The Logic of Comparative Cardinality

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

UC Berkeley Working Papers Title The Logic of Comparative Cardinality Permalink https://escholarship.org/uc/item/2nn3c35x Authors Ding, Yifeng Harrison-Trainor, Matthew Holliday, Wesley Halcrow Publication Date 2020 License https://creativecommons.org/licenses/by-nc-nd/4.0/ 4.0 Peer reviewed eScholarship.org Powered by the California Digital Library University of California THE LOGIC OF COMPARATIVE CARDINALITY YIFENG DING, MATTHEW HARRISON-TRAINOR, AND WESLEY H. HOLLIDAY Abstract. This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections. §1. Introduction. Reasoning about the relative size of infinite sets has been a source of puzzles since at least Galileo [8]. Any consistent extension of the notion of relative size from finite to infinite sets must give us very different principles in the infinite compared to the finite. For two key principles that hold in the finite—that proper subsets are smaller than their supersets, and that sets in one-to-one correspondence have the same size— are inconsistent in the infinite. Cantor’s theory of infinite cardinalities [3] maintains the latter principle at the expense of the former, while the more recent theory of infinite numerosities [12] does the reverse. For logicians, a precise definition of relative size of sets raises an obvious question: can we completely axiomatize reasoning about the relative size of finite sets, of infinite sets, and of arbitrary sets in a formal set-theoretic language? Just as the laws for reasoning about intersection, union, and complementation of sets are captured by the laws of Boolean algebra, what are the laws one must add to Boolean algebra to capture reasoning about the relative size of sets according to the given definition? In this paper, we answer this question for a particular language and definition of relative size. Our language (see Definition 2.1) allows us to build terms using the standard set-theoretic operations of intersection, union, and complementation, and to express that a set s is at least as big as a set t: |s| ≥ |t|. Thus, we work with a comparative notion of size, prior to the reification of sizes as cardinal numbers. The semantics is given by the Cantorian definition: |s| ≥ |t| is true iff there is an injection from t into s. This language has an alternative interpretation in terms of the relative likelihood of events, instead of the relative size of sets. We will exploit The second author was supported by a Banting Fellowship. Preprint of January 2020 Forthcoming in The Journal of Symbolic Logic 1 2 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY this connection to prove some of our main results. In essence, Cantorian reasoning about the relative size of finite size is the same as probabilistic reasoning about the relative likelihood of events, while Cantorian reasoning about the relative size of infinite sets is the same as what is called possibilistic reasoning [6] about the relative likelihood of events. Each type of likelihood reasoning has been axiomatized by itself [18, 9, 1, 5]. If we reinterpret these results in terms of cardinality, then reasoning about the comparative cardinality of finite sets and reasoning about the comparative cardinality of infinite sets have each been axiomatized by themselves. The goal of this paper is to bridge the divide between finite and infinite and axiomatize reasoning about the cardinality of arbitrary sets. In § 2, we define our formal language and its interpretation in fields of sets. We then present our first axiomatization, which uses two extra predicates Fin and Inf to express that a set is finite or infinite. The axiomatization without these predicates is more complicated and saved for later. Both axiomatizations use the so-called finite cancellation axiom schema, which encodes an infinite sequence of axioms of exponentially growing length. In an appendix § A, we show how this schema can be replaced with the combination of a simple axiom and a simple rule. In § 3, we define models based on Boolean algebras to be used later and adapt to this context the representation theorem in the classic paper [11] by Kraft, Pratt, and Seidenberg. We also show the effective finite model property and as a corollary the decidability of our two logics (with or without the Fin and Inf predicates). In § 4, we construct canonical models from maximally consistent sets, as is common in proofs of completeness, leading in § 5 to the completeness of the system with extra predicates. In § 6, we first show in what sense finiteness and infiniteness of a set can be defined in the language with only cardinality comparisons between set terms. Then we finally define the axiomatic system mentioned in § 2 without the two extra predicates and prove its soundness and completeness. Lastly, we end with open problems in § 7. Comparison to related work. Two strands of work related to ours are worth mentioning. The first is the study of computable fragments of set theory, as in [2, 7]. For example, consider the quantifier-free language with intersection and set difference as binary functions and membership, inclusion, and equality as binary predicates. When this language is interpreted on the universe of all sets in the obvious way, the satisfiability problem is decidable; in fact, more functions and predicates can be added without loss of decidability [2]. In particular, a cardinality comparison predicate can be added, resulting in a language very similar to ours [7]. However, the language is still different from ours, due to its lack of set-theoretic complementation. Moreover, the cited works do not provide any axiomatization. THE LOGIC OF COMPARATIVE CARDINALITY 3 Another strand is the work on extending syllogistic logic with cardinality comparison initiated by Lawrence Moss (see [14] for an introduction). In this setting, the language consists of sentences of the form “all x are y”, “some x are y”, “there are at least as many x as y”, and “there are more x than y” with variables interpreted as subsets of an arbitrary set. In [13, 15], axiomatizations of the valid sentences (on finite or infinite domains) are provided. However, in this setting there are no sentential Boolean connectives, nor Boolean set operators except complementation. Thus, the expressivity of the syllogistic language with cardinality comparisons is much weaker than ours, though with the consequent advantage of having a tractable satisfiability problem. §2. Formal setup and statement of main result. Definition 2.1. Given a countably infinite set Φ of set labels, the set terms t and formulas ϕ of the language L are generated by the following grammar: t ::= a | tc | (t ∩ t) ϕ ::= |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ), where a ∈ Φ. The other sentential connectives ∨, →, and ↔ are defined as usual, and we use ϕ ⊕ ψ as an abbreviation for (ϕ ∨ ψ) ∧ ¬(ϕ ∧ ψ). Standard set-theoretic notation may be defined as follows: • ∅ := t ∩ tc ; • t ⊆ s := |∅| ≥ |t ∩ sc |; • t = s := (t ⊆ s ∧ s ⊆ t) and t 6= s := ¬(t = s); • t 6⊆ s := ¬(t ⊆ s) and t ( s := (t ⊆ s ∧ s 6⊆ t). We also use |s| ≤ |t| for |t| ≥ |s|, |s| > |t| for ¬|t| ≥ |s|, and |s| = |t| for |s| ≥ |t| ∧ |t| ≥ |s|. For any ∆ ⊆ Φ, let L(∆) be the fragment of L using only set labels in ∆, and let T (∆) be the set of set terms generated by ∆. Our models consist essentially of a collection of sets, some of which are assigned set labels from Φ. Definition 2.2. A field of sets is a pair hX, Fi where X is a nonempty set and F is a collection of subsets of X closed under intersection and settheoretic complementation. A field of sets model is a triple M = hX, F, V i where hX, Fi is a field of sets and V : Φ → F. The satisfaction relation is defined in the obvious way. Definition 2.3. Given a field of sets model M = hX, F, V i, we define a function Vb , which assigns to each set term a set in F, by: • Vb (a) = V (a) for a ∈ Φ; 4 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY • Vb (tc ) = X \ Vb (t); • Vb (t ∩ s) = Vb (t) ∩ Vb (s). We then define a satisfaction relation |= as follows: • M |= |t| ≥ |s| iff there is an injection from Vb (s) into Vb (t); • M |= ¬ϕ iff M 6|= ϕ; • M |= ϕ ∧ ψ iff M |= ϕ and M |= ψ. Given a class K of field of sets models, ϕ is valid over K iff M |= ϕ for all M ∈ K. In Definition 6.12, we will define the cardinality comparison logic CardCompLogic. Our main result is that this logic is sound and complete. Theorem 2.4. The cardinality comparison logic CardCompLogic is sound and complete with respect to field of sets models. The logic is somewhat complicated, so we will leave its definition for later. For now, it will be helpful to consider the expanded language LFin,Inf that adds predicates Fin and Inf that pick out the finite and infinite sets, respectively. Then the logic CardCompLogic can be obtained by eliminating Fin and Inf. Definition 2.5. Let LFin,Inf be the language extending L with two new unary predicates Fin and Inf using the following grammar: where a ∈ Φ. t ::= a | tc | (t ∩ t) ϕ ::= Fin(t) | Inf(t) | |t| ≥ |t| | ¬ϕ | (ϕ ∧ ϕ), Satisfaction can then be extended from L to LFin,Inf as follows. Definition 2.6. For any field of sets model M = hX, F, V i, define the satisfaction relation |= for LFin,Inf with the following two new clauses: • M |= Fin(t) iff Vb (t) is finite; • M |= Inf(t) iff Vb (t) is infinite. It will be convenient for later use to divide the logic of cardinality comparison with Fin and Inf into two parts, the first of which gives basic comparison principles such as transitivity and the second of which involves additional principles such as that infinite sets are larger than finite sets. Definition 2.7. The basic comparison logic BasicCompLogic is the logic for L (or LFin,Inf ) with the following axiom schemas and rules: (BC1) all substitution instances of classical propositional tautologies; (BC2) ¬ |∅| ≥ |∅c |; (BC3) |s| ≥ |t| ∨ |t| ≥ |s|; 5 THE LOGIC OF COMPARATIVE CARDINALITY (BC4) (BC5) (BC6) (BC7) (BC8) (|s| ≥ |t| ∧ |t| ≥ |u|) → |s| ≥ |u|; |∅| ≥ |s ∩ tc | → |t| ≥ |s|; (|∅| ≥ |s| ∧ |∅| ≥ |t|) → |∅| ≥ |s ∪ t|; if ϕ and ϕ → ψ are theorems, so is ψ; if t = 0 is provable in the equational theory of Boolean algebras, then |∅| ≥ |t| is a theorem. Definition 2.8. The logic CardCompLogicFin,Inf , the cardinality comparison logic with predicates Fin and Inf, consists of the axioms and rules of the basic comparison logic BasicCompLogic together with the following axiom schemas: (A1) (A2) (A3) (A4) (A5) (A6) Fin(s) ⊕ Inf(s); Fin(∅) ∧ ((Fin(s) ∧ Fin(t)) → Fin(s ∪ t)); (Fin(t) ∧ s ⊆ t) → Fin(s); (Fin(s) ∧ Inf(t)) → |t| > |s|; V n i=1 (Fin(si ) ∧ Fin(ti )) → FCn (s1 , · · · , sn , t1 , · · · , tn ) (for all n ≥ 1); (Inf(s) ∧ |s| ≥ |t| ∧ |s| ≥ |u|) → |s| ≥ |t ∪ u|; Here FCn (s1 , · · · , sn , t1 , . . . , tn ) is what we call the finite cancellation axiom. To define this formula, first for each m such that 1 ≤ m ≤ n, define the term Sm as the union of the terms of the form sc11 ∩ sc22 ∩ · · · ∩ scnn where exactly m many ci ’s are c and the rest are empty. Similarly define Tm with s replaced by t. Intuitively, Sn denotes the set of elements which are in exactly m many sets among the sets denoted by s1 , s2 , . . . , sn . Then FCn (s1 , . . . , sn , t1 , . . . , tn ) is defined by n ^ i=1 Si = T i ! → n−1 ^ i=1 |si | ≥ |ti | ! ! → |tn | ≥ |sn | . The first four axioms set up the relations between finite sets and infinite sets—for example, that finite sets are smaller than infinite sets. Axioms (A5) and (A6) describe the distinct