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.

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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.

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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.

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(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.

4 Building up the finite sets

Previale expresses well [12] an intuition that the hereditarily 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”. This intuition underlies an induction schema for finite set theory [12], [6] and also the adjunctive hierarchy [5], but here it motivates the two definitions that follow. We propose to modify Previale’s construction procedure. Suppose, at some stage, we have a pool of already- obtained sets. We are allowed to adjoin to the pool any collection chosen from among the already-obtained sets. So we can gather together a new set, or deem a new collection to be a set, but all of its elements have to have been already there. Definition 4.1 Let x and y be sets. The aura of x is aura(x) :={y | y < x ∧ y ⊆ TC(x)}.

1. By Cantor’s theorem, every set has a non-empty aura. In fact x ∈ aura(x). 2. If x is transitive and y ∈ aura(x) then x; y is transitive.

If x is the set whose elements are all the sets we have obtained so far by our procedure, then x is transitive and aura(x) is the set of options for the next set we can add. But in a sense by naming x in the preceding sentence we have pre-empted the procedure, or stepped outside it, by already introducing this new set x, which cannot have been obtained previously since it cannot be an element of itself. Viewed from within the procedure, the sets in the aura of x, including x itself, can only be regarded as potential sets or “proper classes” unless and until the procedure is extended by adjoining them.

Definition 4.2 A construction sequence for a set x is a finite sequence x0,...xn of distinct sets such that

1. for each i ≤ n,wehavexi ⊆{x j | j < i}, 2. for each i < n,wehavexi < xn, 3. and xn = x.

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The second clause is there to ensure that no sets are constructed along the way which are not used in the end result xn. Of course, x0 = 0 and x1 = 1 ={0}. Any hereditarily finite set has a construction sequence (and in general has many). In fact in infinitary set theory the hereditarily finite sets are characterized by this property. This proposition provides uniform ways of getting construction sequences: Proposition 4.3 Let <∗ be any total ordering of the hereditarily finite sets such that y < x → y <∗ x. Then for any hereditarily finite x the elements of TC(x); x ordered by <∗ form a construction sequence for x. The ordering induced by Ackermann’s [1] bijection between the hereditarily finite sets and ω is an example of such an ordering. This property of the Ackermann ordering was exploited by Peddicord [10] to enumerate the transitive sets x such that |x|=n. Another ordering with this property is in [5]. In fact we can enumerate sets by rank (in any suitable hierarchy), ordering the sets within each rank lexicographically (say) by the first terms naming them in the language with 0 and adjunction.

1. In any construction sequence x0,...,xn for x, {x0,...,xn−1}=TC(x). In particular, n = ν(x). 2. For any construction sequence and any i < n,theset{x0,...,xi } is transitive and

xi+1 ∈ aura({x0,...,xi }).

3. x is transitive iff there is a finite sequence x0,...,xn−1 of distinct sets such that for each i ≤ n, xi ⊆{x j | j < i}, and {xi | i < n}=x.

In the digraph picture of the hereditarily finite universe, a construction sequence for x builds G(x) from the bottom up, against the arrows, one node at a time. The cumulative view of the set theoretic universe is retained, but within each hereditarily finite set. Restricting adjunctions b; p to the case where p ∈ aura(b) gives us control over the digraph parameters of the result: Lemma 4.4 Let p ∈ aura(b) and b = 0. Then |b; p|=|b|+1, ν(b; p) = ν(b) + 1, and η(b; p) = η(b) +|p|+1. Proof. When p ∈ aura(b) and b = 0 the graph of b; p is obtained from the graph of b by adding a single new node p with an edge from the source to p and |p| edges from p to its elements which are all aready in the graph of b.  The remainder of this section is a digression to consider two interesting questions not discussed in the sequel. The first question is: “How many sets x with ν(x) = n are there for a given n?” The answer is an easy consequence of a result of Policriti and Tomescu.

Theorem 4.5 (Policriti-Tomescu, [11]) Let eˆn,s be the number of acyclic extensional digraphs with n nodes and s sources. Then     n−s + 1 n−s−k s k eˆ + , + = 2 eˆ , + − neˆ , , n 1 s 1 s + 1 k n s k n s k=0 with eˆ1,1 = 1 and eˆn,0 = 0. P r o o f . We summarize a proof here since it fits well in our framework. An extensional acyclic graph with − n nodes is isomorphic to G ({x0,...,xn−1}) for a unique transitive set {x0,...,xn−1}.Ifithass + k sources, then we can adjoin a set xn ∈ aura({x0,...,xn− 1}) by making xn have as elements any k of the s + k sources of − { ,..., } s+k n−s−k G ( x0 xn−1 ), which can be done in k ways, along with any of the 2 sets of non-sources, except that when k = 0 the set of elements of xn must differ from any of the n sets x0,...,xn−1 in order to preserve

