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Hypersets Jon Barwise and Larry Moss

I n t r o d u c t i o n : W h a t A r e H y p e r s e t s ?

There were a handfu l of experiences that led me 1 to choose m a t h e m a t i c s as a career . One was a h igh school teacher showing me the cont inued fraction

1 x = 1 + (1)

1 1 + - -

1 + . . .

and then going on to convince me that x is, of all

l + V ~ things, the go lden ratio ~b - - - - - 7 - . On the one

hand was the ellipsis " . . . " descending like Lucifer's band ever deeper into the nefarious denominator . On the other was the light and clarity of the solution of equat ion (1), obta ined by observing that the fraction satisfies the ident i ty

1 x = l + - ,

x (2)

1 + X/-5 which quickly leads to x -

2 I was hooked; I spent several periods on hall mon-

itor du ty making up cont inued fractions and "solving" them. Eventua l ly I s tumbled on some problemat ic ones, which shou ld have led me to the concept of limit, I suppose. But they at least led me to appreciate the not ion w h e n I learned of it a couple years later.

And eventual ly, it led me to appreciate the construction of the reals as Cauchy sequences of rationals. But that was far d o w n the road.

I h a d a s imilar e x p e r i e n c e a few yea r s ago, in read ing a manuscr ip t 2 by Peter Aczel on non-well- f ounded 3 sets.

D e f i n i t i o n 1. A set b is a hyperse t if there exists an infinite descending sequence

�9 . . E a n + l e a h E . . . E a 1 E b.

Otherwise b is we l l - founded . A simple example of a hyperse t is:

x = {1,{1,{1,{1 . . . . }}}} (3)

This set is a member of itself, for the same reason that the cont inued fraction in (1) satisfies (2). Therefore x is

1 For obvious exposi tory reasons , the in t roductory sect ion is wri t ten in the first pe r son , in the voice of J.B., the older a u t ho r a nd the one consequen t ly more incl ined to reminisce. Otherwise , bo th au thors a r e equal ly to b lame for w h a t is conta ined here.

2 This manusc r i p t even tua l ly deve loped into Aczel [2], the definit ive reference on non -we l l - founde d sets. 3 O u r hypersets and Aczel ' s non-well-founded sets are the s ame thing. The former ha s the a d v a n t a g e of not hav ing nega t ive connota t ions , and it r eminds u s of the hyper rea l s of n o n - s t a n d a r d analysis . As we shall see, there is an analogy.

THE MATHEMATICAL 1NTELLIGENCER VOL. 13, NO. 4 �9 1991 Springer-Verlag New York 31

a hyperset. Confronted by such an expression years earlier, I experienced the same sense of vertigo experienced on my first encounter with continued fractions, but without the corresponding uplifting feeling given by the solution of the continued fraction. Sure, you can play the same game, by noting that

x = {1,x} (4)

But then what? In what domain of sets could one solve such an equation? But I was told that such sets do not exist. They are ruled out by an accepted axiom of set theory, the Axiom of Foundation, (FA):

Every set is well-founded. (FA)

So it would seem to be hopeless to try to solve (4).

The Axiom of Foundation

FA states that there are no infinite descending sequences of sets under E. There is an equivalent form of FA that is much more attractive. One of the building blocks of ZFC set theory is the family of sets V~, where e~ is an ordinal number. V 0 is the empty set, Q. V~+ 1 is the set of all subsets of V s. Finally, at limit ordinals we take unions. It turns out that FA is equivalent to the assertion that every set belongs to some V~,. Assuming FA, every set has a place in this well-ordered hierarchy. Roughly speaking, the higher the place, the more complicated the set. By adopting FA, we come to feel that the set-theoretic universe is a richly structured, hierarchical realm.

Set-theorists tend to have one of two attitudes to- wards FA. On the one hand there are those who be- lieve that it is fundamental to our understanding of sets, and so is one of the most basic axioms. The other attitude is that the Axiom of Foundation, unlike the other axioms of Zermelo-Fraenkel set theory (ZFC), does not capture a commonly accepted mathematical principle, one used by the working mathematician when applying set theory. On this view it is a harmless piece of logician's hygiene. (See Box "Some Quotes on The Foundation Axiom" for some examples of these diverging attitudes.)

How do we know it is harmless? Well, start with set theory without FA, known as ZFC-, and consider the collection WF of all well-founded sets. Using the observation that any set of well-founded sets is itself well-founded, it is easy to show (using ZFC-) that all the axioms of ZFC- hold when relativized to WF, and so does FA. Thus, introducing FA cannot introduce any inconsistencies into set theory. And, moreover, we can model familiar mathematical structures (natural numbers, rationals, reals, and so on) within WF. So why not assume FA?