behavior of finite and infinite cardinal Vn arithmetic. To understand (A5), suppose the condition expressed by i=1 Si = Ti is true, assuming that the sets denoted by the si ’s and ti ’s are all finite. Note that to compute the sum K of the cardinalities of the sets denoted by the si ’s, instead of the most straightforward way of adding their cardinalities, we can consider how much each element contributes to K: if an element e is in ke many sets denoted by the si ’s, then the contribution of this e is ke , and then K is the sum of the ke ’s. Thus, n X i=1 |V (si )| = n X i=1 i × |V (Si )| 6 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY as Si is precisely the set of elements that lie in exactly V i many sets denoted n by si ’s. The same holds for the ti ’s. Then given that i=1 Si = Ti is true, n X i=1 |V (si )| = n X i=1 i × |V (Si )| = n X i=1 i × |V (Ti )| = n X i=1 |V (ti )|. Hence it is not hard to see that the consequent ! n−1 ^ |si | ≥ |ti | → |tn | ≥ |sn | i=1 must be true in the same model. Example 2.9. Let A, B, C, D, E ⊆ X be disjoint and finite. Then it follows from • |A| ≥ |B ∪ C|, • |B ∪ E| ≥ |A ∪ C|, and • |C ∪ D| ≥ |A ∪ B|, that |D∪E| ≥ |A∪B ∪C|. To see this, we only need to add the cardinalities of the inequalities on both sides, which leads to |A| + |B| + |C| + |D| + |E| ≥ |A|+|A|+|B|+|B|+|C|+|C|. Hence by cancelling |A|+|B|+|C| since they are finite, we get |D| + |E| ≥ |A| + |B| + |C|. Thus |D ∪ E| ≥ |A ∪ B ∪ C|. In our system, this reasoning is captured by FC4 (a, b ∪ e, c ∪ d, a ∪ b ∪ c, b ∪ c, a ∪ c, a ∪ b, d ∪ e) with a, b, c, d, e ∈ Φ, as the antecedent of FC4 follows from the assumption that these five sets are disjoint, which can be expressed by formulas like |a ∩ b| = ∅. Finally, (A6) captures the distinct absorption property (or non-additivity) of infinite sets. In terms of the analogy with relative likelihood, it is (A5) that matches probabilistic reasoning, as is shown in [11] and [17], while (A6) matches what is called possibilistic reasoning [6]. Theorem 2.10. CardCompLogicFin,Inf , the cardinality comparison logic with predicates Fin and Inf, is sound and complete with respect to field of sets models. Remark 2.11. Admittedly, (A5) is an infinite sequence of axioms that are long and somewhat complicated. We remark here that (A5) can be replaced by the combination of the following axiom and rule: (A7) (Fin(s) ∧ Fin(t)) → (|s| ≥ |t| ↔ |s ∩ tc | ≥ |t ∩ sc |); (A8) where a|t abbreviates |t∩a| = |t∩ac | for a ∈ Φ, if a|t → ϕ is derivable, then ϕ is derivable, assuming that a does not occur in t or in ϕ. THE LOGIC OF COMPARATIVE CARDINALITY 7 Axiom (A7) is sometimes called the quasi-additivity axiom. Intuitively, it means that taking unions with a disjoint set does not change the ordering of sets by cardinality. Rule (A8) is slightly non-standard. Intuitively, a|t says that set a splits t into two parts of the same size. In both [11] and [1], this is expressed by “a polarizes t”. Hence (A8) is called the “polarizability rule” in [1] ((A8) also appeared, but not as a rule in a formal system, in [11]). Proof theoretically, (A8) allows us to assume without loss of generality that any set can be polarized with a fresh set when proving some formula ϕ. Semantically, the idea behind (A8) is the invariance of truth under duplication: for any sets a, b, |a| |b| iff |a × {0, 1}| |b × {0, 1}|, and a is finite iff a × {0, 1} is finite. This warrants the use of (A8), as to show that ϕ is true on all field-of-sets models, it is enough to focus on those models that come from duplication, in which every set in the field-of-sets can be polarized. The power of (A8) lies in the fact that with it, we can simulate the addition of overlapping sets so that we can count overlaps correctly. Hence all instances of (A5) are provable from the system with (A5) replaced by (A7) and (A8). This point is already made implicitly in [11] and explicitly in [1], but we give a direct syntactic proof in the appendix § A. To get a flavor of the strategy, see Figure 1 where the shaded area in the second picture has a cardinality equal to one fourth of the sum of the cardinalities of the three larger sets (say A, B, and C). To see this, first polarize all minimal regions like A ∩ B c ∩ C c and A ∩ B ∩ C into four parts, and then for regions that are contained in m of the sets A, B, and C, select m parts of those regions. For example, as shown in the diagram, three of the four parts of region A ∩ B ∩ C are selected, while only one of the four in A ∩ B c ∩ C c is selected. So we have replaced the non-disjoint union of the shapes A, B, and C by a disjoint union of the shaded areas, while keeping the same area up to a factor of 1/4. For the logic in the language L, without the predicates Fin and Inf, it takes more work to capture the difference between finite and infinite cardinal arithmetic. The cardinality comparison logic CardCompLogic will also extend the same basic comparison logic BasicCompLogic. The key idea for the extra axioms is that of a set being “witnessed to be finite” or “witnessed to be infinite.” For example, if M |= |s| ≥ |s ∪ t| ∧ |s ∩ t| = |∅|, then s must denote an infinite set in M. Since s denotes an infinite set, anything true of infinite sets must be true of s. One can think of the cardinality comparison logic CardCompLogic as being the same as the cardinality comparison logic CardCompLogicFin,Inf except with Fin(t) and Inf(t) being replaced by formulas that witness t to be finite or infinite, respectively. 8 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY A B C A B C Figure 1. Polarization and set addition Of course the definition of a set being finite or infinite would be superfluous if we were to restrict our models so that every set in the field of sets is finite (or infinite except for the empty set). In fact, letting FinCardCompLogic be the result of adding to BasicCompLogic all instances of FCn (s1 , . . . , sn , t1 , . . . , tn ), FinCardCompLogic is sound and complete with respect to all field of sets models where the underlying set is finite. Similarly, the system InfCardCompLogic defined by adding (|s| ≥ |t| ∧ |s| ≥ |u|) → |s| ≥ |t ∪ u| is sound and complete with respect to all field of sets models where all nonempty sets in the field are infinite. We will not formally prove these two completeness results since the strategy we use to prove the completeness of CardCompLogicFin,Inf can be readily adapted. §3. Other types of models. While our interest is in field of sets models, it is convenient to think in terms of an abstract Boolean algebra rather than a concrete field of sets. For then we do not have to worry about which THE LOGIC OF COMPARATIVE CARDINALITY 9 particular elements a set contains, but instead we only have to consider the cardinality of the set. This will help us focus on the structures related to the truth of formulas and to show the effective finite model property. In the following, we use ∧, ∨, and ′ for meet, join, and complementation in arbitrarily picked Boolean algebras. For specifically constructed Boolean algebras, the symbols for the operations may change and we will usualy specify only the complementation and meet operation. The lattice ordering of a Boolean algebra will be denoted by ≤ (below) and ≥ (above1 ), possibly with a subscript to show which Boolean algebra we are talking about. Since the models defined below are all Boolean algebras with extra structure, we call them algebra-based models and call them finite when the underlying Boolean algebra is finite. As a convenient notation, for any models B, C and set L of formulas, we write B ≡L C when for any ϕ ∈ L, B |= ϕ iff C |= ϕ. 3.1. Measure algebra models. The first step is to forget the elements in sets and only keep their Boolean structure and their cardinality. This gives us the following definition. Definition 3.1. A measure algebra is a pair hB, µi where B is a Boolean algebra and µ is a function assigning a cardinal to each element of B such that • if a ∧ b = ⊥, then µ(a ∨ b) = µ(a) + µ(b), and • µ(b) = 0 iff b = ⊥. We call such a cardinal-valued function µ with which hB, µi is a measure algebra a cardinal measure on B. A measure algebra model is a triple B = hB, µ, V i where hB, µi is a measure algebra and V is a function from Φ to B. This V can be extended to a function Vb from T (Φ) to B as in Definition 2.3 but using the Boolean complement and meet in place of set-theoretic complement and intersection. Note that µ is only finitely additive, which is good enough because the language is finitary and unable to express countable additivity. Definition 3.2. Given a measure algebra model B = hB, µ, V i, we define the satisfaction relation |= as follows, where ϕ, ψ ∈ L and s, t ∈ T (Φ): • B |= |t| ≥ |s| iff µ(V (t)) ≥ µ(V (s)); • B |= ¬ϕ iff B 6|= ϕ; • B |= ϕ ∧ ψ iff B |= ϕ and B |= ψ. We also have the following two clauses for LFin,Inf sentences: • B |= Inf(t) iff µ(Vb (t)) is infinite; • B |= Fin(t) iff µ(Vb (t)) is finite. 1 We use “below” and “above” in the weak sense. 10 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY We can turn any field of sets hX, Fi into a measure algebra hB, µi by setting B = hF, X \ · , ∩i and µ(a) = |a| for a ∈ B. It is easy to check that this is a measure algebra. If we have a field of sets model M = hX, F, V i then we get a measure algebra model B = hB, µ, V i using the same valuation V ; moreover, M ≡LFin,Inf B by a simple induction. On the other hand, given a finite measure algebra model B, we can turn it into a field of sets model M such that M ≡LFin,Inf B. Since the cardinal measure functions in measure algebras are only finitely additive, the construction will fail for infinite measure algebra models. Proposition 3.3. For any finite measure algebra model B = hB, µ, V i, there is a field of sets model M = hX, F, V ′ i such that M ≡LFin,Inf B. Proof. Since B is finite, let a1 , . . . , an be S the atoms of B. Let S1 , . . . , Sn n be disjoint sets with |Si | = µ(ai ). Let X = i=1 Si and let F be the field of sets generated by S1 , . . . , Sn under complementation (in X) and intersection. The map f (ai ) = Si extends to an isomorphism between B and F, which maps an element ai1 ∨ · · · ∨ aiℓ to Si1 ∪ · · · ∪ Siℓ . We have µ(ai1 ∨ · · · ∨ aiℓ ) = µ(ai1 ) + · · · + µ(aiℓ ) = |Si1 | + · · · + |Siℓ | = |Si1 ∪ · · · ∪ Siℓ |. Thus, |f (a)| = µ(a). Let V ′ = f ◦ V . Then it is easy to see that for any set term s, |V (s)| = µ(V ′ (s)), since f is an isomorphism preserving cardinalities. A simple induction then shows that hX, F, V ′ i ≡LFin,Inf B. ⊣ Thus, there is no loss of generality when focusing on measure algebra models if we consider only finite models, and as we will see later in this section, there is also no loss of generality in restricting to finite models. 