ON THE AXIOMS OF CHOICE A ND REGULARITY
John H. HARRIS
O. Introduction
In § I a problem is posed which suggests a need fo r a ve ry strong axiom of choice or some extra axiom of set theory. The Axiom of Regularity (REG) is the extra axiom that is usually used. In § 2 some implications of REG are given. In terms of a general definition of "set" given in § 3 it is shown in § 4 that REG seems t o be tru e f o r classical mathematical structures. But the set theoretic mode of thought and modern mathematics are not limited to classical structures. In § 5 evidence against REG is given by introducing a new construction process which can be used to construct (reasonable ?) sets contradicting REG. The construction process of § 5 is clearly open to question and is st ill o n ly at the intuitive level. But the ma in point o f the article is not to prove REG false, but to show (as is done in § 5) that REG is a questionable rather than an obviously true axiom of set theory. So in § 6 a neutral viewpoint is adopted neither accepting REG nor its negation. However some limitation type axiom is necessary in order to eliminate such pathologies as -cycles like x (LIiM ) isx introduced and shown to be equivalent to a "no cyci les axiom". A lso L I M and REG a re shown t o be related to .each other in a ve ry natural wa y wit h L I M being a ve ry weakA and REG being a very strong limitation type axiom. Also sincen REG is no longer available to resolve the difficulty raised in A§ 1, a xsufficiently strong a xio m o f choice (A C*) is in tro - duiced too do the job. m o I. fA motivatin g problem L i mLet NBi G denote Von-Neumann-Bernays-G6del se t th e o ry (spt ecifiacally axioms A, B, C o f [4]). To NBG we need to add t i o n 274 JOHN H. HARRIS some form of the axiom of choice (abbrev.: AC) in order to do mathematics as currently practiced. Let us use the following class form.
AC: Fo r every class X of pairwise disjoint sets, there exists a choice class C wh ich has exactly one point in common with each non-empty x G X: in symbols, (3C)(V x )
If some of txhe classes X formI (in NBC)[ xa *c0o llection having X Iev,e r, Xwe can gintuuitively think of such a collection. 'What if we h, aiveX an „int uitivae cosllect ion {X ) ,eo D f pairwise disjoint clas- e , l e a m r e[ enc t s ; ses ? Intuitively there exists a choice class C which has exactly h p ro o pwn e -r , onte poinht in c=oemmonn wi th e ach X ? ,. S i n c e bew proper, clee{arlyO AC is not strong enough to justify the exis- tsenccoe ofm asuceh an choice class. Let us first discuss h o w such a soitunatiofn oarise ]st ve] ry .naturally. t We hwill deefine an ordXered set to be any ordered pair < x , r > o) f sets, such that r Ç x X x. We think of x as (partly) ordered byc r aond wriute " u rlv" f odr ''< u , v > G r" . For example, x could be quasi, partially, totally, o r well-or- dered b y r. The case when r = 0 corresponds to considering x a s t o t a lly unordered, i.e., considering x abstracted f ro m any ordering on it such as is done when we consider the car- dinality of the set x. Let us define two ordered sets < x , r > and < y , s>as being similar o r order isomorphic (abbrev.: < x , r > < y , s>) if f there is a bijective f:x—>y which preserves ordering, i.e.,
urv f ( u ) s f(v) f o r a ll u, v e x .
The equivalence classes wit h respect t o the equivalence re - lation " = " w i l l be called simila rity classes. Note that in the case when r = s = 0 the notion of order isomorphism reduces to that of 1:1 correspondence as used in cardinality arguments.