And so, until recently, FA has pretty much had its way. While we may have disagreed about the reason,

most set-theorists have been quite content to assume that it is true. And for many of us, there was an almost religious devotion to FA. (See Barwise [3], for example.) But recently Foundation has been shaken, not by contradictions but by the claim that it disallows sets that we might, after all, want to have around. More importantly, an elegant alternative conception of set has been developed. It suggests a universe that en- compasses all the old well-founded sets, but also allows a space in which equations like (4) can be solved. And it is leading to some lovely mathematics. Equally important, this new conception of set has elegant applications outside of mathematics, in computer science, AI, philosophy, and linguistics. Hypersets allow us to model various kinds of real-world phenomena in set theory in a simple and elegant manner, using the machinery long familiar from the theory of sets. As a matter of fact, Aczel's work on non-well- founded sets grew out of a problem in computer science, on the theory of so-called "communicat ing systems." In attempting to understand and simplify work in this area, he was led to develop his universe of sets, and to formulate AFA, the so-called Antifoun- dation Axiom.

Ten years ago we, along with many set-theorists, would have claimed that the Axiom of Foundation was unassailable. But after learning about and using AFA, our feelings about FA and set theory in general have changed. We now recognize that axioms often embody metaphors, and that their adoption colors mathematics. The intuitions that FA is getting at are important, but they are limiting as well. So FA is not just an innocent assumption adopted out of expedience or the wish to be parsimonious. And to adopt FA merely because the iterative conception is attractive is to make a necessity of virtue. Aczel's work has taught us that FA is not a necessary part of a clear picture of the universe of sets. He showed that there is a domain of abstract objects in which equations like (4) have solutions, and that this domain is an elegant alternative to the ordinary universe of sets. Mathemati- cally, the domain is an extension of the well-founded universe, in very much the way that the real numbers are an extension of the rationals. On the other hand, there is a big difference between the two conceptions of set upon which the axioms rest. The point of this article is to describe this domain, and some of its applications.

We are also interested in the hypersets as a case study in the philosophy of mathematics. It gives us a chance to observe and take part in the evolution of a broadening of a mathematical concept, analogous to what took place with the discovery of the irrationals and complex numbers. Our aim will be satisfied if you come to see this domain as an interesting, mathemati- cally and philosophically respectable alternative to the classical domain of set theory.

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We feel that hype r se t s will become impor t an t as people use them in modeling. The fact that they embody an intuit ion about sets is a big plus for them. That this intuit ion differs from what we have for the wel l - founded sets is unfor tunate , but unavoidable. As always with mathematics , the impor tant thing is to be aware of what the axioms are saying, and what is be- h ind them.

The Foundat ion Axiom and Ordered Pairs

The importance of set theory in mathemat ics does not lie in the intrinsic interest in uns t ruc tured sets of objects, so much as in the mathematic ian 's ability to use such sets to represent mathematical s t ructures of enor- mous variety: numbers , functions, relations, and the like, as well as non-mathemat ical structure. When you trace all this back to its roots, it relies crucially on the

ability to represent sets of o rdered pairs, and so ordered pairs themselves. If it were not possible to represent o rdered pairs in set theory, set theory would be of virtually no mathematical interest at all.

The s tandard way of model ing the ordered pair Ix,y) in set theory is by means of the set {{x},{x,y}}. All that mat ters for most purposes is that this model ing satisfy the defining condit ion on o rdered pairs: if Ix,y) = (u,v) then x = u and y = v. For our purposes , it is important tha t this holds no mat te r wha t objects x and y happen to be, even if x and y are hypersets , say if x = {x,y}. A momen t ' s thought shows that the usual proof of this cond i t ion does no t make any a s s u m p t i o n s abou t the na ture of x or y. Once we have o rde red pairs, we can r ep resen t o r d e r e d triples as o rde red pairs (~x,y,z) = (x,(y,z)~), binary relations as sets of or- de red pairs, funct ions as certain k inds of relations, and so forth, as usual.

THE MATHEMATICAL INTELLIGENCER VOL. 13, NO. 4, 1991 33

To see the connect ion be tween o rdered pairs and FA, it is useful to give an equivalent characterization of the hypersets .

Definition 2. A set x is a constituent of a set z, writ ten x E* z, if x is a m e m b e r of z, or a member of a member of z, or a m e m b e r of a m e m b e r of a m e m b e r of z . . . . (That is, the relation E* is the transitive closure of the membersh ip relation E.)

The observat ion is just that a set b is a hyperse t iff there is a sequence Ca n : n = 1,2 . . . . ) in the constituent-of relation E*:

�9 . . ~ * a n + 1 E* a n E * . . . E * a 1 E * b

(One direction ( ~ ) is immediate, the other direction obvious: you just fill out a descending sequence in the E* relation to get a descending sequence in the mem- bership relation.) It is impor tant to note that the sets a, are not required to be distinct. Thus, for example, any cycle in the const i tuent relation in any const i tuent of a set b will cause b to be non-wel l - founded.

Using FA, we can p rove some surpr i s ing general facts about various sorts of mathematical objects. Here are two that will be of use to us.

Proposition 1. Assume the Axiom of Foundation. 1. For any x,y, x ~ Cx,y) and y ~ Cx,y) 2. For any relations R, S, if there is an object a such that

a bears R to S (in symbols, aRS), then there can be no object b such that b bears S to R.