3.2. Comparison algebra models. As one often does to prove the completeness of some logic, we will use a canonical model construction. In building the canonical model, we start with a maximally consistent set of sentences in our language L or LFin,Inf , which encodes only comparisons between set terms and their being finite or infinite in the case of LFin,Inf . Hence it will be convenient to forget about the cardinals we assign to each element of the Boolean algebra and to remember only the comparisons between elements. When we need to work with LFin,Inf , we also need the model to contain a set of distinguished elements. Definition 3.4. A comparison algebra is a pair hB, i where B is a Boolean algebra and is a total preorder on B such that • for all a, b ∈ B, a ≥B b implies a b, and • ⊥B 6 b for all b ∈ B \ {⊥B }. THE LOGIC OF COMPARATIVE CARDINALITY 11 A labeled comparison algebra is a triple hB, , F i where hB, i is a comparison algebra and F ⊆ B. A comparison algebra model is a triple B = hB, , V i where hB, i is a comparison algebra and V a function from Φ to B. Similarly, a labeled comparison algebra model is a labeled comparison algebra model together with a valuation. The valuation V can be extended in the usual way to a valuation Vb from T (Φ) to B. Here the relation is intended to interpret “at least as great in cardinality as” and F is intended to interpret “being a finite set”. Hence we have the required constraints for in the above definition: it is a total preorder, extends the Boolean lattice order (set inclusion relation), and makes the bottom element (the empty set) the strictly smallest set. Formally, the interpretation is given by the satisfaction relation. Definition 3.5. Given a comparison algebra model B = hB, , V i, we define the satisfaction relation |= for L as follows, where s, t ∈ T (Φ): • B |= |s| ≥ |t| iff Vb (s) Vb (t); • usual clauses for ¬ and ∧. Given a labeled comparison algebra model B = hB, , F, V i the satisfaction relation |= can be extended to LFin,Inf with the extra clauses: • B |= Fin(s) iff Vb (s) ∈ F ; • B |= Inf(s) iff Vb (s) 6∈ F . While is intended to compare cardinality and F is intended to include exactly finite elements, the requirements given above are not enough to let us know that it is cardinality that is comparing and that elements in F are precisely those that are finite. Given a measure algebra hB, µi, we can easily build a comparison algebra hB, i by taking a b if and only if µ(a) ≥ µ(b) and further a labeled measure algebra hB, , F i by taking F = {b ∈ B | µ(b) is finite}. But for the other direction, to fix that is comparing cardinality and F captures finiteness, we need some extra conditions. We state this in terms of a representation theorem. Definition 3.6. A comparison algebra hB, i (labeled comparison algebra hB, , F i) is represented by a cardinal measure µ on B if for all a, b ∈ B, we have a b iff µ(a) ≥ µ(b) (and F = {b ∈ B | µ(b) is finite}). We also say hB, i or hB, , F i is represented by a measure algebra B ′ when it is represented by µ and B ′ = hB, µi. A (labeled) comparison algebra model is representable if its (labeled) comparison algebra part is representable. Clearly if a finite (labeled) comparison algebra model hB, V i is represented by a measure algebra hB ′ , V i, then B ≡LFin,Inf B ′ . Hence if ϕ is satisfiable on a finite representable (labeled) comparison algebra model, then, in light of Proposition 3.3, ϕ is satisfiable on a field of sets model. 12 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Before proving the full representation, we recall the following classic theorem on when an ordering is representable by a probability measure. Theorem 3.7 (Kraft, Pratt, Seidenberg [11], Theorem 2). For any finite Boolean algebra B with ⊤ as the top element and ⊥ the bottom element and any binary relation on B, there is a probability measure µ on B such that for all a, b ∈ B, a b iff µ(a) ≥ µ(b), if and only if the following conditions are satisfied: • • • • not ⊥ ⊤; for all b ∈ B, b ⊥; is transitive, and for any a, b ∈ B, a b or b a; for any two sequences of elements a1 , a2 , . . . , an and b1 , b2 , . . . , bn from B of equal length, if every atom of B is below (in the order of the Boolean algebra) exactly as many a’s as b’s, and if ai bi for all i ∈ {1, . . . , n − 1}, then bn an . The fourth condition, known also as finite cancellation, is precisely the truth condition of FCn . Put more algebraically, if we represent elements in B as their characteristic functions over the atoms of B and further identify those functions with vectors of 0’s and 1’s, then the vector sum of a’s being the same as the vector sum of b’s implies that the sum of the probabilities of a’s is also equal to the sum of the probabilities of b’s. This is of course because vector sums count overlaps properly, unlike unions. Dana Scott used this representation in [17] and provided a lucid proof of the above theorem. It can also be observed from the proof in [11] (see Corollary 2) or [17] that the probability measure µ can be turned into an additive function to non-negative rational numbers and then to natural numbers by scaling, since µ is obtained by solving a finite system of (possibly strict) linear inequalities with rational coefficients. With this component dealing with finite elements, we can prove the representation theorem for both finite and infinite elements. Theorem 3.8. A finite labeled comparison algebra B = hB, , F i is represented by some cardinal measure µ on B if and only if the following conditions hold: (1) F is an ideal; (2) elements in F satisfy the finite cancellation condition in Theorem 3.7: for any two sequences of elements a1 , a2 , . . . , an and b1 , b2 , . . . , bn from F of equal length, if every atom of B is below (in the order of the Boolean algebra) exactly as many a’s as b’s, and if ai bi for all i ∈ {1, . . . , n − 1}, then bn an ; (3) for any a, b, c ∈ B such that a 6∈ F , if a b and a c, then a b∨B c; (4) for any a, b ∈ B, if a ∈ F and b 6∈ F , then b a and not a b. THE LOGIC OF COMPARATIVE CARDINALITY 13 It is then easy to see that a finite comparison algebra B = hB, i is represented by µ if and only if there exists an F ⊆ B such that hB, , F i is represented by µ. Proof. For the proof, we use the following definitions. For any Boolean algebra B and b ∈ B, let At(B) be the set of all atoms in B and At(b) the set of atoms below b. Given a preorder on B and b ∈ B, let [b] be {b′ ∈ B | b′ b and b b′ }. Then define Rank(b) for b ∈ B to be the number of atoms strictly smaller than b in the order , modulo equivalence, i.e., the cardinality of the set {[a] | a ∈ At(B), b ≻ a}. Finally, let max At(b) be any one of the -maximal elements in At(b), if there is such an element. Since B is finite, F is a finite ideal and hence principal. Then the quotient B|F becomes a finite Boolean algebra with a binary relation that satisfies all the conditions required in Theorem 3.7 because we required that the finite cancellation condition holds for all elements in F . Hence there is an additive measure function µ0 from B|F to N such that for any b1 , b2 ∈ B|F , we have b1 b2 iff µ0 (b1 ) ≥ µ0 (b2 ). If B = B|F then we are done, so we now consider the case where B 6= B|F . Consider an arbitrary element b outside B|F . Because B is finite, it is atomic and hence At(b) must contain an atom that is not in B|F . Then by (4) this atom is strictly greater in than all atoms in B|F . Hence max At(b) is outside B|F (it exists because B is finite). By definition, max At(b) is an atom under b. Thus b max At(b). But because b 6∈ F , we can also show that max At(b) b using condition (3). To see this, list At(b) as b1 , b2 , . . . , bn . Then we have the following inductive argument: • max At(b) b1 ; Wk • supposing max At(b) i=1 bi , then together with max At(b) bk+1 Wk Wk+1 and condition (3), max At(b) i=1 bi ∨ bk+1 = i=1 bi . Wn Hence at the end of the induction we have max At(b) i=1 bi = b. Now define a measure µ on B as follows: ( µ0 (b) b ∈ B|F µ(b) = ℵRank(max At(b)) b 6∈ B|F . It is not hard to see that this is indeed a measure function on B. Moreover, we now show that for any b1 , b2 ∈ B, we have b1 b2 iff µ(b1 ) ≥ µ(b2 ): • if both b1 , b2 ∈ B|F , then we can use µ0 ; • if both b1 , b2 6∈ B|F , then b1 b2 iff max At(b1 ) max At(b2 ) iff Rank(max At(b1 )) ≥ Rank(max At(b2 )) iff µ(b1 ) ≥ µ(b2 ); • if b1 ∈ B|F and b2 6∈ B|F , then b2 b1 by condition (4), but it is also trivially true that µ(b2 ) ≥ µ(b1 ). ⊣ 14 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY (111) : 4 (011) : 4 (101) : 4 (110) : 4 (001) : 1 (010) : 2 (100) : 3 (000) : 0 Figure 2. A non-representable comparison algebra The above theorem again only works for finite models. Representation of infinite models requires different conditions and techniques to prove. We will review this as an open problem in § 7. Example 3.9. Figure 3.2 presents a comparison algebra that cannot be represented by any cardinal measure. The number after the colon in each node is the rank of that node in ; this determines the preorder , as x y iff the rank of x is at least that of y. We also group the nodes of the same rank into shaded areas. We can then see that if were representable by a cardinal measure, all nodes would be finite. To see this, note that (110) is the join of (010) and (100), but (110) is also strictly greater than both of them, which implies that (110) is finite. Now (110) is of rank 4, which means it is as large as any other set, so all the sets would be finite. However, we can also see that, letting a1 = (101), a2 = (010), b1 = (110), b2 = (001), the finite cancellation condition fails. First, every atom is below exactly one of a1 and a2 and also exactly one of b1 or b2 . Thus, the antecedent of finite cancellation is true. But the consequent is false, as a1 a2 but not b2 b1 . The non-representability of this comparison algebra also implies, by the previous theorem, that the required ideal does not exist. 3.3. Effective finite model property. In the previous two subsections, we saw that our representation theorems (Proposition 3.3 and Theorem 3.8) only work on finite models. However, to make a formula true, THE LOGIC OF COMPARATIVE CARDINALITY 15 we only need finite models. In fact, we can effectively bound the size of satisfying models for any formula that is satisfiable by some (possibly infinite) field of sets model. Since the construction of a finite satisfying model will be used later, we provide a systematic treatment, starting with the following definition. Definition 3.10. Let B be an algebra-based model and ∆ ⊆ Φ. B is adapted to ∆ if Vb is surjective from T (∆) to the underlying Boolean algebra B in B. The importance of this definition is that for any algebra-based model B that is adapted to ∆, every element b ∈ B is named in the sense that there exists t ∈ T (∆) such that Vb (t) = b. It is easy to see that an algebrabased model adapted to a finite set ∆ is finite. To be more precise, when ∆ ⊆ Φ is finite, let T0 (∆) be the set of all distinct terms in ∆ in disjunctive normal form with no repetition of conjuncts or disjuncts. T0 (∆) is finite, and using Boolean identities, for every term t ∈ T (∆), there is a term t′ ∈ T0 (∆) such that Vb (t) is always the same as Vb (t′ ). Thus, in any model, Vb (T (∆)) = Vb (T0 (∆)) is finite. Proposition 3.11. Fix a finite set ∆ ⊆ Φ. For any measure (resp. comparison, labeled comparison) algebra model B, there is a measure (resp. comparison, labeled comparison) algebra model B∆ that is adapted to ∆ and satisfies B∆ ≡L(∆) B. Proof. For any measure algebra model B = hB, µ, V i, define B∆ = hB∆ , µ∆ , V∆ i where the B∆ is the subalgebra of B with Vb (T (∆)) as the carrier set, µ∆ is the restriction of µ to Vb , and V∆ is defined as: ( V (a) a ∈ ∆ V∆ (a) = ⊥B a 6∈ ∆. Similarly, for any comparison algebra model B = hB, , V i, we can define B∆ = hB∆ , ∆ , V∆ i where now ∆ is the restriction of to Vb (T (∆)). It is not hard to see that B∆ ≡L(∆) B. For labeled comparison algebras, we just need to further define F∆ = F ∩ Vb (T (∆)). ⊣ Now we prove the effective finite model property. Theorem 3.12. For any formula ϕ ∈ LFin,Inf , ϕ is satisfied by some field of sets model if and only if it is satisfied by a labeled comparison algebra model hB, , F, V i such that: 1. B is finite with at most 2|∆| many atoms where ∆ is the set of set labels appearing in ϕ, and 2. hB, , F i is representable (i.e., satisfies the conditions listed in Theorem 3.8). 