Math. Log. Quart. 61, No. 4–5, 250–262 (2015) / DOI 10.1002/malq.201400062 Digraph parameters and finite set arithmetic Laurence Kirby∗ Department of Mathematics, Baruch College, 1 Bernard Baruch Way, New York NY 10010, United States of America Received 25 June 2014, accepted 4 November 2014 Published online 20 August 2015 Each hereditarily finite set is associated with a unique extensional acyclic digraph. Three parameters, indicating the size or richness of a set, are associated with its digraph: the cardinality of the set, and the numbers of nodes and of edges in the digraph. We study the effects on these parameters of the operations of the ordinal arithmetic of sets. C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Hereditarily finite sets are endowed with parameters (finite ordinals) that arise naturally when viewing the sets as digraphs or directed graphs: namely, the numbers of nodes and edges in the digraph associated with a set. This article will examine how these parameters interact with the ordinal arithmetic of sets. This article is also a case study in the application of a principle suggested by a finitist intuition to a mathematical result.1 There will be two preliminary sections. § 2 will lay out some basics of the digraph view of the hereditarily finite universe, and § 3 will summarize what will be needed from the arithmetic of sets. § 4 will go into detail about specific finitary constructions of sets which are used in § 5, § 6 and § 7 to find out how the digraph parameters fare under the arithmetical operations. It will be shown how the digraph parameters of a product a · b of sets depend only upon the parameters of a and b (Lemma 5.2 and Theorems 5.5 & 7.4), but that this fails for exponentiation ab (Examples 5.10 & 5.11). For the exponential case, however, we give recurrence relations for the parameters of ab (Theorems 5.9 & 7.7). § 6 is reserved for discussion of the very different properties of exponentiation in the special case where the empty set 0 is an element of the base. 2 Digraph parameters of finite sets The most fundamental operation of finite set theory is adjunction by which a set y is adjoined to another set x to form the set x ∪{y} which we shall denote by x; y or, when disambiguating, [x; y]. In [6], we discussed how this operation can found finitary (more specifically, primitive recursive) set theory, and there will be more about it in §4. Each hereditarily finite set x is uniquely specified by the digraph G(x) of the membership relation restricted to TC(x); x where TC(x) is the transitive closure of x. More precisely, the nodes or vertices of G(x) are the elements of TC(x); x and there is an edge or arc from u to v just when v ∈ u. Cf. [7], and the first pages of [2]. A digraph is extensional iff distinct nodes have distinct out-neighbourhoods (sets of children). Any non-trivial finite acyclic digraph has a sink, and the sink is unique when the graph is extensional. In fact, any finite extensional acyclic digraph with a single source is isomorphic to G(x) for a unique hereditarily finite set x obtained as the Mostowski collapse of the partial order < induced by the digraph. That partial order is the transitive closure of the membership relation, i.e., x < y iff x ∈ TC(y). ∗ Corresponding author: e-mail: [email protected] 1 But it is not intended to present a definition or defence of finitism. C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim Math. Log. Quart. 61, No. 4–5 (2015) / www.mlq-journal.org 251 A related digraph associated with x is the restriction of G(x) to TC(x), which we shall denote by G−(x); this was studied by Milanic,ˇ Policriti, Tomescu, and others [9, 11, 14]. They restrict their attention to graphs of transitive sets. In the framework adopted here, G−(x) is not unique to x, but is shared by all sets that have the same transitive closure as x. How big is a finite set x? The immediate answer is its cardinality |x|, which is equal to the outdegree of the source of G(x). But a set might of course have, for example, elements which are few but large. A truer reflection of the resources needed to construct x is given by ν(x) :=|TC(x)| which is equal to the number of nodes of G(x) minus one, and equal to the number of nodes of G−(x). But there is another parameter suggested2 by the graph description of sets that better captures