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 <nu , yy> , < v , u >, o r < u , u > respectively. Then < x , r> u, <x„,e r„> . Cxlearly F, defined on V -x by F(u) = < x lav 1:u1 fuenction ont the p roper class V-x, hence x> , r u >u , i s R(F) = {f lu ) I u V - x } = < x , u i<s a proper class. But , r „ > I Y u <yE, s >V <-x ,x r>} < y , s > , huence is also a proper class, as claimed. QED. o Consider now the intuitive collection g of all similarity clas- s>es o f well-ordered sets. We ju st showed that each such si- m, ila rity class (with the one exception (< 0 , 0 > )) is a proper colass. Thus we can't use A C to prove the existence of an NBG crlas s C wh ich has e xa ctly one point in common wit h each < But in tu itive ly the class of ordinals, denoted b y "On", isu just such a class C. Likewise we can't use A C to prove the existenceo o f a class K , called th e class o f cardinals, wh ich consist, of exactly one point from each simila rity class o r un- orderedu sets. Wh a t to do ? o In the case of the ordinals we are lucky. We can actually d>efi ne in NB C a particular (proper) class, denoted b y On, o f wr el l-ordered sets and then prove (even without the use of AC) thatb every well-ordered set is simila r to exactly one element oyf O n; p u t another way, class O n intersects each simila rity ctlass of well-ordered sets in exactly one point (cf. [9; p, 1791). hIn the case o f cardinals we are almost as lu cky. Using the tehe ory of ordinals we can actually define in NBG a particular c(proper) class K and prove (with the aid of AC) that every set ocan be put in 1:1 correspondence with exactly one element of Kr ;r pe us tp ao n od ti hn eg rn we aw y , c l a s s K i n t e r s e c t s e a c h s i m i l a r i t y c l a s s o f 276 JOHN H. HARRIS unordered sets in exactly one point. But what about the case of totally ordered sets, o r partially ordered sets ? 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<ri FM) y0wheree n " P ' denotes the power set operation, i.e., P(Y) = { R,yc c ye }. D efine H = U aeon F(u). We will say that x is a regular Eco set or x is regular iff x ( I n the literature these are usually Gof called well-founded sets.) Then REG implies V = H , i.e., every tns set is a regular set. Before proving this let us give a suggestive hte interpretation. We can think of V =H as giving us (to borrow ert a te rm fro m cosmology) a "big-bang" th e o ry o f sets. I n the ras beginning there was only one set, the empty set. edt Seixtage 0: The collection F(0) = P(0) = OP is made into a set. xc„ it1 si„ tn, sg0 axs niu x,c i1h Axt sih una cAt h. t( hU as ti n g A C e v e n t h e c o n - ON THE AXIOMS OF CHOICE AND REGULARITY 277 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.