16 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY The right-to-left direction does not require any bound on the size of B, so long as it is finite. In addition, it is decidable whether a finite labeled comparison algebra model is representable. The complexity of deciding whether a finite labeled comparison algebra model is representable is NP in the size of the underlying Boolean algebra of the labeled comparison algebra model. Proof. The right-to-left direction is immediate by Theorem 3.8 and Proposition 3.3. For the left-to-right direction, suppose M = hX, F, V i|= ϕ, and let ∆ be the set of set labels appearing in ϕ. As we described above, the field of sets model can be naturally turned into a measure algebra model B = hF, µ, V i and then into a labeled comparison algebra model C = hF, , F, V i such that M ≡LFin,Inf B, C. By adapting C to ∆ using Proposition 3.11, we obtain C∆ that satisfies the same formulas in LFin,Inf (∆), which includes ϕ. Note that C∆ is represented by B∆ . Also, since B∆ is adapted to ∆, the size of the Boolean algebra base of B∆ is at |∆| most 22 , and there are at most 2|∆| many atoms. To decide whether a finite labeled comparison algebra hB, , F, V i is representable, the only non-trivial part is to verify whether B|F satisfies the finite cancellation condition. However, rather than using this characterization, it is easier to naively check the definition of representability; this is an integer linear programming problem with at most 2|∆| (the number of |∆| atoms in B) many variables and (22 )2 many inequalities, the coefficients of which are all in {0, 1, −1}. According to a standard result on integer linear programming (see [16]), the complexity is N P in the size of B. ⊣ As a simple corollary, the sets of sentences in L or LFin,Inf valid on all field of sets models are decidable, as to decide whether ϕ is satisifiable (or equivalently whether ¬ϕ is valid) we can enumerate all labeled compari|∆| son algebra models of size up to 22 and for each one first check if it is representable and then check if ϕ is satisfied. If ϕ is never satisfied in this procedure, then ϕ is in fact unsatisfiable. §4. Canonical comparison algebra models. This section is devoted to the construction of canonical comparison algebra models. In Definition 2.7, we formulated a logic that captures our basic intuitions about the comparison of “sizes” of sets. For example, (BC2) says that the complement of the empty set is strictly larger than the empty set, which we should assume as otherwise all set terms would simply be empty. (BC3) and (BC4) together provide the order structure of the “sizes” of sets: it is a total preorder. (BC5) and (BC6) gives two basic interactions between set construction and size comparison: (BC5) says that if a set is a subset of another, then the size of the subset should be no greater than that of the superset, and (BC6) says that the union of two empty sets (those with THE LOGIC OF COMPARATIVE CARDINALITY 17 sizes no greater than the empty set) is still empty. (This is the union of the empty set with itself, though two different set terms denote it.) Those axioms may capture some notion of “size” comparison, but they are not enough to completely capture the notion of “cardinality” comparison. Loosely speaking, we may treat “cardinality” as a special kind of “size”, less general but perhaps more interesting, due to the distinct behaviors of finite and infinite cardinalities. The following theorem says that the basic comparison logic BasicCompLogic captures precisely the notion of “size” comparison in comparison algebra models defined in Definition 3.4. This will be useful, as to show that CardCompLogic captures the notion of “cardinality” comparison, we only need to show that the difference between the two logics captures the difference between the models: the extra properties identified in Theorem 3.8 that make a comparison algebra model representable by a cardinal measure. Lemma 4.1. BasicCompLogic derives the following for terms s, t, u, s′ , t′ : 1. s = t if it is provable in the equational theory of Boolean algebras. 2. ⊆ is a preorder: s ⊆ s, (s ⊆ t ∧ t ⊆ u) → s ⊆ u. 3. = is an equivalence relation: s = s, s = t → t = s, (s = t ∧ t = u) → s = u. 4. ⊆ works as the subset relation: s ⊆ t → tc ⊆ sc , s ⊆ t → (s ∩ u) ⊆ t, (s ⊆ t ∧ s ⊆ u) → s ⊆ (t ∩ u), and (s ⊆ u ∧ t ⊆ u) → (s ∪ t) ⊆ u. 5. = is a congruence relation: s = t → sc = tc and (s = s′ ∧ t = t′ ) → (s ∩ t) = (s′ ∩ t′ ). With axioms in CardCompLogicFin,Inf , s = t → (Fin(s) ↔ Fin(t)) and s = t → (Inf(s) ↔ Inf(t)) are also derivable. 6. Substitution of equal terms: s = t → (ϕ ↔ ψ) for all formulas ϕ and ψ in L where ψ is obtained from ϕ by replacing one or more occurrences of s by t. When using CardCompLogicFin,Inf , this substitution schema is valid for all ϕ ∈ LFin,Inf . Proof. If s = t is provable in the equational theory of Boolean algebras, then so are 0 = s∩tc and 0 = t∩sc . Then by axiom (BC8), both |∅| ≥ |s∩tc | and |∅| ≥ |t ∩ sc | are provable. But they are abbreviated by s ⊆ t and t ⊆ s. Putting them together, we have that s = t is provable. Note that s = s is obviously provable. So we have s ⊆ s as well. Now assume s ⊆ t and t ⊆ u. They abbreviate |∅| ≥ |s ∩ tc | and |∅| ≥ |t ∩ uc |. By (BC6), we have |∅| ≥ |(s ∩ tc ) ∪ (t ∩ uc )|. Note that the following is in the equational theory of Boolean algebras by distinguishing cases t and tc : 0 = (s ∩ uc ) ∩ ((s ∩ tc ) ∪ (t ∩ uc ))c . Hence by (BC7), we have (s ∩ uc ) ⊆ ((s ∩ tc ) ∪ (t ∩ uc )). Then by (BC5), we have |s ∩ uc | ≤ |(s ∩ tc ) ∪ (t ∩ uc )|. Combining this with what we derived by (BC6) above, using (BC4) we have |∅| ≥ |s ∩ uc |, which is just s ⊆ u. 18 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Part 3 follows directly from part 2 with just a few Boolean manipulations. Part 4 is also not hard using the same technique we used in part 2. The congruence over complementation and union in part 5 follows from part 4 by Boolean manipulations. The congruence over Fin is an easy consequence of (A3). Note also that using (A1), (A4), and (BC5), we have (Inf(s) ∧ s ⊆ t) → Inf(t). So we can also easily derive the congruence over Inf. Finally, to show substitution, we need to use two inductions. First, an induction on terms using part 5 will show that for any four terms s, t, u0 , u1 with u1 being the result of replacing some occurrences of s in u0 by t, we can derive s = t → u0 = u1 . Obviously for any terms s, t, we can derive s = t → |s| = |t|. So we proved substitution for atomic sentences in L. When we are in CardCompLogicFin,Inf , part 5 also provides substitution for the rest of the atomic sentences in LFin,Inf . Then a simple induction on formulas will do, since ↔ is again congruential over ¬ and ∧. ⊣ Theorem 4.2. For any set X of L-sentences that is maximally consistent relative to BasicCompLogic, there exists a comparison algebra model C X such that C X |= X. Proof. For terms s and t, define s ⊆X t iff s ⊆ t ∈ X, s X t iff |s| ≥ |t| ∈ X, s =X t iff s = t ∈ X, and s ≃X t iff s X t and t X s, for all s, t ∈ T (Φ). We also write s ≡ t when s = t is provable in the equational theory of Boolean algebras. Note that by the maximality of X and Lemma 4.1, =X is a congruence relation on T (Φ). Let B X = hT (Φ)/=X , ·c , ⊓i be the homomorphic image of the term algebra T (Φ), with the homomorphism [·]=X that sends a term to its equivalence class under =X , and [s]c=X = [sc ]=X , [s]=X ⊓[t]=X = [s∩t]=X . Since =X extends ≡ (by Lemma 4.1 again), B X is a Boolean algebra. The bottom element ⊥BX is obviously [∅]=X as it is the meet of [t]=X and [tc ]=X , but the latter is just [t]c=X . Regarding the Boolean lattice ordering in B X , note that: [s]=X ≤BX [t]=X ⇔ [s]=X ⊓ [t]c=X = ⊥BX ⇔ [s ∩ tc ]=X = [∅]=X ⇔ s ∩ tc =X ∅. It is also not hard to see that s ∩ tc =X ∅ iff s ⊆X t since BasicCompLogic derives s ∩ tc = ∅ ↔ s ⊆ t. Hence ≤BX is just ⊆X . Now we add a comparison structure to B X . Note that by (BC5), ≃X extends =X . So we can take the quotient X/=X , so that [s]=X X/=X [t]=X iff s X t. Let C X = hB X , X/=X , [·]=X i. Now we show that C X is a comparison algebra model: • We have just shown that ≤BX is identical to ⊆X . By (BC5), X extends ⊇X . So X extends ≥BX . THE LOGIC OF COMPARATIVE CARDINALITY 19 • Suppose ⊥BX X/=X [s]=X . Then by definition, ∅ X s. What we need is [s]=X = ⊥BX . To show this, we just need BasicCompLogic to derive |∅| ≥ |s| → s = ∅ as the antecedent is given by ∅ X s. First, ∅ ⊆ s is derivable trivially. To derive s ⊆ ∅, we need |∅| ≥ |s ∩ ∅c | by definition. Obviously s ∩ ∅ ≡ s. So we can substitute and then use the assumption that |∅| ≥ |s|. Finally, we verify that ϕ ∈ X iff C X |= ϕ. For the atomic case, consider a formula |s| ≥ |t| for arbitrary s, t ∈ T (Φ). Then |s| ≥ |t| ∈ X iff s X t ⊣ iff [s]=X X/=X [t]=X iff C X |= |s| ≥ |t|. The induction is trivial. §5. Completeness with predicates for infinite and finite sets. In this short section, we will prove Theorem 2.10, which says that the cardinality comparison logic with Fin and Inf, CardCompLogicFin,Inf , is sound and complete with respect to field of sets models. It is not hard to check that it is sound with respect to field of sets models as well as measure algebra models and labeled comparison algebra models. For completeness, we follow the standard strategy by starting with a consistent formula ϕ, building a canonical labeled comparison model satisfying ϕ, adapting it to the set labels appearing in ϕ so that we obtain a finite model, and finally using the fact that the canonical model must also satisfy all the axioms to show that it is representable. By Theorem 3.12, this means ϕ is satisfied by a field of sets model. Theorem 2.10. CardCompLogicFin,Inf is sound and complete with respect to the class of all measure algebra models and also the class of all field of sets models. Proof. Soundness is almost trivial. For completeness, we show that every formula ϕ that is consistent in CardCompLogicFin,Inf is also satisfied by a measure alegbra model. Since ϕ is consistent, let X be a maximally consistent set that contains ϕ. Since CardCompLogicFin,Inf includes BasicCompLogic ⊆ L, X|L = X ∩ L is a maximally consistent set for the logic BasicCompLogic in L. By the canonical model theorem (Theorem 4.2), C = hhT (Φ)/=X|L , ·c , ⊓i, X|L/=X|L , [·]=X|L i is a comparison algebra model and C |= X|L . Now we need to build an F ⊆ T (Φ)/X|L to interpret Fin and Inf. Define [s]=X|L ∈ F iff Fin(s) ∈ X. For this F to be well defined, we need to show that if s =X|L t, then Fin(s) ∈ X iff Fin(t) ∈ X. As shown in the beginning of the proof of Theorem 4.2, =X|L is extended by ≃X|L . Thus, once s =X|L t, both |s| ≥ |t| and |t| ≥ |s| are in X|L . By axiom (A3) in CardCompLogicFin,Inf and the maximality of X, this implies Fin(s) ∈ X iff Fin(t) ∈ X. 20 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY So we can define F = {[s]=X|L | Fin(s) ∈ X}. Then C F = hhT (Φ)/=X|L , ·c , ⊓i, X|L/=X|L , F, [·]=X|L i is a labeled comparison algebra model. For any s, t ∈ T (Φ), C F |= |s| ≥ |t| iff C |= |s| ≥ |t| iff |s| ≥ |t| ∈ X; and C F |= Fin(s) iff Fin(s) ∈ X. Because of axiom (A1) in CardCompLogicFin,Inf , Inf(s) ∈ X iff Fin(s) 6∈ X. So it follows that C F |= Inf(s) iff s 6∈ F iff Fin(s) 6∈ X iff Inf(s) ∈ X. Then a simple inductive argument on LFin,Inf shows that C F |= X. Hence, in particular, C F |= ϕ. In other words, ϕ is satisfied by the labeled comparison algebra model C F that