Grounding Megethology on Plural Reference (Studia Logica)

Grounding Megethology on Plural Reference Massimiliano Carrara & Enrico Martino Studia Logica An International Journal for Symbolic Logic ISSN 0039-3215 Stud Logica DOI 10.1007/s11225-014-9585-9 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Massimiliano Carrara Enrico Martino Grounding Megethology on Plural Reference Abstract. In Mathematics is megethology (Lewis, Philos Math 1:3–23, 1993) Lewis reconstructs set theory combining mereology with plural quantification. He introduces megethology, a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals. Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, if mereology and plural quantification are ontologically innocent, as Lewis maintains, he achieves an ontological reduction of classes to individuals. Lewis’work is very attractive. However, the alleged innocence of mereology and plural quantification is highly controversial and has been criticized by several authors. In the present paper we propose a new approach to megethology based on the theory of plural reference developed in To be is to be the object of a possible act of choice (Carrara, Stud Log 96: 289–313, 2010). Our approach shows how megethology can be grounded on plural reference without the help of mereology. Keywords: Megethology, Plural reference, Plural quantification. 1. Introduction Although the basic notions of set theory are nowadays very familiar to all mathematicians, the very nature of a set, conceived of as a well-determined entity built up by its members, is quite mysterious. In the famous paper The elusiveness of sets, Black observes: Beginners [of set theory] are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as “three in one” should be child’s play. ([1], p. 616) Black criticizes at length Cantor’s well-known definition of set as well as all other attempts to explain what a set, as an abstract entity, is. He regards all such attempts as a misleading mystification and recommends avoiding any question about the nature of sets. What is important, according to Black, is rather to look at the use of plural reference in natural language Presented by Jacek Malinowski; Received January 23, 2014 Studia Logica DOI: 10.1007/s11225-014-9585-9 c Springer Science+Business Media Dordrecht 2014  Author's personal copy M. Carrara, E. Martino as the appropriate starting point for learning to master the sophisticated language of mathematical set theory; and that is all that a mathematician needs. Black’s conclusion can be satisfactory for a working mathematician who is only interested in the correct use of a set-theoretical talk. But it is unsatisfactory for a philosopher interested in understanding how to think of the objects, if any, that set theory seems to speak of. But even in this perspective, the idea of plural reference is fruitful because it can suggest a way of looking at sets as individuals. In this vein, Stenius (in [18]) has sketched a project of a structuralist set theory where the role of sets is played by arbitrary individuals of a given universe of discourse, which are representative of certain pluralities of individuals of the same universe. In a similar perspective, Lewis reconstructs set theory in Mathematics is megethology [13]. By combining mereology with plural quantification, he introduces a powerful framework, megethology, so called because it turns out to have enough expressive power for expressing interesting hypotheses about the size of reality ([13], p. 3). Within this framework, Lewis develops a structuralist class theory, in which the role of classes is played by individuals. Thus, accepting Boolos’ thesis that second-order logic, interpreted in terms of plural quantification, does not involve the notion of class, Lewis achieves an ontological reduction of classes to individuals. Lewis maintains that both mereology and plural quantification are part of logic, but this claim is highly controversial and has been criticized by several authors.1 In particular, he has shown (in [8]) that the combined use of mereology and plural quantification assures that, if the domain of individuals is infinite, then it is uncountable. It seems to be implausible that such a result holds in virtue of pure logic. The general notion of individual is somewhat vague and inadequate to justify the main axiom of mereology, according to which any plurality of individuals, however given and heterogeneous, form a further individual, their sum. Furthermore, the crucial distinction between pluralities and classes has not been adequately elaborated by Boolos and Lewis to justify their claim that plural quantification is ontologically innocent. They failed to dispel the suspicion that pluralities are nothing but masked classes. Lewis’ ground of megethology, consisting of both mereology and plural quantification, turns out to be overloaded. 