THEOREM 1. I f < x , r > is an ordered set with x t O , then the ON THE AXIOMS OF CHOICE AND REGULARITY 275 similarity class containing < x , r > is a proper class: in sym- bols,
{< y , s > I < x, r > < y , s > } 0 V . PROOF. Choose some element u boe the set obtained fro m x b y replacing u oede xnb.o tey Tth eh ureela. tnio n oAbtalinesd foro m r b y replacing a ll p a irs lf< u oe r t r upairsao I Y u
2. The Axiom of Regularity
There is a candidate for an axiom of set theory, axiom D of [41, ca lle d th e A x io m o f Regularity (abbrev.: REG), wh ic h solves o u r difficulty. On e standard formulation o f REG says that any non-empty class X contains a t least one set wh ich has no element in common with X: in symbols VX[X v l y EX and v 11 X = 01]. The implications of this not very obvious axiom are far reach- ing, but some easily understandable conclusions are now given. (Cf. [9, pp. 201-3] fo r a more detailed treatment.) First, REG implies that there are n o in fin ite ly descending
c c , e -chains ; i . e . , E E X1 E X0 t h e 'rToe p r o ve this, assume t h e contrary. L e t A = {x 0wherei x ,ixs x E iXversen o is. p rovable.) ,Tno Nehxt let us define a function F:0n--*V by eAs ne b—q u F((3) = P( U a
Stage 1: The collection F(1) = P(F(0)) = to, { C M i sa s et.m a d e i n t o
Stage 13: The collection A — U OE<0 F(u ), i f m adne inotot a set. Then any subcollection of A not al- a l r ereadya d a syet is made into a set. Fin a lly a ll subsets of a A are csollecteed and tmade into a set called F((3). i s
Thus V = H = U ccel F(u ) says that a class or collection is a set iff it is made into a set during one of these stages. Put another way, each set is constructed or built up from the empty set via iteration o f the union and p o we r set operations sufficiently many times. (A ctu a lly t h e construction procedure i s mu ch easier at successor ordinals since one can show that F(13+1) = P(F((i))-) In order to prove V = H and several other consequences of REG, le t us define a n ordinal valued function g : H — *g(Ox) n= fibrst yu such that x E FM. We call g(x) the rank of the set x. In terms of our interpretation above the rank of a set x is closely related to the stage at which collection x was made into a set. To prove REGV----H, assume V—H O . Then by REG there is some x E V—H such that x n (V H ) — 0, i.e., x c H . Then {e(u) u e x } i s a set o f ordinals, hence i s st rict ly bounded above b y some ordinal 13, i.e., u E x g ( u ) < (3. Then b y de- finition of e and 13 we have
u cF(g (u ))c U a o Hence F ( u ) f o r a l l u E x x g. U a < f i hence F ( u ) ,
xE P( U Œ<13 F(a)) = F((3)1,.;EI 278 JOHN H. HARRIS contradicting
The rank function has nice properties, the important one for us being xey < e(y) for x, yEH where " < " denotes the standard ordering among ordinals. To see this, assume Q(y) f 3 . Then
xcyEF((3) = P ( U Œx y< Ic3 U Fxa ( Then we use A C to choose one element fro m each o f these disjoint sets X f t In summary, NBG +REG + AC implies the following: , h(I) en on incfineit ely descending -c h a i n s ; f(2) ar "boig-bang" theory of sets (all sets are regular); m(3) n o finite E-cycles: e(4) f oar any equivalence relation R, there exists a choice class c Ch o iemptyt R-equivalence class. f Cwonclu sions (3) and (4) are intuitively true statements of set ttheohry. On the other hand (I) and (2) — which in NBG+ AC are heachi equivalent to REG — are not so obviously true. Ma n y elogician s, however, do consider REG to be obvious (cf. [7; p. 561, R[141h)-. But to discuss whether REG is true o r not is to discuss ethe cqquesution, " Wh a t is a set ?" i vo a l en n c3. Ase def inition o f "set" c i l a Lest uss consider the following intuitive definition wh ich we swillt reefer to as the "generalized big-bang" definition of a set s(abbrev.:s . GBB). One starts wit h certain physical and/or con- ceptoual objects, called atoms. We build n e w sets b y stages. At afn y stage we form new sets by taking collections of objects (i.e.,e atoms and/or sets) formed at previous stages. Also at any stagxe we can add new atoms. We iterate this procedure over all paossible stages (one stage for each ordinal a). The collection of setsc so obtained is the collection of all sets. Ctlassically an urelement is defined as an object which isn't a colllection (hence o f course n o t a set) b u t wh ich can be a n elementy o f a set. We adopt a broader outlook and define an atomo as an object which is discovered or formed at some stage by sno me process o t h e r th a n co lle ctin g objects fo rme d a t preveio us stages. Often atoms w i l l be objects defined b y the propeerties they satisfy. It is even possible that an atom