also satisfies the axioms of CardCompLogicFin,Inf . Now we adapt C F to the finite set ∆ of the set labels appearing in ϕ. F is finite and satisfies the same By Proposition 3.11, the resulting model C∆ F formulas in LFin,Inf as C does. This implies that: F • C∆ |= ϕ. F F • Given that C∆ is adapted to ∆, every element b ∈ C∆ is equal to Vb (t) F for some t ∈ T (∆), where V is the valuation in C∆ . F • C∆ is representable, using Theorem 3.8. As we noted before Proposition 3.11, it is finite. Now we need to check the four conditions listed in Theorem 3.8. Condition (1) is guaranteed since all instances of F (A2) and (A3) are theorems and hence true of C∆ , which means we can apply (A2) and (A3) to every element as they are all named by terms. Hence it is clear that condition (1) is true. Similarly conditions (2), (3), and (4) are guaranteed by axioms (A5), (A6), and (A4), respectively. Then by Theorem 3.12, ϕ is satisfied by a field of sets model. This completes the proof of completeness. ⊣ §6. Completeness without predicates for infinite and finite sets. In this section, we define the logic CardCompLogic and prove Theorem 2.4. Theorem 2.4. The cardinality comparison logic CardCompLogic is sound and complete with respect to field of sets models. By earlier results, it suffices to consider only measure algebra models. The key idea is to find two formulas Fin and Inf in L to replace the two primitive predicates, Fin and Inf, added in LFin,Inf . The general strategy is as follows: 1. Show that there is a way to define formulas Fin(u) and Inf(u) in L, instead of adding the two extra predicates, to capture the finiteness or infiniteness of u on all adapted measure algebra models except a very special class of models, which we call flexible models in § 6.1. We use the name “flexible” because in those models we can change THE LOGIC OF COMPARATIVE CARDINALITY 21 the cardinality of an element to be anything finite or infinite without changing the comparative structure. 2. Then, in § 6.2, we give the axioms for CardCompLogic using the formulas Fin and Inf defined in § 6.1, and we show that any (adapted) comparison algebra model satisfying these axioms can be turned into a measure algebra model. This uses Theorem 3.8 and splits into two cases depending on whether the model is flexible or not. 3. Finally, in § 6.3, given a formula ϕ consistent with CardCompLogic, we use the canonical model construction of Theorem 4.2 to build a comparison algebra model satisfying ϕ. Then, using previous results in this section, we can assume that the model is adapted and hence turn it into a measure algebra model. 6.1. Flexible models. In this subsection, we define flexible models and show how they appear when we try to define Fin and Inf in L. Essentially, flexible models are models where our definition, or in fact any definition to capture Fin and Inf in language L, fails. This is because we can make the cardinality of an element in a flexible model anything we like, be it finite or infinite, without changing the formulas in L satisfied by that model. Definition 6.2. A finite measure algebra model B = hB, µ, V i is flexible if there is an atom a in B whose measure is strictly smaller than the measure of all other atoms in B, and a is the only atom in B with finite measure, if there is any atom with finite measure. The following two propositions show why we call such models flexible. Proposition 6.3. If B = hB, µ, V i is a flexible finite measure algebra model, then for any non-bottom element b ∈ B, we have µ(b) = max{µ(a) | a ∈ At(B), a ≤ b}. Proof. Write b as a finite join of the atoms below it: b = a1 ∨ · · · ∨ an . Then µ(b) = µ(a1 ) + · · · + µ(an ). If b is an atom, then µ(b) = µ(a1 ); otherwise, n ≥ 2 and at least one of µ(a1 ), . . . , µ(an ) is infinite, so µ(b) = max{µ(a1 ), . . . , µ(an )} = max{µ(a) | a ∈ At(B), a ≤ b}. ⊣ Proposition 6.4. For any flexible finite measure algebra model B = hB, µ, V i and cardinal κ, if a0 is the atom of B with the smallest measure, then there is a flexible finite measure algebra model C = hB, ν, V i such that: 1. B ≡L(∆) C; 2. ν(a0 ) = κ. In other words, for any flexible finite measure algebra model, the measure of the smallest atom does not matter, if we are only concerned with the truth of formulas in L(∆). 22 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY |(111)| = ℵ3 |(011)| = ℵ2 |(101)| = ℵ3 |(110)| = ℵ3 |(001)| = κ |(010)| = ℵ2 |(100)| = ℵ3 |(000)| = 0 Figure 3. A flexible measure algebra Proof. First, B ≡L(∆) C when for any b1 , b2 ∈ B, we have µ(b1 ) ≥ µ(b2 ) iff ν(b1 ) ≥ ν(b2 ). But notice that by Proposition 6.3, both µ(b) and ν(b) are calculated by taking maximums of the measures of the atoms below b, given that the hB, ν, V i is flexible. This means the condition for equivalence can be weakened to: for any two atoms a1 , a2 ∈ B, we have µ(a1 ) ≥ µ(a2 ) iff ν(a1 ) ≥ ν(a2 ). So we only need to have an order-preserving map for the measures of all atoms in B while keeping the flexibility. Thus, we can define ν on the atoms of B as follows: ( κ if a = a0 , the smallest atom ν(a) = ℵκ+i if |{µ(a′ ) | a′ ∈ At(B), µ(a′ ) < µ(a)}| = i where to compute ℵκ+i we view κ as an ordinal, i.e., as the least order type of a well-order of size κ. It is not hard to verify that C = hB, ν, V i is still a flexible finite measure algebra model and that for any a1 , a2 ∈ At(b), we have µ(a1 ) ≤ µ(a2 ) iff ν(a1 ) ≤ ν(a2 ). ⊣ Example 6.5. Figure 6.1 displays a particular flexible measure algebra (flexible model without the valuation). The comparison structure (illustrated by shaded areas in the same way as in Example 3.9) is the same regardless of what the cardinal κ is so long as 1 ≤ κ ≤ ℵ1 . Now we capture Fin and Inf in the language L by the following. THE LOGIC OF COMPARATIVE CARDINALITY 23 Definition 6.6. When ∆ ⊆ Φ is finite, define the Fin∆ (u) for any set term u ∈ T (∆) as:   |R| |R| _ ^ [ u = ri ∧ |si ∪ ti | > |si | ≥ |ti | ∧ |si ∪ ti | ≥ |ri |  R⊆T0 (∆) S,T ∈T0 (∆)|R| i=1 i=1 and then define Inf ∆ (u) for any set term u ∈ T (∆) as _ Inf ∆ (u) := (t ⊆ 6 s ∧ |u| ≥ |s| ≥ |s ∪ t|) . s,t∈T0 (∆) Here ri ranges over elements in R, and si , ti range over elements in sequences S and T , respectively. When no confusion arises, we may drop the subscript ∆. To understand this definition, recall that by basic cardinal arithmetic, the distinct feature of infinite sets is so-called absorption: if a set X is infinite, then |X| ≥ |X ∪ Y | whenever |X| ≥ |Y |, even if Y is not a subset of X. On the other hand, when Y is not a subset of X and yet |X| ≥ |X ∪ Y |, then X must be infinite. Hence, we can witness X’s finiteness by a set Y such that |X ∪ Y | > |X| ≥ |Y |. Note that this also shows that Y , and thus X ∪ Y , is finite. Similarly, we can witness X’s infiniteness by a set Y that is not a subset of X yet for which |X| ≥ |X ∪ Y |. Our definitions of Fin∆ and Inf ∆ are based on these simple observations. However, since they are meant to capture as many finite/infinite sets as possible, they must be slightly more complicated than simple cardinal arithmetic, as the intuitions such as “a finite union of finite sets is still finite” must also be incorporated into the definition of Fin∆ . Similarly, Inf ∆ also incorporates the intuition that if a set is no smaller than an infinite set, then they are both infinite. Intuitively, Fin∆ (u) says that u can be expressed as a union of finite sets, the ri ’s, with the finiteness witnessed by si ’s and ti ’s. On the other hand, intuitively Inf ∆ (u) says that there exists a set s whose cardinality is infinite yet u’s cardinality is no smaller, with the infiniteness of s witnessed by t. The following two propositions tell us precisely to what extent these formulas capture Fin and Inf. In sum, their truth forces the respective properties (finiteness and infiniteness) on adapted models, but not vice versa. However, (in adapted models) the other direction fails only on the smallest atom of flexible models. This is the best we can do in L, due to Proposition 6.4 and the existence of flexible models. Proposition 6.7. Fix a finite ∆ ⊆ Φ. For any adapted measure algebra model B = hB, µ, V i: 1. if B |= Fin∆ (u), then µ(Vb (u)) is finite; 24 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY 2. if B |= ¬Fin∆ (u) and yet µ(Vb (u)) is finite, then B is flexible and Vb (u) is the smallest atom in B. Proof. Suppose that B |= Fin∆ (u). Then there are sequences of terms ri , si , and ti in T (∆) that make the disjunct true. Let a = Vb (u), bi = Vb (ri ), ci = Vb (si ), and di = Vb (ti ). Then: 1. for each i, µ(ci ) ≥ µ(di ) but µ(ci ) < µ(ci ∪ di ); 2. for each W i, µ(bi ) ≤ µ(ci ∪ di ); 3. a = i bi . The first item ensures that ci and di have finite measure: di ’s cardinality is no greater than ci ’s, and ci cannot be infinite as otherwise the cardinality of ci ∪ di will not be strictly greater than that of ci . The second item says that bi is smaller than ci ∪ di , which is finite in measure. The third item says that a is a finite union of finite elements. So we have shown that when B |= Fin∆ (u), µ(Vb (u)) = µ(a) is finite. (We have not yet used the fact that B is adapted: Vb (T (∆)) = Vb (T0 (∆)) = B.) For the second claim, assume for contradiction that Vb (u) is not an atom in B. Vb (u) cannot be the bottom element, since then Fin∆ (u) is trivially true. So Vb (u) is neither the bottom nor an atom. This means there are two non-bottom elements a, b ∈ B that are below Vb (u), whose join is Vb (u), whose meet is bottom, and µ(a) ≥ µ(b). Since µ(Vb (u)) is finite, µ(a) and µ(b) must also be finite. Then µ(a) ≥ µ(b) but µ(a) < µ(a∨b) = µ(a)+µ(b), since b is not bottom and µ(b) > 0. From this, we have: • µ(a ∨ b) > µ(a) ≥ µ(b); • Vb (u) = a ∨ b. So B |= Fin∆ (u), a contradiction. Here, we are using the fact that Vb (T (∆)) = Vb (T0 (∆)) = B to get terms s and t with Vb (s) = a and Vb (t) = b. Now assume again for contradiction that Vb (u) is not the only atom finite in measure and in particular that a ∈ At(B), a 6= Vb (u), and µ(a) is finite. Then a ∧ Vb (u) = 0 and µ(a ∨ Vb (u)) = µ(a) + µ(Vb (u)) > µ(Vb (u)), since a is not bottom and µ(a) > 0. Then we again have a witness for Fin∆ (u), depending on which of Vb (u) and a is larger. For example, if a is larger, then Vb (u) is smaller than a, a does not absorb a smaller element Vb (u), and Vb (u) is smaller than the join of Vb (u) and a. Thus, we contradict B |= ¬Fin∆ (u). In summary, we have that Vb (u) is the only finite atom in B, which immediately shows that Vb (u) is the smallest atom in B and B is flexible. ⊣ Proposition 6.8. Fix a finite ∆ ⊆ Φ. For any adapted measure algebra model B = hB, µ, V i: 1. if B |= Inf ∆ (u), then µ(Vb (u)) is infinite; THE LOGIC OF COMPARATIVE CARDINALITY 25 2. if B |= ¬Inf ∆ (u) and yet µ(Vb (u)) is infinite, then B is flexible and Vb (u) is the smallest atom in B. Proof. After expanding the semantics, it is easy to see that Inf ∆ (u) expresses that there is an element b ∈ B such that b absorbs an element c not contained in b, and Vb (u) is no smaller than b in measure. The elements b and c are obtained by first picking out the true disjunct (t 6⊆ s ∧ |u| ≥ |s| ≥ |s ∪ t|) of Inf ∆ (u) and then taking b and c to be just Vb (s) and Vb (t), respectively. Then t 6⊆ s being true means that c is not contained in b, and |s| ≥ |s ∪ t| means c absorbs b. Hence µ(b) must be infinite, and since |u| ≥ |s|, µ(Vb (u)) is infinite too. For the second claim, first suppose that there is an atom a ∈ At(B) with µ(a) finite. Then for any atom b ∈ At(B) that is infnite in measure, a 6⊆ b and µ(b) ≥ µ(b ∨ a). Now