Proof. These all rely on the simple observat ion that, with the above representa t ion of o rdered pairs, both x and y are const i tuents of Cx,y). Every representa t ion of ordered pairs we know of has this proper ty , and this is all that is n e e d e d to p rove these results. Let us prove (2) for the record. Assume by way of contradiction that aRS and bSR, i.e., that Ca,S) E R and that Cb,R) E S. Then we have a cycle in the cons t i tuen t relation E*:

S E* Ca,S) E R E* Cb, R) E S

w h i c h s h o w s t h a t t h e s e se t s a re all n o n - w e l l - founded. �9

This resul t suggests that FA will cause problems w h e n we seek to use w e l l - f o u n d e d sets to mode l various kinds of circular phenomena, and we turn now to a few examples.

ModeIing Circular Phenomena in Set Theory

However , wh en we try to apply these techniques to p h e n o m e n a that involve circular i ty , the Axiom of Founda t ion frequently gets in the way. In this section we examine a few of these. We only go into the simplest two in any detail, since the aim is to give the reader a feel for the sorts of cases where hyperse ts are important .

Streams The notion of a s t ream comes up in computer science. Intuitively, a s tream is an ordered pair s = Ca,s'), the first e l e m e n t of which is some finite value, the second e lement of which is another stream�9 Thus the basic opera t ions on s t reams are taking its first e lement , which gives a finite value, and taking its second element , which produces another stream.

While streams are infinite, it is easy to write computer programs that generate them. For example, if we define:

fin) = (n,fln + 1))

and run this as a program, it will generate the s tream C0,C1,C2 . . . . ))), at least in the ideal limit that interests us as theorists.

Let A be a set of " a t o m s , " character strings, nu- mera ls , wh a t have you , an d s u p p o s e we w a n t to model the collection of streams, each having as first e lement an atom in A. The natural definition that suggests itself is as follows. Let the set St(A) of s treams be the largest set satisfying the following condition: if s E St(A) then s = Ca,s') for some a E A and some s' E St(A).

Intuitively, every infinite sequence of atoms should give rise to a stream (and vice versa). However , we have the following observation:

Proposition 2. Assume the Axiom of Foundation. Then St(A) = (~.

The proof, like the proof of Proposi t ion 1, relies on the observat ion that y is a const i tuent of Cx,y). Thus any s t ream would give rise to an infinite descending chain in the relation E*.

This is a pret ty drastic mismatch be tween the intu- i t ive n o t i o n and o u r ma t h ema t i ca l mode l . Conse - quent ly , if we are going to model streams in zFc, set theory wi th the Axiom of Foundat ion, we cannot do it in the intuitively natural way. We would have to re- sort to some artifice, like treating them as funct ions from the natural numbers into A. This would force us to model the operat ion of taking the second coordinate of a s t r eam by mean s of a shift opera t ion on such functions.

Over the past h u n d r e d years, a weal th of techniques for model ing various sorts of p h e n o m e n a has been deve loped within set theory. These build on the repre- sentation of o rde red pairs, relations, and functions.

Non-hierarchical Databases Intuitively, a database is some sort of syntactic s t ructure that represents pur- po r t ed facts about the wor ld . A relational da tabase represents facts of the form that certain objects s tand

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in certain relations. For our purposes , we can skip the syntactic structure, and worry about the facts themselves. Here is an example of a database that represents facts about the Barwise family:

FatherOf MotherOf

Father Child Mother Child

Brad Casey Nancy Brad Dan David Judith David Dan Alisa Judith Alisa

BrotherOf

Brother Sibling

David Alisa

For ease of exposit ion, let us restrict a t tent ion to databases like this one where all the relations involved are binary, and let us suppose that binary relations are represented as usual in set theory, as sets of ordered pairs. Thus, we def ine a database mode l to be any function ~J~ with domain some set Rel of binary relation symbols such that for each such relation symbol R, R ~ is a finite binary relation.

The basic problem that we want to examine is that it is easy to write d o w n databases that seem correct but which, on this definit ion, have no database model at all. Lets look at a ve ry simple example , taken from Barwise [7].

Assume that our set Rel does not contain the relation symbol SizeOf , and let Rel' = Rel U {SizeOf}. We say that a database model ~J)~ for Rel' is correct if the relation SizeOf ~ consists of all pairs (R,n) where R is a relation of ~ and n is the cardinality of R.

Now, intuitively, every database for ReI can be extended in a unique way to a correct database for Rel'. For example, he re is the ex t ens ion of the da tabase r e p r e s e n t e d above, which shows how the general proof should go.

FatherOf MotherOf

Father Child Mother Child

However , our set-theoretic models do not suppor t this intuition. Indeed, we have the following:

Proposition 3. Assume the Axiom of Foundation. Then there are no correct database models.

Proof. Suppose that ~ is correct, let SizeOf be the size relat ion in ~d~, and let n be its size. Then we have n SizeOf SizeOf. But then we can apply part 2 of Propo- sition 1 (with R = S = SizeOf) and get a contradiction.

In the case of s treams, we saw that we could, at some cost in naturalness, get a round the problem by model ing them by functions on the natural numbers , ra ther than as pairs. Here too we could get a round the problem. For example, we could alter the definit ion of correctness to specify that the interpretat ion SizeOf of S izeOf be a function of the symbols in Rel, not of their interpretat ions. This may not be that objectionable, in this case. We include this example mainly because it is easy to state and gives a feel for the sorts of problems that can arise. At the end of this paper we ment ion some more substant ial examples of desirable circularity. It would take a fuller discussion to do justice to any of these.