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The number of sets x such that ν(x) = n is eˆn+1,1. Some values for small n are given in [11]. The second question is: “How many sets x with η(x) = n are there for a given n?” We don’t know the answer to this beyond simple counting procedures which yield 10 sets for n = 5, 28 for n = 6, and 88 for n = 7. Tomescu [14] answers the related question: how many extensional acyclic digraphs with n edges are there? Or, if you like, how many graphs are there of form G−(x) for some x?

5 How do the graph parameters comport with the Zermelo arithmetical operations?

Throughout this section we work in the Zermelo arithmetic, indicating where this assumption is necessary by the annotation (Z). The behaviour of the graph parameters under addition is easily seen from the digraph representation of the Zermelo sum as a concatenation of graphs [7]: Lemma 5.1 (Z) For any hereditarily finite sets a and b, we have that |a + b|=|b|, ν(a + b) = ν(a) + ν(b), and η(a + b) = η(a) + η(b). Corresponding properties for the cardinality and ν of the product are corollaries of Proposition 3.2:6 Lemma 5.2 For any hereditarily finite sets a and b, we have that |a · b|=|a||b| and ν(a · b) = ν(a) ν(b). For later use, we remark that Lemma 5.2 holds also in the von Neumann arithmetic, as it only uses common properties of the two arithmetics in § 3. The next lemma is also true in both arithmetics: Lemma 5.3 If p ∈ aura(b), then a + p ∈ aura(a + b). If, further, a = 0, then a · p ∈ aura(a · b). If, further, 0 ∈ a and a > 1, then a p ∈ aura(ab). Proof. Let p ∈ aura(b). Since p < b, it follows from Lemma 3.1 that a + p < a + b. We also need to show that if x ∈ a + p then x < a + b. We show this here for the Zermelo arithmetic; the argument is easily modified for the von Neumann case. Suppose x ∈ a + p.Sox = a + y for some y ∈ p. Since p ⊆ TC(b), y < b and so x < a + b by Lemma 3.1 again. For the product, suppose a · p < a · b. By Proposition 3.2, a · p = a · q + r for some q < b and r < a.By uniqueness in Proposition 3.2, p = q, contradicting the assumption that p < b. On the other hand, if x < a · p then x = a · q + r for some q < p and r < a. Since p is in the aura of b, q < b and hence x < a · b. For the exponent, suppose a p < ab. By Proposition 3.4, a p = aq · r + s with q < b, r < a and s < aq .By Proposition 3.6 (and using 0 ∈ a), p = q, a contradiction. An argument similar to the product case shows that if x < a p then x < ab.  We examine the graph parameters of the product in detail in order to extend Lemma 5.2 to η: Lemma 5.4 (Z) Let p ∈ aura(b). Then |a · [b; p]|=|a · b|+|a|, ν(a · [b; p]) = ν(a · b) + ν(a), and η(a · [b; p]) = η(a · b) + η(a) +|a · p|. Proof. Wemayassume a = 0. a · [b; p] = a · b ∪{a · p + r | r ∈ a}=a · b ∪ (a · p + a), the union being disjoint by Proposition 3.2, so |a · [b; p]| is the sum of |a · b| and |a · p + a|=|a| using 5.1. Likewise G−(a · [b; p]) is the union of G−(a · b) and G−(a · p + a), although this union is no longer disjoint. Now if x < a · p + a, i.e., x is a node of G−(a · p + a), then either x < a · p in which case x < a · b because a · p ∈ aura(a · b) by Lemma 5.3, or x = a · p + r for some r < a in which case x < a · b (since p ∈ b, and

6 They can also be obtained from the first two parts of Lemma 5.4 below.

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Fig. 1 Schematic diagram of G(a · [b; p]) in the Zermelo arithmetic when a = 0, b = 0, and p ∈ aura(b) (cf. Lemma 5.4).