µ(Vb (u)) is infinite, so Vb (u) must have an infnite atom b below it. Then Inf ∆ (u) is witnessed by b and a, a contradiction. Thus, there is no finite atom in B. Now suppose there is no strictly smallest atom—for any atom b, there is another atom a such that µ(b) ≥ µ(a). Then a 6⊆ b, and because they are both infinite in measure, µ(b) ≥ max{µ(b), µ(a)} = µ(b ∨ a). So by the same reasoning as the previous case, Inf ∆ (u) is witnessed, and we have a contradiction. Thus, there is also a strictly smallest infinite atom a0 in B. Finally, suppose Vb (u) 6= a0 . Then Vb (u) must be above another atom b, as it is infinite in measure and cannot be bottom. But then a0 6⊆ b and µ(b) ≥ max{µ(a0 ), µ(b)} = µ(b ∨ a0 ). So Inf ∆ (u) is still witnessed. In sum, Vb (u) is the strictly smallest infinite atom in B, so B is flexible. ⊣ 6.2. Representation using axioms of the language. It is now time to give the axioms for cardinality comparison in L that are not already in BasicCompLogic. Definition 6.9. Where ∆ ⊆ Φ is finite, define Axiom(∆) as the set containing all of the following formulas for all u, s, t ∈ T0 (∆): (C1) (C2) (C3) (C4) (C5) ¬(Fin∆ (u) ∧ Inf ∆ (u)); V (¬Fin∆ (u) ∧ ¬Inf ∆ (u)) → t∈T0 (∆) (|u| ≥ |t| → (t = ∅ ∨ t = u)); Vn i=1 (Fin∆ (si ) ∧ Fin∆ (ti )) → FCn (s1 , . . . , sn , t1 , . . . tn ); Inf ∆ (u) → ((|u| ≥ |s| ∧ |u| ≥ |t|) → |u| ≥ |s ∪ t|); (Inf ∆ (s) ∧ Fin∆ (t)) → |s| > |t|, where n ≥ 1, and s1 , . . . , sn , t1 , . . . , tn ∈ T0 (∆) are also all arbitrary. Given that Fin and Inf do not fully capture Fin and Inf, we cannot use Fin∆ (u) ⊕ Inf ∆ (u) like axiom (A1) for Fin and Inf, since it is outright invalid among all adapted measure algebra models. Instead, we have (C1) and (C2) here. Put together, they ensure that the only case when Fin∆ (u) 26 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY and Inf ∆ (u) fail to capture Fin and Inf is when we are in a flexible model and u is the smallest atom. As we have done for CardCompLogicFin,Inf , we prove a representation theorem using Axiom(∆). In other words, we show that Axiom(∆) is enough to force the comparison relation in an adapted comparison algebra model to be a comparison of cardinalities. To start, we need the following straightforward lemma. Lemma 6.10. Fix a finite ∆ ⊆ Φ. For any comparison algebra model B and s, t ∈ T (∆): 1. B |= |∅| ≥ |t| iff V (t) = ⊥B ; 2. B |= t ⊆ s iff V (t) ≤B V (s); 3. B |= t = s iff V (t) = V (s); 4. if B |= Fin∆ (t) ∧ s ⊆ t, then B |= Fin∆ (s); 5. if B |= Fin∆ (s) ∧ Fin∆ (t), then B |= Fin∆ (s ∪ t); 6. if B |= Inf ∆ (t) ∧ |s| ≥ |t|, then B |= Inf ∆ (s). Proof. The first item follows from the requirement that ⊥B 6 b for all b ∈ B \ {⊥B }. The second and third follow easily. For the fourth item, suppose B |= Fin∆ (t) ∧ s ⊆ t. Then by definition, we have terms ri as the witnesses of the finiteness of s. It is easy to see that ri ∩ s’s are witnesses of the finiteness of s. Similarly for the fifth item, the witnesses of s ∪ t are just the union of witnesses for s and witnesses for t. The sixth item is even easier, as the same witness works. ⊣ Theorem 6.11. Fix a finite ∆ ⊆ Φ. Let B be an adapted comparison algebra model such that B |= Axiom(∆). Then there is a µ such that m(B) = hB, µ, V i is a finite measure algebra model representing B and hence B ≡L(∆) m(B). Proof. Since B is adapted to ∆, every element in B is named by some term in T (∆). Thus, given b ∈ B we may write ϕ(b) for the sentence ϕ(tb ) where tb ∈ T (∆) and V (tb ) = b. By axiom (C1) in Axiom(∆), there are two cases: Case 1: there is a b ∈ B such that B |= ¬(Fin∆ (b) ∨ Inf ∆ (b)); Case 2: for any b ∈ B, B |= Fin∆ (b) ⊕ Inf ∆ (b). In both cases, we want to obtain an F ⊆ B, so that using F as the labeling set, hB, , F, V i is a labeled comparison algebra model satisfying all the conditions in Theorem 3.8. Case 1: First, we show that there is a unique atom a0 ∈ At(B) such that B |= ¬(Fin∆ (a0 ) ∨ Inf ∆ (a0 )), and that for all other atoms b, we have b ≻ a0 . Suppose for a0 , a1 ∈ B that we have both B |= ¬Fin∆ (ai ) ∧ ¬Inf ∆ (ai ) for i ∈ {0, 1}. Then by axiom (C2) and Lemma 6.10, for any b ∈ B, if a0 b, then b = a0 or b = ⊥B . But for any b ∈ B that is below a0 in the Boolean THE LOGIC OF COMPARATIVE CARDINALITY 27 algebra, i.e., b ≤ a0 , it is also true that b a0 , by the definition of a comparison algebra. So whenever b ≤ a0 , we have b a0 and hence b is a0 or ⊥B . This means that a0 is an atom in B, as it must not be ⊥B , since B |= Fin∆ (⊥B ). In exactly the same fashion we can show that a1 is also an atom. Now is a total preorder. So either a0 a1 or a1 a0 . But in either case, they must be equal, as they cannot be ⊥B , but axiom (C2) says they are either equal or are bottom. Thus, there is a unique atom a0 such that B |= ¬Fin∆ (a0 ) ∧ ¬Inf ∆ (a0 ). Building on the previous conclusion, for any a ∈ At(B) \ {a0 }, we have B |= Fin∆ (a)⊕Inf ∆ (a). The second step is to show that in fact B |= Inf ∆ (a). Consider a ∪ a0 . If B |= (a ∪ a0 ) ≻ a, then we have B |= (a ∪ a0 ) ≻ a a0 ∧ (a ∪ a0 ) a0 . It is not hard to see that then B |= Fin∆ (a0 ), a contradiction. Hence B |= a (a ∪ a0 ) instead. However, B |= a0 6⊆ a since a0 and a are distinct atoms. So B |= Inf ∆ (a). Thus we see that there is no element in B satisfying Fin except the bottom element. So define F = {⊥B }. We can show that Theorem 3.8 can be applied to hB, , F, V i. In fact, the only nontrivial condition is the third condition: for any a, b, c ∈ B with a 6∈ F , if a b and a c, then a b ∨ c. There are two cases: • a = a0 : then a is the smallest atom, and b, c are either ⊥B or a0 and so is b ∨ c; hence a b ∨ c; • a 6= a0 : as we have shown, now B |= Inf ∆ (a); thus by axiom (C4) in Axioms(∆), a b ∨ c. Hence we can invoke Theorem 3.8 to build µ. Case 2: Define F = {b ∈ B | B |= Fin∆ (b)}. By Lemma 6.10, F is an ideal. Axioms (C3), (C4), and (C5) in Axioms(∆) ensure conditions (2), (3), and (4) in Theorem 3.8. Thus, Theorem 3.8 applies again. ⊣ 6.3. Completeness. Finally we are in a position to present the logic of cardinal comparison CardCompLogic. Definition 6.12. Let CardCompLogic be the logic for L with the following axioms and rules: 1. all axioms and rules in BasicCompLogic; 2. for any finite ∆ ⊆ Φ, all formulas in Axioms(∆); Now we show that CardCompLogic is sound and complete with respect to all measure algebra models and also field of sets models, completing our proof of Theorem 2.4. Theorem 6.13 (Soundness). CardCompLogic is sound with respect to the class of all measure algebra models. Since every field of sets model can be equivalently turned into a measure algebra model in an obvious way, CardCompLogic is also sound on the class of all field of sets models. 28 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Proof. The only non-trivial axioms are in Axioms(∆) for an arbitrary finite ∆. Given an arbitrary measure algebra model B = hB, µ, V i, by Proposition 3.11 there is a measure algebra model B∆ which is adapted to ∆ and which satisfies B∆ ≡L(∆) B. So to show that B |= Axioms(∆), it is enough to show B∆ |= Axioms(∆). Hence we only need to show that all sentences in Axioms(∆) are true on any adapted measure algebra model B. Fix an arbitrary such B = hB, µ, V i; we check that all the sentences in Axioms(∆) are true in B: • ¬(Fin∆ (u) ∧ Inf ∆ (u)): By Propositions 6.7 and 6.8, B |= Fin∆ (u) implies that µ(Vb (u)) is finite, and B |= Inf ∆ (u) implies that µ(Vb (u)) is infinite. But µ(Vb (u)) cannot be both finite and infinite. Therefore, B |= ¬(Fin∆ (u) ∧ Inf ∆ (u)).V • (¬Fin∆ (u) ∧ ¬Inf ∆ (u)) → t∈T0 (∆) (|u| ≥ |t| → (t = ∅ ∨ t = u)): Suppose B |= ¬Fin∆ (u) ∧ ¬Inf ∆ (u). Then by the second part of Proposition 6.7, if µ(Vb (u)) is infinite, then Vb (u) is the strictly smallest atom in B. Then it follows that ^ B |= (|u| ≥ |t| → (t = ∅ ∨ t = u)). t∈T0 (∆) Similarly, by Proposition 6.8, if µ(Vb (u)) is finite, the above statement holds Vn as well. So indeed the formula is valid. • i=1 (Fin∆ (si ) ∧ Fin∆ (ti )) → FCn (s1 , . . . , sn , t1 , . . . tn ): By Proposition 6.7, when B |= Fin∆ (si ) ∧ Fin∆ (ti ), Vb (si ) and Vb (ti ) are indeed finite. But as we have explained both in § 2 and immediately after Theorem 3.7, the consequent (finite cancellation axiom) is clearly valid for elements of finite cardinality. • Inf ∆ (u) → ((|u| ≥ |s1 | ∧ |u| ≥ |s2 |) → |u| ≥ |s1 ∪ s2 |): By Proposition 6.8, Vb (u) is infinite, and the consequent expresses a simple property of elements of infinite cardinality. • (Inf ∆ (s1 ) ∧ Fin∆ (s2 )) → |s1 | > |s2 |: Using Propositions 6.7 and 6.8 again, this says that when µ(Vb (s1 )) is infinite and µ(Vb (s2 )) is finite, then µ(Vb (s1 )) is greater than µ(Vb (s2 )), which is trivial. ⊣ Theorem 6.14 (Completeness). CardCompLogic is complete with respect to the class of all field of sets models: every valid formulas is derivable in CardCompLogic. Proof. We show that any formula that is consistent is satisfied by a measure algebra model. By Proposition 3.3, it is also satisfied by a field of sets model. THE LOGIC OF COMPARATIVE CARDINALITY 29 Suppose ϕ is consistent. Then let X be a maximally consistent set of CardCompLogic containing ϕ. Using the canonical model theorem (Theorem 4.2), we obtain a comparison algebra model C |= X. Now let ∆ be the set of all set labels appearing in ϕ. Then ∆ is finite. By Proposition 3.11, there is a comparison algebra model C∆ that is adapted to ∆ and that satisfies C∆ ≡L(∆) C. Since Axioms(∆) ⊆ X, C |= Axioms(∆), and since Axioms(∆) ∪ {ϕ} ⊆ L(∆), C∆ |= Axioms(∆) ∪ {ϕ}. Now the representation theorem (Theorem 6.11) can be applied to C∆ , and we obtain a measure algebra model B such that B ≡L(∆) C∆ . Thus B |= ϕ. So ϕ is satisfied on a measure algebra model. ⊣ Thus, the question with which we began—what are the laws one must add to Boolean algebra to capture reasoning about the relative size of sets according to Cantor’s definition?—is answered by the laws of CardCompLogic. §7. Open problems. In our proofs, we quickly passed to finite models, that is, models with only finitely many sets (some of which may of course be infinite). For example, our representation theorem (Theorem 3.8) applies only to finite models, and in Theorems 2.10 and 2.4, we proved completeness rather than strong completeness. Problem 7.1. Find a logic that is sound and strongly complete with respect to field of sets models. Problem 7.2. Prove a representation theorem for infinite comparison algebras. We will give some examples that show the difficulties that arise here. First, such a logic cannot be compact. Indeed (in the language LFin,Inf ), with distinct set terms hsn in∈ω and t, the following set of formulas is finitely satisfiable in field of sets models, but not satisfiable: {|sn | < |sn+1 | | n ∈ ω} ∪ {|sn | ≤ |t| | n ∈ ω} ∪ {Fin(t)}. One can give similar examples in the language L. Then to obtain a strongly complete logic, one might add an infinitary rule stating that if the sentences in {|sn | < |sn+1 | | n ∈ ω} ∪ {|sn | ≤ |t| | n ∈ ω} are derivable, then so is Fin(t). Another interesting example comes from the fact that the relation of cardinality comparison must be well-founded (assuming the axiom of choice). Thus, if hsn in∈ω is a sequence of distinct set terms, then the set of sentences {|sn+1 | < |sn | | n ∈ ω} is not satisfiable in field of sets models, but it is again finitely satisfiable. 