1 See [15] and [16] for a criticism to second order logic, and [10] for a criticism to the thesis that mereology is part of logic. Author's personal copy Grounding Megethology on Plural Reference In fact, in [6] it has been shown how megethology is developable using mereology without plural quantification. In the present paper, we will show how megethology is developable using plural quantification without mereology. In order to better understand the crucial differences between pluralities and classes, we will propose an approach to the notion of a plurality based on the distinction between acts and entities. We will restrict ourselves to a particular sort of individuals, taking as individuals an idealized team of infinitely many agents. We will then show that our framework is appropriate for a reconstruction of megethology. In fact, it turns out to be adequate—as Lewis’ framework—to express the hypothesis that the universe of individuals is so large, that these can play the role of sets.2 2. Plural Reference Boolos (in [2,3]) has proposed an interesting reinterpretation of secondorder monadic logic in terms of plural quantification. In Boolos’ perspective second-order monadic logic is ontologically innocent: second-order variables do not range over sets of individuals but over individuals plurally. By contrast, first-order variables range over individuals singularly. Boolos’ basic idea consists in interpreting the atomic formulas of the form Xy as “y is one of the X s,” and the existential formulas of form ∃X . . . as “There are some individuals X s such that...”. The universal quantifier ∀X is expressible in terms of the existential one in the usual way. Boolos’ treatment of plural quantification bypasses the notion of plural reference and tries to directly discuss plural quantification. His explanation rests entirely on the intended meaning of the locution “there are some objects such that. . . ” in the natural language. But the meaning of this locution is somewhat ambiguous, strictly depending on the context of discourse. In some context its meaning is the same as the first-order quantification “there is at least an object such that. . . ”. And when such locution is not reducible to a first-order quantification, as in Geach–Kaplan’s proposition “Some critics admire only one another,” it is not clear if it is a sloppy way of referring to some class of individuals. Besides, the natural language does not suggest a direct interpretation of the universal quantifier: “for each indi- 2 For other approaches to a construction of set theory see [4, 5], and [17]. Author's personal copy M. Carrara, E. Martino viduals. . . ” is ungrammatical and “for all individuals” is indistinguishable from the first-order quantification “for each individual.”3 Boloos provides also a formal semantics for his language in Nominalistic Platonism [3]. But Boolos’ semantics has no explicative power. It involves, in the metalanguage, a quantification over relations among individuals and the problem arises how to understand that in turn in terms of plural quantification over individuals. For these reasons, Boolos’ view, although very attractive, is highly controversial. It has met the criticism of several philosophers of mathematics (see for example [15,16]). Quine’s old claim that second-order logic is “set theory in disguise” (metaphorically “a wolf in sheep clothing”) does not seem to have lost all its advocates. We think that, independently of the use of plural quantification in the natural language, the role of plural quantification in logic and mathematics can be better understood within the frame of a highly idealized notion of reference, which can be seen behind the mathematical use of arbitrary reference (on this see [14]). In To be is to be the object of a possible act of choice [9] we have argued that mathematicians use in their reasoning expressions such as “let a be an arbitrary object of the universe of discourse,” for instance “let a be an arbitrary real number.” Observe that there is no link between the letter “a” and the number that it is supposed to be indicating. However, after introducing a with that locution, a working mathematician, accustomed to argue informally through meaningful propositions, reasons about a as if “a” designated a well-determined individual. This means that he reasons under the implicit assumption that someone has associated to “a” a certain (unknown) individual. For instance, she reads a formula of form F (a) as “the individual a falls under the concept F ”. Lacking that assumption, F (a) would be meaningless. If so, mathematical reasoning would seem to presuppose, at least ideally, the possibility of indicating any object of the universe of discourse, even when, as in the case of real numbers, not every object has a name in the language.4 The possibility, in principle, of referring to any 3 Lewis has suggested an interpretation of the universal plural quantifier as “whenever there are some things, then. . . ” ([13], p. 11). But, we have some doubt that such locution is appropriate. “Whenever there are some numbers, then. . . ” sounds somewhat strange and seems to suggest that it might happen sometimes that there are no numbers. 