Fig. 2 Example illustrating Figure 1 with a ={{1}, {0, 1}}, b ={{0, 1}, {0, 1, {0, 1}}},andp = 2 ={1}.Left to right: G(a), G(b), G(a · b), G(a · [b; p]). using Proposition 3.2 again). Thus the nodes of G−(a · [b; p]) consist of those of G−(a · b) together with ν(a) nodes of the second type a · p + r, r < a,inG−(a · p + a). Cf. Figure 1 and the example shown in Figure 2. Now for the third equation, G(a · [b; p]) is obtained from G(a, b) and G(a · p + a) as combined above by identifying their source nodes. Its edges are those of G(a · b) together with two kinds of additional edges. The first kind are the η(a) edges interconnecting the nodes of form a · p + r, r ≤ a,inG(a · p + a): the induced subgraph on these nodes is isomorphic to G(a). The second kind are the |a · p| edges from a · p to its elements, all of which are nodes of G−(a · b).  Theorem 5.5 (Z) For any hereditarily finite sets a and b, we have that η(a · b) =|a| η(b) + (η(a) −|a|) ν(b).

P r o o f . We begin by proving this in the case when b is transitive. So b can be written as {b0,...,bn−1} with each bi in the aura of {b0,...,bi−1}. It will suffice to fix a which we may assume non-empty, and prove by induction on n that the theorem holds for any transitive b obtained by such a construction sequence of length n.

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The induction step is as follows: suppose the statement of the theorem is true for b, and p ∈ aura(b):weshow that it remains true when b is replaced by b; p. But in these circumstances, η(a · [b; p]) = η(a · b) + η(a) +|a||p| by Lemmas 5.4 & 5.2 =|a| η(b) + (η(a) −|a|) ν(b) + η(a) +|a||p| by inductive hypothesis =|a| (η(b) +|p|+1) + (η(a) −|a|)(ν(b) + 1) =|a| η(b; p) + (η(a) −|a|) ν(b; p) by Lemma 4.4 and the induction step is completed. It remains to show that the theorem still holds for intransitive b. But then η(a · b) = η(TC(a · b)) − ν(a · b) +|a · b| by Lemma 2.2 = η((TC(a · TC(b)) − ν(a) ν(b) +|a||b| by Corollary 3.3 and Lemma 5.2 = η(a · TC(b)) + ν(a · TC(b)) −|a · TC(b)|−ν(a) ν(b) +|a||b| by Lemma 2.2 We can break down the first term because we have proved the theorem for transitive sets such as TC(b), and we also keep applying the same lemmas as above on the next terms, so η(a · b) =|a|η(TC(b)) + (η(a) −|a|)ν(TC(b)) + ν(a)ν(TC(b)) −|a|ν(b) − ν(a)ν(b) +|a||b| =|a|(η(b) + ν(b) −|b|) + (η(a) −|a|)ν(b) + ν(a)ν(b) −|a|ν(b) − ν(a)ν(b) +|a||b| =|a|η(b) + (η(a) −|a|)ν(b). 

Notice that if instead of η(x),weuseμ(x) := η(x) − ν(x) (the cyclomatic number of the undirected graph underlying G(x)), then Theorem 5.5 can be written in a slightly simpler form: Theorem 5.6 (Z) For any finite sets a and b, we have that μ(a · b) =|a| μ(b) + μ(a) ν(b). We now turn to exponentiation. Lemma 5.7 (Z) For any p, |a0;p|=|a p||a|, ν(a0;p) = ν(a p)ν(a), and η(a0;p) = η(a)|a p|+ν(a)(η(a p) −|a p|). Proof. Thata0;p = a p · a follows directly from the definitions of product and exponentiation.7 The lemma follows from Lemma 5.2 and Theorem 5.5.  Lemma 5.8 (Z) Let a = 0, 0 ∈ a, b = 0, and p ∈ aura(b). Then |ab;p|=|ab|+|a p||a|, ν(ab;p) = ν(ab) + (ν(a) − 1)ν(a p), and η(ab;p) = η(ab) + η(a)|a p|+(ν(a) − 1)(η(a p) −|a p|). Proof. Wehavethatab;p = ab ∪{a p · q + r | q ∈ a ∧ r ∈ a p}=ab ∪ (a p · a), the union being disjoint by Proposition 3.6, so |ab;p| is the sum of |ab| and |a p · a|. Now apply Lemma 5.2. The graph G−(ab;p) is the union of G−(ab) and G−(a p · a). Suppose x < a p · a,sox = a p · q + r for some q < a and r < a p.Ifq = 0 then x = r < a p and so x < ab because a p is in the aura of ab (Lemma 5.3). If q = 0 then a p ≤ a p · q ≤ x. So the nodes of G(ab;p) which are not already nodes of G(ab) can be partitioned

7 Cf. [8, Lemma 4.4]. Note that because we are working in the Zermelo arithmetic, 0; p = p + 1, and that a p+1 = a p · a is a case of one of the “high school” laws of exponentiation. But both these equations are false in general in the von Neumann arithmetic, cf. [8].