Rethinking These Negative Results Let's rephrase the results of this section in a way that does not de- p en d on FA. Let us call a s t ream s E St(A) well-founded if it is in the class WF of wel l - founded sets. Similarly, let us call a database model ~JJ~ hierarchical if it belongs to WF. Proposit ions 2 and 3 can be restated as follows:

Proposition 4 (Working in zFc-.) 1. There are no well-founded streams. 2. No hierarchical database is correct.

Put this way, these results are nei ther surprising nor upset t ing. They simply show that we have been at- t e m p t i n g to mode l n o n - w e l l - f o u n d e d p h e n o m e n a with wel l - founded structures. It suggests that if we had a workable replacement for FA, we might be able to prove the sorts of existence results we want. And this turns out to be the case.

Brad Casey Nancy Brad Dan David Judith David Dan Alisa Judith Alisa

BrotherOf SizeOf

Brother Sibling Relation N u m b e r

David Alisa FatherOf 3

MotherOf 3 BrotherOf 1

SizeOf 4

The Antifoundation Axiom

It is, of course, not much use giving up the Axiom of Foundat ion wi thout someth ing with which to replace it. It is not enough to s imply d rop FA and deduce that there might be hypersets . If you are going to prove that the right sorts of hyperse ts do exist, you need to assume some general principle which, together with the o ther axioms, implies this. And we need to do this in a way that is compatible with all the other principles of set theory.

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There is another problem as well. The usual Axiom of Extensionality tells us that sets with the same members are identical. But what about sets a = {a} and b = {b}? Is a necessarily identical with b? The usual formulation of the Axiom of Extensionality (sets with the same members are themselves the same) does not help us answer this question. Without a clear identity criterion for hypersets, they are not of much use. It is not even clear that they deserve the name "set."

So a theory of hypersets must be based on new axioms. How are new axioms of set theory born? For us, axioms are conceived in an attempt to capture the essence of some important phenomenon. We want to abstract from a repeated pattern in the physical world or the world of mathematical ideas and say something true and fundamental . It is not enough that a new axiom not lead to a contradict ion (as sometimes happens in set theory). Even if it were "obviously" in- oculated against the plague of paradox, a new axiom will not survive unless it is based on a solid intuitive understanding of something. Otherwise it would die stillborn for lack of interest. And indeed, the literature is strewn with stillborn theories without an intuitive conception on which to rest.

There have been a number of set-theorists exploring in the wilderness of set theory wi thout FA, pretty much ignored by their peers. These visionaries include M. Boffa, M. Forti, L. Gordeev, and F. Honsell. How- ever, it has only been recently, with the work of Peter Aczel, that the theory has really jelled into a coherent body of work that gives a clear picture of a universe of hypersets. Perhaps the reason that the earlier work wasn't received well was that it wasn't clear that the many axioms proposed were connected to the modeling of circular phenomena or how they related to the deep and important enterprise of finding new conceptions of set.

There are at least two quite distinct metaphors that can inspire a theory of sets. One is that a set is like a box of things, and that forming a set is like putting things in a box. Quite a distinct metaphor is that set formation is the result of forgetting components in favor of structure. Take any sort of structured object, an object with " 'components ," and forget the particular components and the particular "glue" that holds them together. What you are left with is abstract set-theoretic structure. The difference between these metaphors is apparent when we ask: Where do the objects come from in the first place, and what objects are there?

The box metaphor is the motivating intuition that gives rise to the Axiom of Foundation, by means of the cumulative hierarchy. Starting with some set of atoms, you can form sets of these. This gives you a new collection of things to use in forming sets. And so on. Any set formed in this way will be well-founded, since any set of well-founded sets is itself well-founded.

The structure-forgetting metaphor, by contrast, is reflected in the " forge t fu l func tors" of category theory. It is also the intuition behind a conception of set that admits of hypersets. First, it is worth noting that every set b, well-founded or not, can be represented as a structured object, namely as a directed graph. For nodes, take b and all its constituents. For edges, draw an edge x ~ y from node x to node y whenever y E x. This graph is what is known as an accessible pointed graph, or apg. The "point" is the distinguished node b. It is accessible because every node in the graph can be reached by some path starting from the point.

Figure 1 shows four sets and the apg's they deter- mine. In each case, the distinguished node of the apg is the uppermost node. For the moment, consider only Figure l(a) and (b). In Figure l(a), we see the apg de- termined by {0,{0}}. This set is the standard represen- tation of the natural number 2 as a v o n Neumann ordinal. Figure l(b) shows the apg for 3.

(a) { r162 (b) {~ , { r162162

(c) s = {~} (d) ~ again

Figure 1. Four pictures of sets.