30 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Note also that finite cardinalities are just natural numbers, whose ratios are all rational. However, with infinitely many formulas, we can express that the sizes of two sets are of irrational ratio. To do this, define the following set of formulas: A = {|ai | = |aj |, |bi | = |bj | | i, j ∈ ω} ∪ {|ai ∩ aj | = ∅ ∧ |bi ∩ bj | = ∅ | i, j ∈ ω, i 6= j}. Intuitively, this says that the ai ’s are disjoint and of the same size, and the same holds for the bi ’s. Then we can approximate any ratio by using ai ’s and bi ’s. For example, consider sequences √ hli ii∈ω , hri ii∈ω , hni ii∈ω of natural numbers such that li /ni approaches 2 from below and ri /ni from above. Then let [ [ [ ak | | i ∈ ω}. bk | < | ak | < | B = {| k<li k<ni k<ri The set A ∪ B is then finitely √ satisfiable, but not satisfiable as it forces the ratio of |a0 | and |b0 | to be 2. As the last example of non-compactness, suppose that we allow more than countably many set labels in the language. Let |Φ| = ℵ1 . Then the set of sentences {|a| = 6 |b| | a, b ∈ Φ, a 6= b} ∪ {Fin(a) | a ∈ Φ} is not satisfiable. However, any countable subset is satisfiable. A natural extension of our language is to add the powerset operation. In this case, one must replace the complement operation with the relative complement operation s \ t. Then a field of sets model (with powerset) is a collection F of sets closed under intersection, union, relative complement, and powerset, together with a valuation of the set labels. Problem 7.3. Axiomatize the logic of cardinality comparison with the powerset operation. In this language, one can consider principles such as |s| < |t| → |P(s)| < |P(t)|, which is true under GCH but is independent of ZFC [10]. It would be interesting to have a logic for comparing such principles. Acknowledgement. We wish to thank Johan van Benthem, an audience at BLAST 2018, and the anonymous referee for The Journal of Symbolic Logic for their helpful feedback. REFERENCES [1] John P. Burgess, Axiomatizing the logic of comparative probability, Notre Dame Journal of Formal Logic, vol. 51 (2010), no. 1, pp. 119–126. THE LOGIC OF COMPARATIVE CARDINALITY 31 [2] Domenico Cantone, A survey of computable set theory, Le Matematiche, vol. 43 (1988), no. 1, 2, pp. 125–194. 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[10] Thomas Jech, Set theory, Springer, Berlin, 2003. [11] Charles H. Kraft, John W. Pratt, and A. Seidenberg, Intuitive probability on finite sets, The Annals of Mathematical Statistics, vol. 30 (1959), no. 2, pp. 408– 419. [12] Paolo Mancosu, Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?, The Review of Symbolic Logic, vol. 2 (2009), no. 4, p. 2009. [13] Lawrence S Moss, Syllogistic logic with cardinality comparisons, J. michael dunn on information based logics, Springer, 2016, pp. 391–415. [14] Lawrence S Moss and Charlotte Raty, Reasoning about the sizes of sets: Progress, problems and prospects, Proceedings of the fourth workshop on bridging the gap between human and automated reasoning (Claudia Schon, editor), CEUR Workshop Proceedings, 2018, pp. 33–39. [15] Lawrence S. Moss and Selçuk Topal, Syllogistic logic with cardinality comparisons, on infinite sets, The Review of Symbolic Logic, (2018), pp. 1–22. [16] Christos H. Papadimitriou, On the complexity of integer programming, Journal of the ACM (JACM), vol. 28 (1981), no. 4, pp. 765–768. [17] Dana Scott, Measurement structures and linear inequalities, Journal of Mathematical Psychology, vol. 1 (1964), pp. 233–247. [18] Krister Segerberg, Qualitative probability in a modal setting, Proceedings of the Second Scandinavian Logic Symposium, vol. 63 (1971), pp. 341–352. Appendix A. Polarizability rule and finite cancellation axiom schema. Let CardCompLogic′ Fin,Inf be the system obtained by adding axiom schemas (A1)–(A4), (A6), (A7), and (A8) to BasicCompLogic. In this appendix, we discuss how (A5) can be derived in CardCompLogic′ Fin,Inf . First, we verify that the rule is sound in the sense that if the premise a|t → ϕ is valid, then the conclusion ϕ is also valid. Proposition A.1. The polarizability rule (A8) is sound on field of sets models. 32 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Proof. Suppose ϕ is not valid, so ¬ϕ is satisfiable. Our goal is to show that a|t ∧ ¬ϕ is satisfiable. Let ∆ be the set of set labels in ϕ or t. Then by the constraint of (A7), a 6∈ ∆. The strategy is simple: take a model M of ¬ϕ; construct the disjoint union N of two copies of M with a valued to exactly one copy of M; then we have that both a|t and ¬ϕ are true in N . Formally, take a field of sets model M = hX, F, V i that makes ¬ϕ true. Let X ′ = X × {0, 1}. Define a duplication function d : ℘(X) → ℘(X ′ ) by d(S) = S × {0, 1}. Then d ◦ V is a valuation on hX ′ , ℘(X ′ )i. Define V ′ such that if a ∈ ∆, then V ′ (a) = d(V (a)), and otherwise V ′ (a) = X × {0}. Let N = hX ′ , ℘(X ′ ), V ′ i. A simple induction shows that for any term c′ (t) = d(Vb (t)) = Vb (t) × {0, 1}. This implies that for any two t ∈ T (∆), V terms s, t ∈ T (∆), M |= |s| ≥ |t| iff N |= |s| ≥ |t|. In addition, a set S is finite iff S × {0, 1} is finite. Another simple induction then shows that M and N satisfy the same formulas in LFin,Inf using only labels in c′ (a) = X × {0} while ∆. In particular, N |= ¬ϕ. Since a 6∈ ∆, we have V ′ c b V (t) = V (t) × {0, 1}. Thus, N |= a|t. ⊣ While it is not hard to understand the content of the polarizability rule itself, it is harder to see what it can prove and how it can be used. Kraft, Pratt, and Seidenberg famously observed in [11] that without the polarizability rule (A8), the remaining system does not capture all valid reasoning patterns for finite sets, contrary to a conjecture of de Finetti [4]. For compact notation, we use the standard set theoretical definition of n = {0, 1, . . . , n − 1} and do not distinguish a sequence of length n and a function with domain n. We let n 2 denote the set of such functions/sequences with codomain 2. Then FCn can be defined under this notation by the following. Definition A.2. For each n, m ∈ N, sequence ~s = hs0 , · · · , sn−1 i of n terms, and f ∈ n 2, define the term \ \ ~s[f ] = {si | f (i) = 1} ∩ {sci | f (i) = 0} and the term Nm (~s) = [ {~s[f ] | f : n → 2 and |f −1 (1)| = m}. For each f ∈ n 2, ~s[f ] is intuitively a “definable” atom (in the Boolean algebra of terms constructible from ~s), and Nm (~s) is then the union of atoms that appear in exactly m terms in ~s. Given two sequences ~s and ~t of n terms, we can then define the formula ^ ~s E ~t = (Ni (~s) = Ni (~t)). 0≤i≤n THE LOGIC OF COMPARATIVE CARDINALITY 33 Recall that equality between terms is defined in Definition 2.1. Then ^ |si | ≥ |ti |) → |tn−1 | ≥ |sn−1 |). FCn (~s,~t) = ~s E ~t → (( i<n−1 Consequently (A5) is now (A5) (Fin(~s) ∧ Fin(~t)) → FCn (~s,~t), where Fin is extended to sequences of terms in the obvious way. Now we show that we can derive (A5) in CardCompLogic′Fin,Inf . The main strategy is to repeatedly use (A8) so that (A7) can be applied. In fact, Kraft, Pratt, and Seidenberg already sketched a proof of this in [11] for their Theorem 5. More specifically, the idea is the following, assuming that we are dealing with only finite set terms (for convenience we often speak loosely of terms as sets, say that one set is a subset of another when the relevant formula involving terms is provable, etc.): 1. Given two sequences ~s and ~t of length n, use the polarizability rule (A8) to keep polarizing atomic terms (minimal regions in the Venn diagram) definable from the terms in ~s and ~t until each atomic term is split into 2n ≥ 2n pieces of equal size. 2. Now each si and ti are unions of definable atomic terms. For each si , define s′i to be the union of the ith piece of the definable atomic terms that are subsets of si (so for example, if there are just s1 and s2 , then s′1 is the union of the first piece of s1 ∩ s2 and the first piece of s1 ∩ sc2 ). Similarly define t′i by using the (i + n)th pieces. 3. Then intuitively s′i and t′i are disjoint representatives of si and ti : for any i 6= j, s′i , s′j , t′i , and t′j are all disjoint, and for each i, |s′i | = 21n |si | and |t′i | = 21n |ti |. P P 4. Recall that P intuitively, when ~s E ~t, we have i<n |si | = i<n |ti |. P This means i<n |s′i | = i<n |t′i | as we just need to scale both sides by 21n . Also, since now the primed versions of si and ti are disjoint, the sum of S the sizes is S just the size of the union. So intuitively we ′ ′ s s E ~t. = should get i<n ti . Indeed, this is derivable from ~ i<n i 5. Using (A7), which deals with disjoint unions, we can then derive V ( i<n−1 |s′i | ≥ |t′i |) → |t′n−1 | ≥ |s′n−1 |. But recall that intuitively |s′i | and |t′i | are just 21n of |si | and |ti |. Formally, this means that |si | ≥ |ti | ↔ |s′i | ≥ |t′i | is derivable for any i < n. So the real consequent of FCn (~s,~t) is derivable. The rest of this section implements the sketch above in the formal system CardCompLogic′Fin,Inf . Now we start with a lemma showing that for disjoint finite sets, cardinality comparison works as intended. Note that we have proved that theorems are closed under substitution in Lemma 4.1. Hence we will use substitution freely without explicit reference. 34 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY Lemma A.3. For any sequence ~s of n terms, define the disjointness of terms in ~s by ^ D(~s) := (si ∩ sj ) = ∅. 0≤i<j<n CardCompLogic′Fin,Inf Then terms: derives the following with ~s a sequence of 2n ^ [ [ (1) D(~s) ∧ Fin(~s) → si+n | ; si | ≥ | |si | ≥ |si+n | → | i<n (2) D(~s) ∧ Fin(~s) → (3) D(~s) ∧ Fin(~s) ∧ | ^ [ i<n i<n |si | = |si+n | → | si | = | ^ i<n−1 [ i<n i<n i<n [ i<n si | = | [ i<n si+n | ; si+n | → |si | ≥ |si+n | → |sn−1 | ≤ |s2n−1 | . Proof. Note that (2) follows directly from (1). Also, we need only prove the case for n = 2, as the general formula can then be derived inductively. Suppose now that D(~s) ∧ Fin(~s) holds with n = 2. Then consider the following three terms: s01 = s0 ∪ s1 , s12 = s1 ∪ s2 , and s23 = s2 ∪ s3 . Using BasicCompLogic, we have s01 ∩ sc12 = s0 , s12 ∩ sc23 s12 ∩ sc01 = s2 , s23 ∩ sc12 = s3 . = s1 , So by (A7), we have |s01 | ≥ |s12 | ↔ |s0 | ≥ |s2 |, |s12 | ≥ |s23 | ↔ |s1 | ≥ |s3 |. Hence, we get (|s0 | ≥ |s2 | ∧ |s1 | ≥ |s3 |) → |s01 | ≥ |s12 |. This shows (1). Also, when |s0 | ≥ |s2 |, suppose further that ¬|s3 | ≥ |s1 |, that is, |s1 | > |s3 |. Then |s01 | ≥ |s12 | > |s23 |. Hence |s01 | > |s23 |, contradicting |s01 | = |s23 |. For induction, we just need to consider the union of the first n − 1 sets, the nth set, the next n − 1 sets, and the last set as a four-set sequence. ⊣ Proposition A.4. CardCompLogic′Fin,Inf derives (A5). Proof. Take an arbitrary sequence ~s of 2n terms. Let ~s< be the sequence of the first n terms in ~s and ~s> that of the last n terms. Similarly, for any function f ∈ 2n 2, define f< to be the restriction of f on n and f> the restriction of f on {n, n + 1, · · · , 2n − 1}. Our final goal is to derive ^ (4) (Fin(~s) ∧~s< E ~s> ) → |si | ≥ |si+n | → |sn−1 | ≤ |s2n−1 | . i<n−1 THE LOGIC OF COMPARATIVE CARDINALITY 35 As we mentioned above, our strategy will be to “disjointify” ~s so that we can use (3) in Lemma A.3. This is done by constructing in each si a subset s′i so that hs′i ii<2n is a sequence of pairwise disjoint sets while each s′i is 21n of si . Then Lemma A.3 can be applied. More formally, our plan is to use the polarizability rule (A8) to construct a term s′i for each i < 2n so that the following three formulas are derivable: ^ (5) s′i ⊆ si ) ∧ D(hs′i ii<2n ); ( i<2n (6) (7) Fin(~s) → ( ^ i<n ~s< E ~s> → | (|s′i | ≥ |s′i+n | ↔ |si | ≥ |si+n |)); [ i<n s′i | = | [ i<n s′i+n |. Once the three formulas are derived, it is then quite obvious that the system can derive (4) with the