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Fig. 3 Schematic diagram of G(ab;p) in the Zermelo arithmetic when a = 0, 0 ∈ a, b = 0, p ∈ aura(b),and 0 < q < a. Cf. Lemma 5.8. Note that the additional edges from a p · q to the elements of a p within the body of G(ab) are only present when 0 ∈ q.

Fig. 4 Example illustrating Figure 3 with a ={{1}, {0, 1}}, b ={{0, 1}, {0, 1, {0, 1}}},andp = 2 ={1}.Left to right: G(a p), G(ab), G(ab;p). (For illustrations of G(a) and G(b), cf. Figure 2.) as follows. Assign to each 0 < q < a the set {a p · q + r | r < a p}. These sets are disjoint, there are ν(a) − 1of them, and each has cardinality ν(a p). Cf. Figures 3 & 4. Likewise, G(ab;p) is the union of G(ab) and G(a p · a), so long as we identify the source nodes of these two graphs. This analysis of G(ab;p) shows that its edges are those of G(ab) together with two kinds of additional edges. The first kind are the |a p| edges connecting a p with its elements (which are nodes of G(ab)). The rest are the edges of G(a p · a) which do not lie below a p. There are η(a p · a) − η(a p) edges of this second kind. Thus η(ab;p) = η(ab) +|a p|+η(a p · a) − η(a p) = η(ab) +|a p|+|a p| η(a) + (η(a p) −|a p|) ν(a) − η(a p) by Theorem 5.5 = η(ab) + η(a)|a p|+(ν(a) − 1)(η(a p) −|a p|).

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The next theorem gives recurrence relations for the graph parameters of ab.  Theorem 5.9 (Z) Let a = 0, 0 ∈ a, and b = 0. Then  |ab|=|a| |a p|, p∈b  ν(ab) = (ν(a) − 1) ν(a p) + 1, and p

6 Exponentiation when the empty set is an element of the base

The results on Zermelo exponentiation in § 5 only work when 0 is not an element of the base. This brief section will examine what happens when 0 is in the base, which gives rise to the following degenerate behaviour:

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Fig. 5 Left to right: graphs of c, a,2c,2a in the Zermelo arithmetic. Cf. Example 5.10.

Fig. 6 Left to right: graphs of d, e, dd , de in the Zermelo arithmetic. Cf. Example 5.11.

Theorem 6.1 (Kirby, [8]) If 0 ∈ a, then ab = a(b).

Thus the only parameter of b that matters in this case is (b). To put it another way, we only need to consider ordinal exponents when 0 is in the base. Ordinal exponents are dealt with in the following two results, which hold for any base.

Lemma 6.2 (Z) If n is an ordinal, then for any a, |an|=|a|n and ν(an) = ν(a)n.

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P r o o f . This follows from Lemma 5.2 and the fact that the “high school algebra” laws of exponentiation hold in the Zermelo arithmetic [8], so that an really equals the product of n copies of a.  As for the number of edges η(an), this is stated most simply in terms of the related parameter μ = η − ν,cf. Theorem 5.6: Theorem 6.3 (Z) Suppose a > 1, and n > 0 is an ordinal. Then n−1 μ(an) = μ(a) ν(a)n−1−i |a|i . i=0 P r o o f . Inductively on n: μ(an+1) = μ(an · a) (the “high school” laws again) =|an| μ(a) + μ(an) ν(a) by Theorem 5.6

n−1 = μ(a) · |a|n + ν(a) ν(a)n−1−i |a|i by inductive hypothesis and Lemma 6.2 i=0 n = μ(a) ν(a)n−i |a|i i=0 and the induction step is completed.  Noting that a is transitive just when ν(a) =|a|, we can reformulate this result without the need for the summation: Corollary 6.4 (Z) Suppose a > 1, and n > 0 is an ordinal. If a is transitive, then μ(an) = n μ(a) ν(a)n−1. On the other hand, if a is intransitive, then ν(a)n −|a|n μ(an) = μ(a) . ν(a) −|a| Together with Theorem 6.1, these results completely specify the digraph parameters of ab when 0 ∈ a.Soin contrast with the situation when 0 is not in the base, here the digraph parameters of ab depend directly and only upon the digraph parameters of a together with (b), without need of recurrence relations.