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The forgetful func to r me t apho r sugges ts that we should think of sets as arising f rom such directed graphs by the process of abstract ion, forget t ing the nature of the nodes themselves. That is, to each directed graph we should be able to " forge t" the exact nature of the nodes and so assign to each node n a set d(n) so that

d(n) = {d(m)ln ~ m} (5)

Here , and above , "n ~ m" is read "n is a pa ren t o f m . "

Aczel puts it s o m e w h a t different ly , in terms of a picturing metaphor . Sets, on this metaphor , are just the sorts of s t ructures that are pictured by accessible po in ted graphs , whe re the deno ta t ion funct ion between a node and the set it represents mus t satisfy (5). This leads us to the following formulat ion of AFA, the Ant i foundat ion Axiom.

Def in i t ion 3. (Working in ZFC-.) 1. A directed graph (G,---~) consists of a set G of ob-

jects called nodes, and a set ~ of pairs of nodes, these pairs being called edges.

2. An accessible pointed graph (G,---~,p) consists of a directed graph together with a dis t inguished node p w i th the p r o p e r t y tha t e v e r y n o d e can be reached by some finite path from p.

3. A decoration for a directed graph G consists of a se t -va lued func t ion d wi th d o m a i n the set of nodes, satisfying equat ion (5).

4. An apg G = (G,---~,p) pictures the set b if there is a decorat ion d of the graph so that d(p) = b; that is, so that b is the set that decorates the top node.

We have already seen how to picture any set whatsoever with an apg. The quest ion is about the con- verse. Which apg 's picture sets?

It is a theorem of ZFC- that every wel l - founded apg (an apg with no infinite descending chains) pictures a un ique set. If one assumes FA, then one can prove that only wel l - founded apg's can picture sets, since any infinite descending chain in an apg gives rise to an infinite descend ing chain in the m e m b e r s h i p relation. Hence, assuming FA, the apg's that picture sets are the wel l - founded apg's .

Aczel's proposal is that we think of sets as just those objects that can be pictured by any apg whatsoever. This gives rise to the Ant i foundat ion Axiom, AFA.

Every apg pictures a unique set. (AFA)

There are severa l equ iva len t ways to s tate this axiom. We have chosen the one that is closest to the intuit ion that motivates it. Another way to state it is tha t eve ry d i rec ted g raph has a un ique decorat ion. Still ano ther equivalent version is k n o w n as the Solu- t ion Lemma, s ta ted below. It is this vers ion that is mos t usefu l for ac tua l ly p rov ing resu l t s abou t hypersets.

The s imples t h y p e r se t is called fL It satisfies the equat ion f~ = {f~}, and it is pictured in Figure 1(c). It should be no ted that any apg in which every node has a child is a picture of ft.

There are two sides to AFA, existence and uniqueness. Existence gives us man y hypersets . Uniqueness settles the problems about ident i ty of sets. Consider, for example, sets a = {a} and b = {b}. Both of these sets are pictured by the simplest cyclic graph, depicted in Figure 1(c). But according to AFA, there is only one such set, so a = b = fk More generally, AFA tells us that a set is completely de te rmined by any graph that pictures it. This has the effect of making sure that sets are equal wh en ev e r they possibly could be. This is ex- t r emely useful in mathemat ica l mode l ing with AFA since it forces one to be explicit about wha t it is that makes distinct objects distinct.

The Consistency of AFA We saw earlier (see Box "Aren ' t Contradict ions Lurking in Hyperse ts?") that hyperse ts do not give rise to any obvious paradoxes. But h o w do we know for sure that the notion of set u n d e rwr i t i n g AFA is coheren t , so that no contradic- t ions can arise? The reply to this challenge is to show h o w to cons t ruc t a d o ma in wh e re all the axioms of ZFC- together with AFA are true. This would show that ZFC- plus AFA is consistent.

Of course we know by the famous G6del Incomple- teness Theorems that we cannot prove such a result out r ight . No mathemat ica l t heo ry wor th its salt is s t rong e n o u g h to p rove its o w n consis tency. H o w - ever, what we can do is to prove a relative consistency result, by showing that if ZFC- is consistent, so is the result of adding AFA. The proof shows more. It shows h o w to take any domain W of sets satisfying ZFC- and extend it in a canonical way to get a domain V satisfying all of zF c - + AFA. 4

We will not give the proof in detail, bu t we will sketch it. The basic idea is similar to the model ing of the real numbers as equivalence classes of Cauchy se- q u e n c e s of ra t ional n u m b e r s . But here , ins tead of using sequences, we use apg's, wi th their own equiva- l ence re la t ion . A bisimulation b e t w e e n two a p g ' s (G1,---~l,Pl) and (G2,---~2,P2 ~ consists of a relation R C G 1 x G 2 satisfying the fol lowing conditions:

1. plRp2 2. if xRy then

�9 for every x 0 ~--1 x there is a Yo ~--2 Y such that

xoRyo

�9 for every Yo <--2 Y there is a x 0 *--1 x such that xoRyo

We say that two apg's are bisimilar, and write G 1 - G 2,

4 This observa t ion s h o w s w h y it is tha t AFA does not yield an y n e w t h e o r e m s tha t speak only about we l l - founded sets.


if there is a bisimulation relation be tween them. As an example, the g raphs in Figure 1(c) and l (d) are bisimilar: the bisimulation relates the unique point in (c) to every point in (d).