is itself a set.l However, an atom x can't be a proper class since if x is introeduced in stage a, then x is a n element o f sets formed m e n t f r o m e a c h n o n - 280 JOHN H. HARRIS during stage a +1 . Hence proper classes a re n o t considered to be objects. One commonly seen version of GBB is to allow starting with certain atoms which are urelements but not allow adding new atoms at future stages (cf. [13; p. 233]). One gets what might be called regular set theory if the collection o f a ll atoms is the empty set. Then the only objects are sets and GBB reduces to a vague wording of the simple "big-bang" axiom V—H, which is equivalent to REG. Thus we can conclude that REG is true for the concept of set embodied in GBB if we can show that we can assume without loss of generality that there are no atoms. In GBB we said that atoms would be physical or conceptual objects. We will also consider sets as conceptual objects. To say that object X is a conceptual object is not meant to imp ly that we are assuming X exists only in the mind of the conceiver and has no existence independent of thought. In this paper we do n o t take sides between conceptualism and platonism (o r realism). Rather we mean that a conceptual object is a non- material object whose existence has been or can be discovered by the use of pure reason. 4. Evidence for REG One problem with using physical objects such as elementary particles or hamburgers as atoms is that it gives a time depen- dent set theory, wit h sets changing as objects are created o r destroyed. I t is better to th in k of a fixed universe consisting only o f conceptual objects; then if desired, one can set up a time dependent in fo rma l correspondence between p h ysica l and conceptual objects. What can be done in a set theory with physical atoms can be duplicated u p to "conceptual isomor- phism" in a set theory without physical atoms. Hence without loss o f generality we can assume that there are no physical atoms. But wh a t about atoms wh ich a re conceptual objects; e.g., what about natural numbers ? Conceptually a natural number isn't necessarily a collection and isn 't thought o f as having ON THE AXIOMS OF CHOICE AND REGULARITY 2 8 1 any internal structure. The natural numbers are in fact good examples o f atoms wh ich are introduced conceptually a t a n early stage in GBB and are defined formally by the properties they satisfy. Thus instead of bothering with such foundational questions as " Wh a t is the number 1 ?", the mathematician will usually ju st assume that he is g ive n a collection o f objects (which we are calling atoms) satisfying the second order axioms of number theory and will proceed from there. Fo rma lly this corresponds t o extending the fo rma l theory NB G b y adding new symbols "N", " 0 " , "S", + " , "." and at least the following new proper axioms. (i) N G ( N is the set of natural numbers) (ii) O N (iii) V x [x N --> Sx N ] (iv) (N i(N x()v, ( ) (vz[o ( v x ) Y ) (vN(ii) x + y and x.y N for all x, y N (v[[viixi)S x +z0 —= x and x + Sy = S(x + y)for all x, y N x>(xix) sx.0x — 0 and x.Sy = x.y + x for all x, y EN. —Z) , 1 — ) S•NLe tN us ccallZ thi]s extended theory NB G+ N. ysThus th e mathematician ju st leaves t h e natural numbers 0,1xx — SO , 2 = SSO, a s unspecified objects or atoms satisfying = ce*rta in properties. Whether these atoms are also sets is usual- lyy left unspecified. I f we are working in ordinary NBG where th]e0re are no urelements and X EV implies X is a set, then a ll the integers are sets, which sets however being left open. If we are working in a modified NBG where we allow urelements as well as sets and classes, (cf. [12]), then we don't even kn o w whether the integers are urelements or sets. Logicians have discovered several ways of defining an NBG set o.) and NBG set functions f , f , , f. such that if N, 0, S, + , are interpreted as u.), 0, fs, f , and f .a rer tehesorpemesc otf i(av deelfinyit,io nal extension of) NBG (cf. [9; pp. t h e n ( i ) ( i x ) 282 JOHN H. HARRIS 175-81 and [10; Ch. IV ]; (when referring to [10] we mean the results of Ch. I V translated into NBG). In fact for such defini- tions as given in [9] and [101 it is provable in NBG that w itself, the elements o f w, and the sets f s(i.e., elements o f H). Put simply, N, 0 , S, + , c a n be defined ,in tferm s, of reagulnar sdets. Thfus in summary, what can be done .in a seta theorry weit h t h e natural numbers as atoms ca n be rdupelicategd (iun falct ian mrany ways) in some definitional exten- sion o fe NBG. t s The average mathematician when studying number theory still probably works conceptually in what would be formalized as N B G + N rather than