the richness or fullness of a set, namely the number of edges of G(x), denoted by η(x). Because η(x) is the sum of the outdegrees of nodes of G(x), we note that: η = | | Lemma 2.1 For any finite set x, we have (x) y≤x y . The following lemma relates the graph parameters of TC(x) with those of x. Lemma 2.2 For any finite set x, we have |TC(x)|=ν(TC(x)) = ν(x) and η(TC(x)) = η(x) + ν(x) −|x|. P r o o f . The graph G(TC(x)) is obtained from G(x) by adding ν(x) −|x| new edges from the source to each element of TC(x)\x. We conclude this section by noting that a familiar ordinal parameter of a hereditarily finite set x, its rank (x) in the cumulative hierarchy, is expressed in digraph terms as the length of the longest path through G(x). 3 Arithmetics of sets How to represent the natural numbers in the universe of hereditarily finite sets? The classical method uses the von Neumann ordinals, obtained from the empty set 0 via the successor function α → α; α. Ordinal arithmetic can then be extended naturally to all hereditarily finite sets, defining addition by x + y = x ∪{x + r | r ∈ y}. (vN) This definition is due to Tarski [13]3 but we call this the von Neumann arithmetic of sets [7] to distinguish it from a different arithmetic of sets which arises from another way to represent the natural numbers as sets: the Zermelo ordinals, obtained via the Zermelo successor function α →{α}=0; α (Zermelo [15]). In the Zermelo arithmetic of sets, addition is extended to all sets by defining4 x + 0 = x, x + y ={x + r | r ∈ y} if y = 0. (Z) In [7] and [8], we argued that this is more parsimonious and perspicuous, both as a representation of the natural numbers and as an arithmetic of finite sets.5 In both arithmetics, multiplication and exponentiation are obtained from the respective additions by x · y ={x · q + r | q ∈ y ∧ r ∈ x}, x0 = 1, x y ={x p · q + r | p ∈ y ∧ q ∈ x ∧ r ∈ x p} if y = 0. 2 Cf., e.g., some of the sets illustrated in [8], such as 6 and 6z . 3 Cf. [4] for historical remarks. 4 In [7] and [8], we distinguished the Zermelo arithmetic from the von Neumann by using the subscript z. In the present paper, to avoid a proliferation of subscripts, it will be left to the context to make clear which arithmetic is intended. 5 Especially in the infinite case the von Neumann arithmetic gives a simpler and cleaner account of well-ordered order types and cardinalities so its hegemony is well justified [7], but in the finite case this advantage is perhaps outweighed by its greater complication and computational requirements. These issues do not concern Benacerraf [3] but to a finitist, or one implementing the arithmetic on a machine, they become crucial. In any case we are not talking here of a metaphysical identification of numbers with certain sets, merely a formal representation for which it is appropriate to ask its efficiency. www.mlq-journal.org C 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 252 L. Kirby: Digraph parameters and finite set arithmetic (These definitions are due to Scott.) In each case these operations, when restricted to the approprate ordinals, agree with the standard natural number operations. In the presence of exponentiation, another advantage of the Zermelo arithmetic emerges: unlike the von Neumann arithmetic, it obeys the “high school algebra” laws of exponentiation [8]. The remainder of this section is devoted to statements of some basic properties that are true in both the von Neumann and Zermelo arithmetics, as shown in [7] and [8]: Lemma 3.1 For any sets x, y, and z, we have x + y ∈ x + z if and only if y ∈ z and x + y < x + z if only only if y < z. Proposition 3.2 For sets x, a, and b, we have that x < a · b if and only if there are q < b and r < a such that x = a · q + r. Furthermore such q, r are unique. Corollary 3.3 For any sets a and b, we have that TC(a · b) = TC(a · TC(b)). Proposition 3.4 (Kirby, [8]) Suppose a > 1 and b = 0. Then in any additive arithmetic of sets, x < ab ←→ ∃ p < b ∃q < a ∃r < a p (x = a p · q + r). Corollary 3.5 For any sets a and b, we have TC(ab) = TC(aTC(b) ). The uniqueness of the representation in Proposition 3.4 requires that 0 is not an element of the base: Proposition 3.6 (Kirby, [8]) If 0 ∈ a and a = 0, then for any p, q, r, s, t, u, 0 < q < a ∧ 0 < t < a ∧ r < a p ∧ u < as ∧ a p · q + r = as · t + u −→ p = s ∧ q = t ∧ r = u.