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A Fine Structure for the Hereditarily Finite Sets

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A Fine Structure for the Hereditarily Finite Sets A fine structure for the hereditarily finite sets Laurence Kirby Figure 1: A5. 1 1 Introduction. What does set theory tell us about the finite sets? This may seem an odd question, because the explication of the infinite is the raison d’ˆetre of set theory. That’s how it originated in Cantor’s work. The universalist, or reductionist, claim of set theory — its claim to provide a foundation for all of mathematics — came later. Nevertheless I propose to take seriously the picture that set theory provides of the (or a) universe and apply it to the finite sets. In any case, you cannot reach the infinite without building it upon the finite. And perhaps the finite can tell us something about the infinite? After all, the finite is all we have to go by, at least in a communicable form. The set-theoretic view of the universe of sets has different aspects that apply to the hereditarily finite sets: 1. The universalist claim, inasmuch as it presumably says that the hereditar- ily finite sets provide sufficient means to express all of finite mathematics. 2. In particular, the subsuming of arithmetic within finite set theory by the identification of the natural numbers with the finite (von Neumann) or- dinals. Although this is, in theory, not the only representation one could choose, it is hegemonic because it is the most practical and graceful. 3. The cumulative hierarchy which starts with the empty set and generates all sets by iterating the power set operator. I shall take for granted the first two items, as well as the general idea of gener- ating all sets from the empty set, but propose a different generating principle. In infinitary set theory, the cumulative hierarchy and its many variants (from G¨odelonwards) are basic to the picture of the set-theoretic universe. The power set operator ensures explosive growth in the cumulative hierarchy and is really a crude tool. Indeed, many of the related hierarchies, such as the constructible hierarchy, can be viewed as ways of putting brakes on the growth in order to see some fine structure. At the finite level, the power set is also fast and crude. I shall consider a natural candidate for a slower growing hierarchy whose gen- erating principle is the binary operation of adduction hx, yi7→x ∪{y} which adds a single new element to an already existing set. I shall suggest some things that are naturally expressed in this new way of describing sets. Adduction, as a generator, will be seen to be intrinsically finitist in character. An interesting contrast between arithmetic and set theory is that the former is based upon functions (successor, addition, multiplication) whereas the latter, in its usual formalization, is based upon a relation (the membership relation). By making the notion of adduction fundamental, we can base set theory upon a function which is a generalization of the successor function of arithmetic. This brings out in a more direct way the parallels between finite set theory and arithmetic. 2 In fact, the fundamental generator for arithmetic, the successor function, will be seen as the diagonalization of the adduction operator. §2 introduces the appropriate formal language for the purpose, with a binary function symbol to represent adduction instead of the usual binary relation symbol for membership, and defines and establishes some basic properties of the adductive hierarchy. §3, which can be omitted without prejudice to the succeeding sections, de- fines a canonical ordering on the hereditarily finite sets and bases upon it some visualizations of the small classes in the adductive hierarchy. It then calculates the sizes of small classes and subclasses. In §4, building on work of Flavio Previale, I define a first order version of an induction principle based on adduction which is due to Alfred Tarski and Steven Givant. I then define in our language Peano set theory PS which is equivalent to both Peano arithmetic and to ZF with the axiom of infinity replaced by its negation. §5 is concerned with some more philosophical arguments for basing consider- ation of finiteness upon the idea of adduction, bearing in mind Hilbert’s concept of finitism as applying to systems of signs. Any actual communication consists of a finite series of adductions, of bringing things to your attention. I also argue for the provisional nature of the empty set. And I am led, via W. W. Tait’s thesis that finitist reasoning is essentially primitive recursive reasoning, to the primitive recursive set functions. In §6 I survey the primitive recursive set functions, establish their basic properties and give some