The following observat ions are each pre t ty easy to check, using just ZFC-:

1. The relation G 1 - G 2 is an equivalence relation on apg's.

2. If G 1 - G 2, then G 1 and G 2 picture the same sets, if any.

The basic idea of the construction of V from W is to take equivalence classes [G] of apg's G E W, unde r the relation - , and use them to model sets. The member- ship relation is def ined on them in the natural way:

[H] E [G] iff there is a node n that is a child of the top node in G so that H - G n, where G n is the apg got ten from G by snipping it off just above n. The effect of this construct ion is just to identify any two apg's that should be identified in that they will be picturing the same sets. Of course one m u s t check that all the axioms hold in the result ing structure. To embed W isomorphical ly into V we go from any set b to the canonical picture G b of it, described earlier, and on to

[Gb]. This construct ion does more than show consistency

of AFA. It also shows that, in a certain sense, the AFA sets are " f ina l" in the ca tegory of universes of set theory . But we will not go into the cons t ruc t ion in


greater detail here , instead referr ing the reader to Aczel [2] for deta i ls and historical r e m a r k s on the proof.

The Solut ion Lemma We ment ioned above that there are a number of different ways to state AFA. One of the most useful ones in the applications is the Solution Lemma, a vers ion of which we state here. It lets us solve systems of equat ions such as

x = {x ,y}

y = {2,3,y,z} z = {x,y,f~}

The intuitive content of the Solution Lemma is that any such system of equat ions is solvable in the universe of hypersets . In stating the result, we use "9X" for the power set of X.

Lemma 5 (The Solu t ion Lemma) Assume AFA, and let X be any set, let b be a function from X to VX, and let c be a function with domain X. Then there is a unique function d with domain X such that for all x E X,

d(x) = {d(y) : y E b(x)} U c(x).

In fact, this Solution Lemma is equivalent to AFA; if we assume only ZFC- and the Solution Lemma, we can deduce AFA. (This is a good exercise.)

As a s imple example , he re is h o w the Solut ion Lemma allows us to solve the system above. Let X be any three-e lement set {x,y,z}. Let b(x) = {x,y}, b(y) = {y,z}, and let b(z) = {x,y}. Let c(x) = 0 , c(y) = {2,3}, and c(z) = {12} = 12. Then the Solution Lemma gives us sets d(x), d(y) and d(z) that solve the system. It also impl ies tha t the so lu t ions of sys t ems like this are always unique.

Let us look again at the set St(A) of s treams on a set A. Under FA, we saw that this set was empty . Using the Solu t ion L e m m a , we can s h o w tha t eve ry sequence a I, a 2, a 3 . . . . of a toms gives rise to a s tream ( a l , ( a 2 , ( a 3 . . . ) ) ) . N a m e l y , c o n s i d e r the fo l l o win g system of equations:

x 0 = (al,xl) X 1 = ( a 2 , x 2 )

x2 = (a3,x3)

(We leave it to the reader to convert this intuitive de- scription into the form demanded by our version of the Solution Lemma.) If we take the solution to this set of equations, then obviously the value assigned to the u n k n o w n x 0 is the desired stream.

Next, recall our example concerning databases. We assume a finite set Rel of binary relations together with an ass ignment R ~ R ~ of finite binary relations to the members of Rel. Let S izeOf be a new relation symbol. Consider the equat ion

x = {(R~a, I R ~ I ) : R E Rel} U {(x, IRel I + 1)},

where the bars I I denote the cardinality of the set ins ide . To e x t e n d ~ to a cor rec t mo d e l , we set SizeOf ~ to be the solution to this equation.

We should point out that AFA is not magic: not every "equa t ion" involving sets has a solution. You must be able to cast the equat ion in the form given by the Solu- tion Lemma. For example, it is a theorem of Cantor, p r o v e d us ing only ZFC-, that there is no set which contains its own powerset . It follows that we cannot solve the equat ion x = I0x, no mat ter what axioms we add to ZFC-. This observat ion is a parallel to the fact that some cont inued fractions do not converge to real numbers .

L i k e w i s e , a n d s o m e w h a t d i s a p p o i n t i n g l y , we cannot use AFA to build nontrivial sets or topological spaces equal to their o w n funct ion spaces. True, 12 is a solution to X = X x. However , this is the only solution AFA gives us. Indeed, we have the following stronger observat ion, due to Aczel.

Proposition 6. Assume AFA. If X C X X, then either X = Q o r X = 12.

P roof . Let X C X x. A s s u m e X # •. Cons ide r the system

x = {(y,y(x)) : y E X},

where we have an equat ion for each x E X. The Solu- tion Lemma implies that this sys tem has a unique solution. But the constant funct ion fl is a solution, so the only solut ion. Thus , eve ry y E X is 12 so X = {fl} =12. m

Hypersets as Limits of Well-founded Sets

We began this paper with an analogy, be tween con- t i n u ed f rac t ions and n o n - w e l l - f o u n d e d sets. This analogy suggests that we should be able to look at a set like

x = {1,{1,{1,{1 . . . . }}}} (6)

as some sort of limit of the wel l - founded sets:

{1}, {1,{1}}, {1,{1,{1}}}, {1,{1,{1,{1}}}} . . . . (7)

The in tu i t ion is that the sets fu r the r out in this sequence are harder and harder to distinguish from the given set x.

