz ➔ en). Furthermore (I) is equivalent to each of the following: n) . (Ia) lzFu (WZ ➔ D , n (Ib) (W ➔ w ) , ~u z (Ic) lzFu (AZ (X) ➔ An (X) ) _. It can be sh~3 that T( ), fails. Tarski has shown 2 4 4 that C 4 C · is provable in ZFU. Hence it follows from (2} the above theorem that (I) is not always necessary for (II). Another main result involves Mostowski's condition ~ , z= For every decomposition of: n into a sum of (not necessarily distinct) positive primes, n = p + ••• + pr, 1 there exist non-negative integers k , ••• ,kr such that 1 n Theorem: M is sufficient fd'r (III) r,;;; (C ~ W). n,Z ZFU Z Mostowski has proven that if ZF is consistent, M n,Z n is necessary for (IIIa) ~ D). Hence, the following lz'Fu (c 2 result: Theorem M: If ZF is consistent, M Z is necessary and n, sufficient for (III) and also for ·(IIIa). It follows from Theorems T and M that there is an effective procedure for determining whether (I) holds and whether (III) holds. ACKNOWLEDGMENTS I would ·like to express my appreciation to Professor Carol Karp for her invaluable guidance of my studies of "··· the modern developments of Axioma t .ic Set Theory t and for her many technic~l suggestions and continual encouragement during the preparation of this· work. Special thanks are due to Professor James Owings, Jr. for numerous stimulating'conversations on the foundations of mathematics and for the continued interest he has shown for my research efforts. I am indebted to Professor E. G. K. Lopez-Escobar . for his valuable lectures and discussions on mathematical logic. I would like to/ express my gratitude to· Professor Leonard Gillman for a lively early introduction to set theory. I would also like to thank the following persons for enlightening and encouraging conversations: Professor A. Bernstein, Mr. -.:(. Gregory, Professor R. Jensen, Mr. A. Mathias, and Professor A. 'Mostowski. ii TABLE OF CONTENTS Chapter Page ACKN'OWLEDGMENTS • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ii INTRODUCTION • . • . • • • • . • . • . • • . • . • . • . • • . • • . • • . 1 PRELIMINARIES 2 I. FINITE AXIOMS OF CHOICE AND THE THEORY ZFU ....• 4 1. Finite Axioms of Choice 4 2. The Role of Well-Orderings ••..••.•••....•.• 6 3 . The Ma in The or ems . • . • • • . • . • • . • . • 9 4. Proofs of Sufficiency . 12 5. Models of ZFU . -. 21 6. Proofs of Necessity . 26 II.· CORRESPONDING RESULTS IN THE THEORY ZF . 33 BIBLIOGRAPHY . 46 iii INTRODUCTION For a set theory without the Axiom.. of Choice it is natural to ask what sort of implications involving various restricted versions of the Axiom of Choice are provable. Mostowski [2] considers implications among axioms of the n form C : "Every collection of n-element sets has a choice function" where n is a natural number. Mostowski proved the necessity of a certa{n number theoretic condition M n1 nr n for a proof of C &... & C ~ C and raised the question of its sufficiency. Mostowski actually shows that Mis n1 nr n n necessary for a proof of C &... & C ~ n· where D ,.., .. means· "Every denumerable collection of n-element sets has a choice function". One of the main results of this paper is that Mis also sufficient for a proof of nl cnr n C & ... & ~ D. The principal result of this paper is that a certain group theoretic condition Tis necessary nl nr n and sufficient for a proof of D & ••• & D ~ D . 1 PRELIMINARIES Let ZFU be the modification of ..ZF (Zermelo-Fraenkel set theory) which permits the existence of atoms (also I'•., called urelemente or individuals). The axioms of ZFU are given by Suppes [4]. Statements and proofs will be 11 given informally, and the symbol. uL_2FU will frequently be omitted where a proof involves only conventional mathe- matical methods. However, except where another theory is specified,. all statements and proofs may be formalized in ZFU. We let O denote the empty set. We say x isan atom if x I 0, but x has no elements. · xis a set if xis not an atom. A pure set is a set having no atoms in its transitive closure. We let w be the set of finite ordinals. A natural number n is an el~ment of w, so we have n = [O, ..• ,n-1}. We write x ~ y if there is a 1-1 mapping from x onto y. For finite x, we write Ix\ for the cardinal of x. We write UX for the set of all elements of elements of X. A function-Fis called a choice function if for each x in the domain of F we have F(x) € x. Fis called - a subset function if for each x in the domain of F, F(x) . ~ .. 2 3 is a non-empty proper subset of x. A collection Y is said to have a choice function (subset function) if Y is the domain of some choice function (subset function). r,. ._. CHAPTER I FINITE AXIOMS OF CHOICE AND THE. THEORY ZFU 1. Finite Axioms of .Choice · For each natural number n we let An(X} denote the sentence "For ev_ery collection Y of n-element sets with n Y ~ X, Y has a choice function''. (more precisely A (X} is some formula of ZFU, with one free variable X, having the same meaning as the giv~n sentence}. Let en be the statement "Ev.ery collection of n-element sets has a choice . n function", let W denote "Every well-orderable collection of n-element sets has a choice function" and let Dn denote "Every denumerable collection of n-element sets has a choice function". Letting a. vary over ordinals, we then n n n n n n have C H ('<fX}A (X}, W ~ (Va}A (a} and D H A (w}. For each finite set Z = fn ~ •.• ,nk} .£ LU define A (X} as the 1 n1 nk ~ conjunction A (X} &... & A (X}. Similarly we define con- junctions C , w · and D so we have c ~ (VX}A (X}, z z z 2 2 W H (Va}A (a} and D HA (w}. Throughout the paper n z z . z z will vary over .na tura 1 numbers, Z will vary over finite sets of positive natural numbers and a will vary over ordinals. 4 5 2 4 Lemma 1 (Tarski): c "-? c . 2 Proof: Suppose C holds and Y is a collection of 4-element sets. We wish to show that Y has a choice 2 function. We have from C that there is a choice function f on the set of 2-element subsets of UY. For each y € Y, y has exactly 4 elements so the set y* of 2-element subsets 4 of y has exactly (2)=6 elements. Each x € y* is in the do- main of f so f(x) € y and hence y* determines exactly 6 choices from y (i.e./ \ [(x,v) € f: X € y*, V €y}j=6). Let g(y) be the set of elements of y chosen most often (i.e., g (y) = [ V € y: I [(x,v) € f: X € y*} I is maximum}). Since there are exactly 4 elements in y, they cannot be chosen equally often so g(y) '-/ y. Hence\ g(y) \ = 1, 2 or 3. We now define a choice function Fon Y as follows: for each y € Y, if g(y) has one element, let F(y) be that element, if g(y) has 2 elements, let F(y) = f(g(y)) and if g(y) has 3 elements, let.F(y) be the element of y ~(g:(y).