some definitional extension o f NBG. He wo u ld (o r should) lo o k a t th e wo rk o f [9] a n d [10] a s providing models f o r N B G + N in NBG, hence demonstrating that he can wo rk in NB G + N free of worries of contradiction if he already accepts NBG. It is true that anything that can be done in N B G + N can be done in a definitional extension o f NBG, but the converse is false; e.g., one can prove such "ac- cidental" properties o f the natural numbers as 1 3 in [9] o r 1e2 and 2 e 3 but 1çt3 in [10]. Let us give another example of a mathematical object which conceptually isn't necessarily a collection. Specifically, given any two sets x and y, we have in mind the ordered pair < x , y > . Even if we wanted to, we couldn't start off with a ll the ordered pairs taken as atoms at some fixed stage a: it makes sense to talk about the ordered pair < x , y > o n ly during some stage after the conceptual construction o f the sets x and y themselves. Thus if we were to treat ordered pairs as atoms, we would need to allow the addition of new atoms at every stage. Instead o f bothering wit h the foundational question " Wh a t is the ordered p a ir < x , y > ?", wh a t the mathematician con- ceptually does corresponds f o rma lly t o extending NB G b y adding a new symbol " < , > " and the following proper axioms: ( (ii) (Vx, y)[ 5. Evidence against REG General set theory and modern mathematics, however, are not limite d to th e study o f classical structures and objects. Are there any non-classical conceptual objects which can't be expressed in terms of a set theory with no atoms ? Good can- didates f o r such objects seem to be objects x othat x ,n x GX2 GX1 EX0 . i= ,x x 2n , s u c h 1 1 f o r a l l n . T h i s g i v e s u s 284 JOHN H. HARRIS One o f the main goals o f this paper is to describe a general construction procedure wh ich can be used to ju stify the in - tuitive existence of infinitely descending G-chains of concep- tual objects. Let us start b y discussing G-descending chains in general. If we are not excluding the possibility of infinitely descending G-chains, it seems that we must allow the possibility that for any ordinal a we can find an e-descending chain of length a. But wh a t wo u ld we mean b y a chain o f length even w + 1, denoted by x, e x,, e ...Gx t G x The o n ly reasoonable condition seems to be to require (in ad- dition to the o?bvious condition that x„ , 1xo , bce xin, x,„ for fallo n beyondr asomel m:l in symbols, n < w ) t h a (am)t (Vn) [n m x ] . This suggests the following generalization. An - chaind e ofs lecnegthn bd isi definedn g as a sequence { x a } (i) 3 4 , a < (i1i) f o r every limit ordinal ? s s u c h t h a t ,c xx _a< b) , t h e r e Weef sax,y tihast zt iss a t th e bottom o f a n -d e sce n d in g chain ( x„). ao G < Even i f in fin ite ly descending e -ch a in s e xist th e situation (r x3 < ? , 8 isns'at toao mu essy csince, has sho wn in the next theorem, any two o f settsl cofnnehcted bay such an G-chain are also connected b y a lfineite G-chain. tl o n a r gTHtEOREM 1. I f x + a h2 x and ending at x8, then there exists a finite se- 1 l 6} quence o f ordinals < l +a 0 ----- ao PROOF. Assume proposition i s t ru e f o r a l l G-descending chains o f length less than b and consider a n y e-descending chain of length 8. I f b is a successor ordinal, say b = (i+ 1, we have Xs = xt3. 1 If 8 is a limit ordinal, then for some fi<8 we have e x 5 e EXx „ fo r a ll ti--aO• BDy theoirem f1 wef can always choose ( x b . ayt8is s oshmue fcienihte rordineal n . )aet „ L, hset xUas (Z)uit o rc Ush Zx denote t t s hthe auniotn o f the sets in the class Z. bWefo then define =U" Z by in duction as follows: x , i U Z= Z. nx s U " I i= Z Ttzhe uniUon a xiom of NBG just says that if Z is a set, then so is Z henUce a lso U BZ for all ea n n d Z s ) u . b c h a i n , g i v i n g u s t h e f i n i t e 286 JOHN H. HARRIS For a ll n?„-0 we can easily show b y induction that there exists sets xo, xi, , 1 such that x,,,, ex Fort if true for case n = k, then for case n = k + 1 we have , Xk.,4 eX1,1C X1 3C0 e<=>Xk,, 4 1,,1 Ukx x4=>X,k * T,e hus clearly z is an -a n ce st o r of a set x iff z U "X for some n?xE-0. We can now even define the ancestor function A where forU an y set x A (x) equals the set of a ll G-ancestors o f x: in symok bols , <+ A(x)I = (Unx) Unxin<00. =X TH>.EORE M 2. Assume that there are no finite G-cycles. Then forx each set x there is an ordinal 6, such that all G-descending chains starting at x are of length less than b PxR,OOF. Fo r any set x let 6, be the first cardinal number greater t.ha,n the cardinality o f A (x) — in the standard development of ,NB G + AC cardinal numbers are ordinal numbers. No w con- sider any G-descending chain x . 