A n u m b e r of people have pu r sued this line of inves- tigation. Every set x is pictured by some apg G. By consider ing a closely related apg, we may assume that G has no cycles ( though it might have infinite paths). Thus G is a tree. The basic idea is that this tree should be the "l imit" of its finite (hence well-founded) sub- trees. The set x we started with should be the limit of the fami ly of sets p i c tu red by these w e l l - f o u n d e d trees. So in order to consider hyperse ts as limits f rom WF requires the right not ion of convergence of trees.


Under the most natural notion of "limit" one can ob- tain all of the hypersets in this way. However, limits in this sense are not unique, and taking the limits of all of the trees in the well-founded universe gives a collection of objects that is larger than the universe of hypersets.

An offshoot of this work has been the introduction of several different types of partial sets, objects which are set-like but somehow incompletely specified. Sev- eral different notions of partial set have emerged, and these are related to different intuitions about the nature of possible memberships. As with work on AFA, this work has led to interesting mathematics.

Epilogue: Where Does the Stuff of Mathematics Come From?

For thousands of years philosophers have been won- dering about the source and home of mathematical objects. It is easy to make the question seem puzzling indeed.

As should be clear from the above, we are mathematical realists: we take mathematics to be about real stuff. Numbers and other patterns are real. So are sets. The challenge is to talk sense about the stuff of mathematics while steering a course between the Scylla of Naive Platonism and the Charybdis of formalism. For the Naive Platonist is committed to a world of mathematical objects independent of humankind, discon- nected from the world of experience. As a result, the applicability of mathematics to the world of matter and man becomes rather puzzling, a matter of "partic- ipating in ideal forms." The formalist, on the other hand, seems committed to a world without mathematical objects at all, only meaningless symbols. Here the applicability of mathematics is even more of a mys- tery.

To us, one of the exciting aspects of recent development in hypersets and their theory is that it serves as a case study in this age-old debate. It seems that neither Platonism nor formalism gives an adequate insight to the history as it is actually unfolding. Indeed, we think this unfolding shows how we can be realists without being Platonists.

Where did hypersets come from? A mixture of places. In some ways, they arise out of the desire to use a certain set of tools in modeling significant real- world phenomena in the areas of computer science, cognitive science, and philosophy. It is of course fundamental to science that mathematical models of real- world phenomena can enrich our understanding of the phenomena. But more important for mathematics is the flip side. The need to create such models sometimes enriches mathematics by bringing to light new abstract patterns that need to find a home in the universe of mathematical objects. And this is certainly a major part of what is going on here.

A different and more subtle source of mathematical ideas is the whole array of assumptions, metaphor, taste, and analogy that mathemat ic ians (we are people, after all, in spite of what the public thinks) bring to the experience of mathematics. It is easy to neglect this aspect or to pretend that it isn't there, but when we do this we lose much of the motivation for and human essence of mathematics. In our case study, this neglect can obscure the nature of conventional set theory. But when push comes to shove, we must admit that the cumulative notion of a set is deeply rooted in a metaphor, one that can guide us but is simply not rich enough to settle all questions of set theory. And it is not the only metaphor around. Hy- persets are based on an alternative metaphor, one that gives rise to an alternative conception of set.

The question of taste comes up here, too. Some set- theorists find the cumulative picture ultimately more satisfying. Others find the elegance and versatility of the Solution Lemma irresistible. (And of course category-theorists find set theory more or less beside the point.) Choosing between them is both a matter of taste and a matter of expedience.

Finally, there is the matter of analogy. Analogy plays a large role in the development of mathematics. We know how things go in one domain and try to make things go analogously in another. In the case of hyperse ts , a mot ivat ing analogy is the different number systems: rationals, reals, and complexes. Each is an extension of the ones that come before. Certain nice properties are lost in the extension. The complex numbers have no nice ordering, for example. On the other hand, each extension allows us a space in which to solve equations that were not solvable before, and so to solve certain real-world problems that were not solvable before. Similarly, the hyperreals of infinites- imal analysis constitute an extension of the reals where certain systems of equations and inequalities that don't have "real" solutions do have solut ions:

What the extensions of the number systems and set theories have in common is that in making the extension, we lose none of the original objects, but we gain solut ions to interest ing sys tems of equations. In making the extension in the set-theoretic case, we lose the nice hierarchical picture, 6 but instead we have the view that sets are exactly the objects pictured by apg's. In this way, we gain sets that allow us to solve new equations, and so model new phenomena. And just as the reals can be modeled as equivalence classes of Cauchy sequences of rationals, so too the hypersets

5 See the p resen ta t ion in Keisler [14], for example . 6 Actual ly, there is a way to have aspec t s of both views. By the Axiom of Choice, every apg is i somorph ic to an apg w h o s e n o d es are we l l - founded ( indeed ordinals). So one can have the view that the apg ' s are built in s tages, and only after all of t h e m have been created, the sets appear as decorat ions .


can be modeled as equivalence classes of apg's. In both cases, these constructions convince us that there is nothing really ontologically suspicious going on.