4 As we have observed in our To be is to be the object of a possible act of choice [9] a number of arguments in the literature seem to be in disagreement with the above observation. Probably, the reason is a misunderstanding of the notion of arbitrariness. One may argue that considering an arbitrary number is nothing but a way of speaking, which by no means Author's personal copy Grounding Megethology on Plural Reference individual of the universe of discourse is not confined to informal reasoning. It is also required for justifying the quantification rules in a formal system of natural deduction. Take the introduction rule of the universal quantifier (I∀). It allows the inference of ∀xAx from the premise Ab in the usual way: . A(b) . ∀xA(x) where “b” is an arbitrary name (or a free variable) not occurring in any assumption on which A(b) depends. The soundness of the rule is grounded on the consideration that if one has proved that b satisfies the formula A, without any specific piece of information about b, then any individual enjoys the property in question. This justification clearly presupposes that b can denote any individual in the range of the variables (otherwise the universal quantifier should be restricted to the denotable individuals). Similar considerations hold for the elimination rule of the existential quantifier (E∃). The possibility, in principle, that any individual of the universe of discourse is capable of being picked out by an act of reference plays an essential role even in the semantics of quantifiers. Consider the usual inductive definition of truth (or satisfaction) for first order logic. Take, for simplicity, the Footnote 4 continued involves the possibility of actually singling out such a number because, for the very same arbitrariness, it is irrelevant which number one is speaking of. Indeed, when reasoning about an arbitrary number a, there is no need to know it. Yet, as observed, ignorance of the number one is referring to has the desired effect of granting generality to the reasoning: what is provable for a completely unknown number holds necessarily for all numbers. That is right. However, the lack of information about a cannot avoid the assumption that the letter “a” designates a precise number; lacking that assumption, it would make no sense talking about a, not even to say it is unknown. Perhaps one could object that speaking of an arbitrary number amounts to speaking of all numbers simultaneously. But this is not the case. When reasoning about a, a mathematician can exploit the assumption that “a” has a well-determined referent, which is kept fixed in the whole course of the reasoning. Besides, when, for instance, arguing about a, one proves F (a) and later ¬F (a), she gets a contradiction. But, of course, no contradiction would follow if the two occurrences of a might designate different individuals. For these reasons a is to be thought of as a well-determined individual. Nor could F (a) be understood as expressing the fact that F is instantiated by some individual (without any reference to a particular individual), like in Frege’s interpretation of the existential quantifier. For, according to such interpretation, ¬F (a) would say, in turn, that the complement of F is instantiated as well. Again, no contradiction would follow from F (a) and ¬F (a). On the same topic see also [7]. Author's personal copy M. Carrara, E. Martino universal quantification of an atomic formula. If P is a predicate letter, the truth-value of ∀xP x is defined by the clause: ∀xP x is true if P x is true relative to every assignment of an individual to variable x. The definition of the truth-value of ∀xP x presupposes the definition of the truth-value of P x relative to an arbitrary assignment. One can easily realize that the idea that any object of the universe of discourse is capable of been picked out by an act of reference is masked behind the familiar use of the notion of an assignment, defined as a function from variables to individuals. One can see here a twofold role of this idea; it guarantees: • that any object can be the value of some assignment, and • that one can pick out an arbitrary assignment, understood as a mathematical function. Problem: how can one refer to an arbitrary individual? Perhaps, one might think, by means of some characterizing property, but that, unfortunately, would involve a problematic universe of properties, suitable for characterizing any individual. Besides, this option faces the problem of how to refer to an arbitrary property. Therefore, it seems that the notion of reference to an arbitrary individual, presupposed in mathematical reasoning, is more basic than any linguistic notion of reference via a definite description. We think that the most appropriate idealization for justifying arbitrary reference should be grounded on the ideal possibility of a direct access to any individual.5 We shall invoke an ideal agent who is supposed to be able, by means of an arbitrary act of choice, to single out any individual by ostension. In such a conceptual frame, the introduction rule of the universal quantifier (I∀) is justified as follows. Let us imagine an ideal agent who arbitrarily chooses an individual b about which we have no information at all. If we are able, just by reasoning about b, to conclude that it satisfies the formula A(x), because, as far as we know, any individual could be the chosen one, we can conclude that each individual has the property in question and, therefore, infer ∀xAx. Now, consider the usual set-theoretical semantic of second-order logic. Since second-order variables are intended to range over all classes of individuals, the problem arises: How can the ideal agent have direct access to any class of individuals? Since all we know about an arbitrary class is that 5 The notion of arbitrary reference can be seen as a highly idealized version of Kripke’s notion of direct reference for natural language [12]. Author's personal copy Grounding Megethology on Plural Reference it is an entity determined by its members (or by absence of