1 „, s with x aor e1 d=iff erexnt:. fo r oAthelrwisl e the et -dehsceneding chain connecting x.x Œand „xo where x . = xo and ct c o u l d be replaced b y a finUite c-ch a in , giving us a finite e -cycle . Also each x. is an G-ancestor of x toh e"re= a re xele.me nts of A(x), hence b Then if we denote f(a) b y x„ we have x ox x i Figuarativnely spdeaki ng we see that in constructing -a s c e n - e,ding cvhainse we nconceptually construct each x5 i n te rms o f cpr eviously completely constructed x xOrecEursive process. I n contrast, t o conceptually construct e - di( ea s yk +I = { N ( k +1), yo, • • • Yk}• Finally, le t x„ = y N Clearly the y „ are regular sets since they are b u ilt up b y r,e cuf rsoionr starting with the regular set NI ). Hence the x„ are regular. In o u r construction o f th e same x „ b y th e simultaneous process we will use stages 0, N , N +1. We first use sugges- tive wording to aid the conceptual visualization of the process. At stage 0 we make an open necked (conceptual) balloon and label it x oby passing it through the open neck of x .so ures that we will have Om)( W(.Assume, Tewh h iwese nhac veo cownmpdleetei dt stagesi o n 0, 1, n 1 where tfiAt hsi tangne ni w-es inthrod uce and start to work with x„ and continue eawithn oul r conlstruct ion activities on the x poiwe f moarkue aino in terms of conditions imposed on the xi„ during various stages, is as follows: Stage n?-0: require x i 1Sta gee cxo: iclosure step. That„ f f oo lr l o w s trivia lly from the definition of stages en anda to. cCleahrly a ll the x vi1ious prob lems. What is the difference between say xi °1„if ia* jnr ? edT.h esnxe otiwon o-bjects appear merely to be copies of each °rotheerg, buut lwae cra.n't prove that they are equal. For that matter, xBi u t ot For ha n y see t x r l e t A.*(x) -= x U A(x), o fte n ca lle d t h e etransit ive closure of x. We say that two sets x and y have the vsaame E-strructure if f A''(x) and ik*(y) are - • eitheres o mexois trsp ah 1:1i c fu, n ctiion. f ferom. A'K, (x) onto AL*(y) such that no b u e v f ( u ) e f (v) fo r a ll u, v A * ( x ) . s- eWe abbreviate this b y " A *(x) i k * ( y ) under f". e Intuitively one feels that two sets with the same e-structure mare equal. Strong support fo r his feeling is given b y the fo l- slo wing result, provable in NBG without REG. t oTH EOREM 3. Let x and y be regular sets. Then x and y have bthe same -s t ru c t u re iff x = y. PeR OOF. Use the proof of the Shepherdson-Mostowski theorem, acf. [3; p. 73]. Since x and y are assumed to be regular and since cREG restricted to the class H of all regular sets is provable in oNBG, the proof can be carried out without assuming REG. QED. p Now we don't know what part of our intuition or knowledge yof regular sets should ca rry o ve r to non-regular sets, i f the loatter exist in any reasonable sense. But it does seem falacious fto recognize the existence of two o r more distinct sets wh ich can'tx b e conceptually distinguished b y ju st looking a t th e ir e-structures. And we do want to construct as reasonable non- l r„eg ular sets as possible. So we propose the following axiom. STRUCTUREf A X I O M: I f two sets have the same e-structure, tohen they are equal: in symbols r a A*(x) A*(y) x y. n y n > 0 . ON THE AXIOMS OF CHOICE AND REGULARITY 291 THEOREM 4. Th e re is no sequence ( x 1 x ) n 0PR OsOFu (inc NhBG + tStruhctuare +t No G -cycles): Assume there is such a =s equence of sets. Then A*(x,) and A * (x iG-struc{ture: the G-isomorphism f is given by ) h ax v e t h e „f (x s a i m e Hence „according to the Structure Axiom x , ix = 1x k io We s}a y that x is a non-regular set if we can find sets xi, x2, ) .E suTchf hthaut s w e o= hx a cvx e rx ° 2 ni W, heneverG we construct a sequence {x.}a Stage n,>-0: require n e x nSt agae: na):d cl osxure, . „ For1 a ny ne thex on.ly stage. at which any potential element is ad- ded to x pno tei nstia l element x ns t a x ,g e n lTnhe proofs of properties (i) - (iii) are similar to but simpler than = .t,h e Tp rohofus osf s,im ilar properties for the next theorem which we { bwa ill pyrove in detail. Q E D . n sn Out r naext tgheorem shows one wa y to construct an e -descen- , edding c hain of length 8, given any 8>0. x at ) „ wThHEOREMe 6. B y t h e simultaneous process, f o r a n y ordinal , bhe>0 theare exists an e-descending chain {x}a < 6 such that 1 vn e x„ { x }, I a Now to discuss the regularity of the x„ for a<8. I f 8 isn't a limit ordinal, then = A.+N-F1 where A = 0 o r a limit ordinal depending on whether 8 X x + are regulai r sets as in example 1. I f 8 ? is a limit orr dinal, then for any a