The analogy even extends into the sociological di- mension. There were times when V'2 and i were viewed with great suspicion, even rancor. Nowadays we don't think twice about them, except in teaching wary undergraduates. More recently, the discovery of the infinitesimals led to some name-calling by otherwise sensible people. Similarly, sets like f~ and the stream (0,~1,~2 . . . . ~ were until recently viewed as non-entities. Nowadays we are coming to understand them for what they are, perfectly good sets under a richer conception of set.

The two of us feel that as more and more applications of hypersets are found, they are going to find their way into everyday mathematics. In twenty years, introductory books on set theory will have to have a chapter on them. They are no more irrational than the irrational numbers and no more imaginary than the complex numbers. They are darned useful. And better still, the universe of hypersets is a charming one in which to pursue some lovely mathematics.

Some Remarks on the Literature

The theory of non-well-founded sets is studied in detail in Aczel's book [2]. This book also contains a detailed bibliography and a discussion of their history. A more elementary introduction can be found in Chapter 3 of [8]. Some more recent theory of hypersets can be found in Chapter 12 of [5], and theories of approximations to hypersets can be found in [1] and [17].

For a general discussion of the two metaphors for sets, and the need for non-well-founded sets for modeling circular phenomena, see Chapter 8 of [5]. At the present time, we are only just beginning to see applications of hypersets . We expect to see many more in the years to come. Here are some references:

�9 To the theory of concurrency: Chapter 7 of [2]. This is a mathematical model of processes that communicate with each other in the course of a computation, inspired by large parallel computers composed of many small parts. The notion of a bisimulation can be found in this area, first in the work of Park, though it had antecedents in the set-theoretic and model-theoretic literature.

�9 To database theory: [6] �9 To the semantical paradoxes: [8]. The semantic

paradoxes include the Liar Paradox ("This sen- tence is false."), and also Richard's Paradox ("the first non-definable number"). Like the set-theoretic paradoxes, the semantic paradoxes seem to involve circularity. The standard modern treat- ment is to impose some sort of hierarchy on the world, and thereby claim that the circularity is only apparent. In contrast, [8] advances the view

that the circularity is genuine, and that it can be understood coherently once we have mathematical tools like AFA for modeling it.

�9 To the theory of common knowledge: Chapter 9 of [5]. Suppose two people are making bids at a public auction. Their behavior is different than it would be if they made their bids privately, because each is aware that the other is aware of the entire situation. The analysis of common knowledge has been controversial and problematic in the philosophy of language. It also comes up in theoretical economics.

�9 To theoretical and computational linguistics: Bar- wise [5] and Rounds [18].

References

1. Samson Abramsky, Topological aspects of non-well- founded sets, to appear.

2. Peter Aczel, Non-well-founded Sets, CSLI Lecture Notes, Chicago: University of Chicago Press (1988).

3. Jon Barwise, Admissible Sets and Structures, New York: Springer-Verlag (1975).

4. Jon Barwise (ed.), Handbook of Mathematical Logic, Am- sterdam: North-Holland (1977).

5. Jon Barwise, The Situation in Logic, CSLI Lecture Notes, Chicago: University of Chicago Press (1989).

6. Jon Barwise, review of [12], Journal of Symbolic Logic 54 (June, 1989).

7. Jon Barwise, Consistency and logical consequence, Truth or Consequences, Dunn and Gupta, eds., Dordrecht: Klu- wer Academic Publishers (1990).

8. Jon Barwise and John Etchemendy, The Liar: An Essay on Truth and Circularity, New York: Oxford University Press (1987).

9. Paul Bernays, Axiomatic Set Theory, Amsterdam: North- Holland (1958).

10. Paul Cohen, Set Theory and The Continuum Hypothesis, New York: W. A. Benjamin (1966).

11. Abraham A. Fraenke|, Yehoshua Bar-Hillel, and Azriel Levy, Foundations of Set Theory, Amsterdam: North-Hol- land (1973).

12. Barry Jacobs, Applied Database Logic I, Englewood Cliffs, NJ: Prentice-Hall (1985).

13. Mark Johnson, Attribute-Value Logic and the Theory of Grammar, CSLI Lecture Notes, Chicago: University of Chicago Press (1988).

14. H. Jerome Keisler, Elementary Calculus, Boston: Prindle, Weber and Schmidt (1976).

15. Kenneth Kunen, Set Theory: An Introduction to Indepen- dence Proofs, Amsterdam: North-Holland (1980).

16. James D. McCawley, Everything Linguists Want to Know about Logic but are Afraid to Ask, Chicago: University of Chicago Press (1981).

17. Michael W. Mislove, Lawrence S. Moss and Frank J. Oles, Non-welMounded sets modeled as ideal fixed points, to appear in Information and Control

18. William C. Rounds, Complex objects and morphisms I. A set-theoretic semantics, Situation Theory and its Applica- tions II, to appear.

19. Joseph Shoenfield, Axioms of set theory, in [4], 322-344.

Department of Mathematics Indiana University Bloomington, IN 47405 USA

THE MATHEMATICAL INTELLIGENCER VOL, 13, NO. 4, 1991 41

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