primitive recursive definitions of basic set-theoretic operators. §7 introduces primitive recursive generalizations to sets of the arithmetical operations of addition and multiplication. This suggests an arithmetic of sets.I shall also provide evidence that the adduction operator, as generating principle, is intrinsically more finitist than the power set operator. Once we have started tinkering with the language of set theory, we can go further and question the role of the constant symbol for the empty set. Drop- ping this allows us to construct an extension of a model of finite set theory in which the empty set of the original model is no longer empty in the extension. In §8 I shall show that every model M of PS has such an extension which is isomorphic to M. The proof uses addition of sets. I am grateful to Arthur Apter and Akihiro Kanamori for their helpful com- ments. 2 Basics. We work in a language L(0 ;) for set theory which differs from the usual language L(0 ∈). Instead of the membership relation ∈ we employ as basic operation ad- duction, a binary operation denoted by [x; y]. So L(0 ;) has a binary function symbol for adduction as well as a constant symbol 0. The intended interpreta- 3 tion of [a; p]isa ∪{p}. Frequently we shall drop the brackets, especially the outermost, where no ambiguity results. Notation. a; p, q =[a; p]; q and thus a; p1,...,pn =[...[[a; p1]; p2]; ...]; pn. This is a notational abbreviation for convenience, and not in the formal lan- guage. The symbol 0 will be used in two ways: as the constant symbol in the formal language, and for the empty set which is the standard interpretation of this symbol. Likewise the semi-colon does duty both as the function symbol and as its interpretation in sets. Thus 0; p1,p2,...,pn = {p1,p2,...,pn}. Also p ∈ a is defined to mean a; p = a. So in fact L(0 ∈) and L(0 ;) are equivalent: each of the symbols is definable in terms of the other in a common extension L(0 ∈ ;). As a background theory, assume a familiar set theory: ZF will do. As W. W. Tait [12] emphasizes, when we describe a finitist operator in general terms, we are doing so from an infinitist standpoint. Following the usual development of set theory, the finite (von Neumann) ordinals are defined by iterating the operator n 7→ [n; n] and we temporarily and informally use the notation n + 1 for [n; n]. The successor operation for numbers has become the diagonalization of the adduction operator. Vω is the set of hereditarily finite sets, defined as usual by means of the cumulative hierarchy V0 =0,Vn+1 = PVn,Vω = [{Vn | n ∈ ω}, where Px is the power set of x. In the iterative conception of set theory, the hierarchy of the Vn is the way of building up the hereditarily finite sets starting from 0. But if we examine this process of generation in detail, especially taking into account considerations of feasibility, it becomes evident that it does not conform with Ockham’s razor. The number 5, for example, doesn’t occur until V6, by which time we have also obtained 265536 − 1 other sets, including many complicated structures. This number is far more than the number of protons in the universe of physics, indeed in 265270 such universes. This seems a grossly inefficient way to generate sets, in as much as long before we get to small, simple, useful sets such as 6 we have generated more sets than we can ever possibly use or name in practice. Of course this is a version of a familiar complaint about exponentiation which, especially along with iteration of functions, results in unfeasible growth too soon and too fast. In the language L(0 ;) we can define a natural, slower-growing hierarchy for the hereditarily finite sets, the adductive hierarchy: A0 = [0; 0] = {0} =1,An+1 = An ∪{[x; y] | x, y ∈ An}. (The subscripts n and the names An are here in the metalanguage.) 4 It is intuitively clear that this hierarchy generates all hereditarily finite sets: Vω = S{An | n ∈ ω}. And it conforms well with the following intuitive picture of the hereditarily finite sets given by Flavio Previale ([9], p. 214): Hered. finite sets are exactly what can be obtained by starting from the empty set and then iterating the operation of adding, to a set already obtained (by the procedure itself), an element taken in its turn among the sets already obtained. Note that 5 ∈ A5, and we shall see that the cardinality of A5 is 11680, so we have a more reasonable amount of baggage along with the number 5. The situation has not improved sufficiently, however, to be called feasible. A6 has become unwieldy at 135717904 elements. How fast does the cardinality of An grow? Clearly an upper bound for 2 |An+1| is |An| + |An| , so that the growth of the size of An, as a function of n n, is double-exponential (i.
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