help of (3). Hence the rest of this proof is devoted to the construction of s′i and s′i+n and the derivation of (5)–(7) above. Indented passages marked with a vertical line give details that may be skipped on a first reading. Polarization and construction. By repeated use of (A8), for any f ∈ 2n 2, we can also assume that ~s[f ] is polarized into 2n many pieces. Let us enumerate the partitions of~s[f ] by ~s[f ][i] with i < 2n . Let us also generalize the notation of ~s[f ][i] to ~s[F ][I] where F ⊆ 2n 2, I ⊆ 2n , defined by [ {~s[f ][i] | f ∈ F, i ∈ I}. Then we abbreviate ~s[{f }][I] as ~s[f ][I] and ~s[F ][{i}] as ~s[F ][i]. Now define Ci = {f ∈ 2n 2 | f (i) = 1} for i < 2n. The equation ~s[Ci ] = si is in the equational theory of Boolean algebras and hence is derivable in our system. Then for any i < 2n, our s′i used in the outline above is defined by ~s[Ci ][i] (note that for any n ≥ 1, 2n ≥ 2n). In Figure 4, we use a grid to illustrate the partition resulting from polarization. Each column is an ~s[f ] for some f ∈ 2n 2. Each cell is then an ~s[f ][i]. We shade ~s[Ci ][i] for i = 0, 1, 2, 3, each in its own row; note that they are disjoint, and each is 1/4 the size of the corresponding s~i . This is essentially Figure 1 but since there are 4 sets, we choose not to draw a Venn diagram in the usual way. The indented passage provides more details on the construction of ~s[f ][i]: We can prepare for each ~s[f ] and each natural number l < n a set of 2l many fresh set labels. For convenience, we can just use functions in l 2. Now, we can first assume that the empty function ε polarizes ~s[f ]: ε|~s[f ]. This gives us two sets: ~s[f ] ∩ ε and ~s[f ] ∩ εc . Then we can inductively polarize the generated sets. 36 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY ~s[C3 ][3] ~s[C2 ][2] ~s[C1 ][1] ~s[C0 ][0] 3 2 1 0 11 11 0 1 11 1 0 11 0 0 11 1 1 10 0 1 10 1 0 10 0 0 10 1 1 01 0 1 01 1 0 01 0 0 01 1 1 00 0 1 00 1 0 00 0 0 00 Figure 4. Polarization and construction when n = 2. Squares in the same row can be of different sizes. But squares in the same column must be of the same size. In fact, for each function g ∈ l 2 with l < n, we can define \ ~s[f ][g] = ~s[f ] ∩ g<k c·g(j) , k<l where g<k is g restricted to k and c·x is c if x is 1 and empty otherwise. Then (A8) allows us to assume that for all 0, 1 sequences g of length at most n − 1, g|~s[f ][g], or equivalently by our definition, |~s[f ][hg, 0i]| = |~s[f ][hg, 1i]|. Of course, due to the restriction of (A8), we need to arrange those formulas so that S those with shorter variables come first. Fix an enumeration hgi i of l<n l 2 so that if gj extends gi then j ≥ i. Then formally we are using (A8), so that it suffices to prove g2n −1 |~s[f ][g2n −1 ] → (· · · (gi |~s[f ][gi ] → (· · · (g0 |~s[f ][g0 ] → ϕ) · · · )) · · · ) when we want to prove ϕ. Hence, from now on, we have that each ~s[f ] is polarized into 2n many pieces, enumerated by ~s[f ][g] with g ∈ n 2. Deriving formula (5). Since for all f ∈ Ci , ~s[f ][i] ⊆ si is obviously derivable, we have ~s[Ci ][i] ⊆ si . Hence the first part of (5) can be derived. Disjointness is slightly less trivial. Recall that by our definition of ~s[f ][i], for any f ∈ 2n 2, ~s[f ][i] ∩ ~s[f ][j] = ∅ is derivable when i 6= j. Thus when relativized to each ~s[f ], ~s[Ci ][i] and ~s[Cj ][j] are disjoint for i 6= j. Some simple Boolean equational theory will then show that ~s[Ci ][i] and ~s[Cj ][j] themselves are disjoint. Deriving formula (6). Assume Fin(~s). Note that for any f ∈ 2n 2 that is not constantly 0, there is an i < 2n such that~s[f ] ⊆ si is derivable: just pick i with f (i) = 1. Hence, using (A2) and (A3), for any I ⊆ 2n and F ⊆ 2n 2 with the constantly 0 function not in F , the system derives Fin(~s[F ][I]). Then, by repeated use of Lemma A.3, the system derives that for any i, j < 2n and f ∈ 2n 2 with f not constantly 0, |~s[f ][i]| = |~s[f ][j]|. THE LOGIC OF COMPARATIVE CARDINALITY 37 Recall how we defined ~s[f ][i] by polarization. We can in fact use a simple induction on 0 < l < n to show that for each l and g0 , g1 ∈ l 2, |~s[f ][g0 ]| = |~s[f ][g1 ]| is derivable. The base case is when l = 1 and g0 = h0i, g1 = h1i. Here what we need to show is already assumed when we apply (A8): ε|~s[f ], as this is defined precisely as |~s[f ][h0i]| = |~s[f ][h1i]|. To go from l to l + 1, note that any function in l+1 2 is obtained by appending a 0 or 1 to functions in l 2. So it is enough to show that for any g0 , g1 ∈ l 2, the four sets in the sequence ~t = h~s[f ][hg0 , 0i],~s[f ][hg0 , 1i],~s[f ][hg1 , 0i],~s[f ][hg1 , 1i]i are of equal size. In the previous (unindented) paragraph, we have derived Fin(~s[F ][I]) for any F and I, and hence we have derived Fin(~t). It is also obvious that the system can derive D(~t) using the equational theory of Boolean algebras. By the induction hypothesis, we also have that the union of the first two and the last two are of equal size. Hence we can apply (3) to ~t and obtain ~s[f ][hg0 , 0i] ≥ ~s[f ][hg1 , 0i] → ~s[f ][hg1 , 1i] ≥ ~s[f ][hg0 , 1i]. By switching the first two and the second two sets in ~t and applying (3) again, we get ~s[f ][hg0 , 0i] ≤ ~s[f ][hg1 , 0i] → ~s[f ][hg1 , 1i] ≤ ~s[f ][hg0 , 1i]. Now |~s[f ][hg0 , 0i]| = |~s[f ][hg0 , 1i]| and |~s[f ][hg1 , 0i]| = |~s[f ][hg1 , 1i]| are derivable since we have assumed when using the polarizability rule (A8) that g0 |~s[f ][g0 ] and g1 |~s[f ][g1 ]. With the transitivity of ≥ encoded by axiom (BC3), we can derive that the four sets involved are all equal in size. This shows that the 2n subsets of ~s[f ] obtained by polarization are of equal size whenever f is not constantly 0. Since Ci does not contain the constantly 0 function and ~s[Ci ][j] is a disjoint union of ~s[f ][j] with f ∈ Ci , using (2) we have |~s[Ci ][j]| = |~s[Ci ][k]| for any i < n and j, k < 2n . Now we can start to derive the consequent of (6). Fix an i < n. The idea is simple: |~s[Ci ][i]| > |~s[Ci+n ][i + n]| iff for any j, |~s[Ci ][j]| > |~s[Ci+n ][j]|. Summing over j, this is equivalent to |~s[Ci ]| > |~s[Ci+n ]|. Of course, the equivalences must be derived by Lemma A.3 and in particular (3). First, since both Ci and Ci+n do not include the constantly 0 function, we have Fin(~s[Ci ][j]) and Fin(~s[Ci+n ][j]). With (A7), we have for all j < 2n , |~s[Ci ][j]| ≥ |~s[Ci+n ][j]| ↔ |~s[Ci \ Ci+n ][j]| ≥ |~s[Ci+n \ Ci ][j]|. Let ~t be the sequence of 2 × 2n terms with the first 2n terms being h~s[Ci \ Ci+n ][j]ij<2n and the rest being h~s[Ci+n \ Ci ]ij<2n . Also let ~t′ 38 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY be the same as ~t except that the first 2n terms and the last 2n terms are switched. Then D(~t) and D(~t′ ) are derivable. This is because for any two terms, if they do not share the same second coordinate, then they are certainly disjoint. But if they do share the same second coordinate, then they are of the form ~s[Ci \ Ci+n ][j] and ~s[Ci+n \ Ci ][j], which are disjoint. Obviously we also have Fin(~t) and Fin(~t′ ). Now, from left to right, suppose |~s[Ci ][i]| ≥ |~s[Ci+n ][i + n]|. Then, for any j < 2n , we have |~s[Ci ][j]| ≥ |~s[Ci+n ][j]|. By (A7), this implies |~s[Ci \ Ci+n ][j]| ≥ |~s[Ci+n \ Ci ][j]|. Then we can apply (2) to ~t and obtain |~s[Ci \ Ci+n ]| = |~s[Ci+n \ Ci ]|. But by (A7) again, this gives us |~s[Ci ]| ≥ |~s[Ci+n ]|. From right to left, assume |~s[Ci ]| ≥ |~s[Ci+n ]| and suppose for contradiction that |~s[Ci+n ][i + n]| > |~s[Ci ][i]|. Then for any j < 2n , we have |~s[Ci+n ][j]| > |~s[Ci ][j]|. By (A7), this implies |~s[Ci+n \ Ci ][j]| > |~s[Ci \ Ci+n ][j]|. Thus, in sequence ~t′ the first 2n terms are strictly larger than the last 2n terms, respectively. By (BC3), > implies ≥. Hence, by (1), |~s[Ci+n ]| ≥ |~s[Ci ]|, as they are the unions of the first and last 2n terms, respectively. Together with the assumption, we have |~s[Ci+n ]| = |~s[Ci ]|. At this point, we can apply (3) and obtain |~s[Ci+n \ Ci ][2n − 1]| ≤ |~s[Ci \ Ci+n ][2n − 1]|. With (A7), this contradicts |~s[Ci+n ][2n − 1]| > |~s[Ci ][2n − 1]|, which is derived from |~s[Ci+n ][i + n]| > |~s[Ci ][i]|. Deriving formula (7). First, note that [ ~s[Ci ][i] = i<n (8) [ i<n ~s[Ci+n ][i + n] = [ [ ~s[f ][i] = i<n f ∈Ci 2n−1 [ [ i=n f ∈Ci [ −1 ~s[f ][f< (1)], f ∈2n 2 ~s[f ][i] = [ −1 ~s[f ][f> (1)]. f ∈2n 2 S Now S assume ~s< E ~s> . Recall that our goal is to derive | i<n ~s[Ci ][i]| = | i<n ~s[Ci+n ][i + n]|. Our strategy is the following. When we assume ~s< E ~s> , we can show that for any f ∈ 2n 2, treated as a sequence of 0’s and 1’s, if the number of 1’s in the first n places of f and the number of 1’s in the last n places of n are not equal, then ~s[f ] = ∅ can be derived. We can call f “balanced” when this condition is satisfied; when f is not balanced, ~s[f ] = ∅Scan be derived. However, for S those balanced f , when restricted to ~s[f ], i<n ~s[Ci ][i] and ~s[Ci ][i]| = | i<n ~s[Ci+n ][i + n]| are of the same size. For a simple illustration, see Figure 5. Then summing over all balanced f , we obtain the required formula. THE LOGIC OF COMPARATIVE CARDINALITY 39 ~s[C3 ][3] ~s[C2 ][2] ~s[C1 ][1] ~s[C0 ][0] 3 2 1 0 11 11 0 1 10 1 0 10 0 1 01 1 0 01 0 0 00 Figure 5. The grid with n = 2 when ~s< E ~s> . Recall that squares in the same column are of the same size. It is not hard to see then that |~s[C0 ][0] ∪~s[C1 ][1]| = |~s[C2 ][2] ∪ ~s[C3 ][3]| by comparing them in each column. −1 −1 Pick an arbitrary f ∈ 2n 2 and let k< = |f< (1)|, k> = |f> (1)|. Then it is easy to see that the system can derive the following through the equational theory of Boolean algebras: ~s[f ] ⊆ Nk< (~s< ) ∧~s[f ] ⊆ Nk> (~s> ). Also by the definition of N in Definition A.2 and by using the equational theory of Boolean algebras, Ni (~s> ) ∩ Nj (~s> ) = ∅ and Ni (~s> ) ⊆ (Nj (~s> ))c are derivable when i 6= j. Hence, if k< 6= k> , then ~s[f ] ⊆ Nk< (~s< ) and also ~s[f ] ⊆ (Nk< (~s> ))c . Since we have assumed ~s< E ~s> , we have Nk< (~s< ) = Nk< (~s> ). This means that we can derive ~s[f ] ⊆ Nk< (~s< ) ∧~s[f ] ⊆ (Nk< (~s< ))c and then ~s[f ] = ∅. −1 −1 Now we derive that |~s[f ][f< (1)]| = |~s[f ][f> (1)]|. When k< 6= k> , −1 −1 we derive ~s[f ] = ∅. Then trivially |~s[f ][f< (1)]| = |∅| = |~s[f ][f> (1)]|. If k< = k> , then let k = k< = k> and consider the sequence ~t where the terms are h~s[f ][i]if ∈Ci : • D(~t) is derivable using the equational theory of Boolean algebras. −1 • ~t has 2k terms; the union of the first k terms is ~s[f ][f< (1)] and −1 the union of the last k terms is ~s[f ][f> (1)]; • we showed when we derived (6) that for any i, j, |~s[f ][i]| = |~s[f ][j]|; hence for any i < k, |ti | = |ti+k |. Given these three points, we can apply (2) to ~t and derive the equation −1 −1 |~s[f ][f< (1)]| = |~s[f ][f> (1)]|. −1 In sum, we have derived for any f ∈ 2n 2 the equation |~s[f ][f< (1)]| = −1 |~s[f ][f> (1)]|. But then we can apply (2) to the sequence where the first −1 −1 22n terms are h~s[f ][f< (1)]if ∈2n 2 and the last 22n are h~s[f ][f> (1)]if ∈2n 2 . are just Hence it is derivable that the unions of each, which by (8) S S ~ s[C ][i + n], are of equal size. ~ s[C ][i] and i+n i i<n i<n This completes the whole proof. ⊣ 40 Y. DING, M. HARRISON-TRAINOR, AND W. H. HOLLIDAY GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 E-mail: yf.ding@berkeley.edu SCHOOL OF MATHEMATICS AND STATISTICS VICTORIA UNIVERSITY OF WELLINGTON NEW ZEALAND 6140 E-mail: matthew.harrisontrainor@vuw.ac.nz DEPARTMENT OF PHILOSOPHY AND GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 E-mail: wesholliday@berkeley.edu

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