7 How do the graph parameters comport with the von Neumann arithmetical operations?

In this section we work in the von Neumann arithmetic, indicating this by the annotation (vN). The graph of the von Neumann sum a + b looks like that of the Zermelo sum with additional edges from the source to each element of a [7], so: Lemma 7.1 (vN) For any sets a and b, we have |a + b|=|a|+|b|, ν(a + b) = ν(a) + ν(b), and η(a + b) = η(a) + η(b) +|a|. For the product and exponentiation we shall modify the arguments in § 5. In each case we shall find that, in the passage from b to b; p with p in the aura of b, the graph of the von Neumann construction consists of the graph of the Zermelo construction along with some additional edges (but no additional nodes). Thus if f (x) is any composition of additions, multiplications, and exponentiations then for any hereditarily finite sets a, the graph of f (a) in the von Neumann arithmetic is the graph of f (a) in the Zermelo arithmetic along with a “festoon” of additional edges. In particular, it will emerge from the proofs below that:

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Theorem 7.2 For any sets a and b, the values of ν(a + b), ν(a · b), ν(ab), |a · b|, and |ab| are identical in the von Neumann and Zermelo arithmetics. So for the von Neumann case we only need to count edges. Lemma 7.3 (vN) Let p ∈ aura(b).Then η(a · [b; p]) = η(a · b) + η(a) +|a| ν(a) |p|. P r o o f . We obtain the graph of a · [b; p] from the graph of G(a · b) in the von Neumann arithmetic just as in the Zermelo arithmetic (cf. the proof of Lemma 5.4 and Figure 1) except that there are now edges from a · p + r, for each r < a, to the elements of a · p. There are ν(a) such nodes a · p + r, each endowed with |a||p| new edges.  Theorem 7.4 (vN) For any hereditarily finite sets a and b, we have that η(a · b) = (η(a) −|a|) ν(b) + |a| ν(a)(η(b) −|b|) +|a||b|. P r o o f . First note that when b is transitive, i.e., |b|=ν(b), the statement of the theorem is equivalent to

η(a · b) = η(a) ν(b) +|a| ν(a)(η(b) −|b|). (3)

As in the proofs in § 5, the theorem will be proved for transitive b if we can assume (3) and show that it survives the passage to b; p with p ∈ aura(b). By Lemma 7.3 and inductive hypothesis,

η(a · [b; p]) = η(a) ν(b) +|a| ν(a)(η(b) −|b|) + η(a) +|a| ν(a) |p| = η(a)(ν(b) + 1) +|a| ν(a )(η(b) −|b|+|p|) = η(a) ν(b; p) +|a| ν(a)(η(b; p) −|b; p|), as required.

Now for b intransitive, the following equation still holds in the von Neumann arithmetic because the arguments for it in the proof of Theorem 5.5 work in both arithmetics:

η(a · b) = η(a · TC(b)) + ν(a · TC(b)) −|a · TC(b)|−ν(a) ν(b) +|a||b|.

By earlier results, the second and fourth terms here cancel each other out, and since TC(b) is transitive we can use (3) to write

η(a · b) = η(a) ν(TC(b)) +|a| ν(a) η(TC(b)) −|a| ν(a) |TC(b)|−|a| ν(b) +|a||b| = η(a) ν(b) +|a| ν(a)(η(b) + ν(b) −|b|) −|a| ν(a) ν(b) −|a| ν(b) +|a||b| by Lemma 2.2 = η(a) ν(b) +|a| ν(a) η(b) −|a| ν(a) |b|−|a| ν(b) +|a||b| which is equivalent to the statement of the theorem.  Since a0;p = a p · a we can apply Theorem 7.4 to get: Lemma 7.5 (vN) For hereditarily finite sets a and p, we have

0;p p p p p p . η(a ) = ν(a)(η(a ) −|a |) + (η(a) −|a|) |a | ν(a ) +|a||a | Lemma 7.6 (vN) Let a = 0, 0 ∈ a, b = 0, and p ∈ aura(b). Then

η(ab;p) = η(ab) + (ν(a) − 1)(η(a p) −|a p|) + (η(a) −|a|) |a p| ν(a p) +|a||a p|

P r o o f . Just as for the Zermelo case in Lemma 5.8, G(ab;p) in the von Neumann arithmetic is the union of G(ab) and G(a p · a) where we identify the source nodes, so

η(ab;p) = η(ab) +|a p|+η(a p · a) − η(a p) = η(ab) +|a p|+ν(a)(η(a p) −|a p|) + (η(a) −|a|) |a p| ν(a p) +|a||a p|−η(a p) by Lemma 7.5 which reduces to the statement of the lemma. 

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Theorem 7.7 (vN) Let a = 0, 0 ∈ a, and b = 0. Then   η(ab) = (ν(a) − 1)(η(a p) −|a p|) + (η(a) −|a|)|a p| ν(a p) +|a| |a p|. p

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