members, in the case of the empty set), it seems that one can have access to a class only through its members. And since a class may be infinite, it seems that a single agent is unable to perform a simultaneous choice of all members of any class. A choice of infinitely many individuals, one at a time, would face the hard problem of dealing with undetermined classes (as it happens in intuitionistic mathematics). Black himself describes a “plural pointing” as a “simultaneous reference to several things at once.” We propose of extending the idealization of a single agent introducing an infinite team of agents, consisting of a number of agents equal to the number of individuals. To reconstruct megethology in sect. 3 we will introduce suitable megethological axioms for determining the size of the infinity of the individuals. Each agent is supposed to be able to arbitrarily choose one or finitely many individuals. More precisely, our agents can perform the following acts of choice: (1) Singular selecting choice (s.s.c.): one of the agents chooses an individual ad libitum; (2) Plural selecting choice (p.s.c.): it is performed by all agents simultaneously; each agent chooses an individual ad libitum (independently one of the others) or refrains from choosing (refraining from choosing serves the purpose to simulate the empty class); (3) Plural relating choice (p.r.c.) of degree n > 1; it is performed by all agents simultaneously, each agent chooses n (not necessarily distinct) individuals in a certain order or refrains from choosing. We imagine that the team is guided by ourselves (the real mathematician), who can order at will the execution of one of the described acts of choice. Boolos’ approach to plural quantification provides an interpretation only of monadic second-order logic, because his appeal to natural language does not help to simulate relations. Lewis, in turn, extends the interpretation to full second-order logic, by using mereological sums of individuals for codifying ordered pairs of individuals. A merit of our approach is that, thanks to our p.r.c., we do not need mereology in order to recover full second-order logic. By means of such acts of choice, we can refer to a single individual, to a plurality of individuals or n-tuples of individuals, without committing to abstract entities the job of collecting and correlating individuals. In this way any talk about pluralities of individuals or relations among individuals is to be understood in terms of appropriate acts of choice. A locution as “Let X Author's personal copy M. Carrara, E. Martino be an arbitrary plurality of individuals” is to be rephrased as “Imagine to have ordered a p.s.c. and call X the chosen individuals.” So the locution in question is to be understood as an act of reference performed through the execution of a certain plural choice. Similarly, a universal quantification “for every plurality X . . . ” is to be read as “however a p.s.c. of certain individuals X is performed. . . ” ; an existential quantification “there is a plurality such that. . . ” is to be read as “it is possible that such a p.s.c. of certain X s be performed that. . . ”. Formally we use a full second-order language L with identity, with firstorder variables x, y, z . . . and second-order variables X n , Y n , Z n , . . . (of any degree n ≥ 1). We omit the superscripts for variables of degree 1. Let us explain the semantics of acts of choice. An assignment to a formula A is obtained by ordering, for every free variable v (of any sort) in A, an appropriate act of choice, i.e., a s.s.c. for every first-order variable, a p.s.c. for every second-order variable of degree 1, a p.r.c. of degree n for every variable of degree n ≥ 2. With reference to an assignment to a formula A, if v is a free variable of A of any sort, we indicate by v* the relative act of choice. We define inductively the truth of a formula relative to an assignment: x=y is true if x* and y* choose the same individual; Xy is true if the individual chosen by y* is one of the individuals chosen by X* ; X n y1 . . . yn is true if the if the individuals chosen respectively by y1∗ . . . yn∗ are chosen in the order by X n∗ . Usual clauses for the propositional connectives; ∀vB is true if, however the assignment may be extended to B by an appropriate act of choice v* for v, B turns out to be true; ∃vB is true if it is performable an act of choice v* for v such that B turns out to be true. Now, it is clear that a p.s.c. does not create any entity that collects the chosen individuals. Speaking of pluralities as if they were genuine entities is a mere façon de parler, paraphrasable in terms of plural choices. In particular, a talk about the empty plurality is to be understood as a talk about a p.s.c. where all agents refrain from choosing. The fictional Platonist flavour of our theory consists in the fact that individuals and agents are treated as if they actually existed. That is essential Author's personal copy Grounding Megethology on Plural Reference in order to make acts of plural simultaneous choices performable. In this respect, our perspective could be labelled as a “fictional Platonism about individuals and agents”. In contrast, acts of choice, unlike individuals and agents, are to be understood in a mere potential way. In this respect, our perspective could be labelled as a “fictional possibilism about acts of choice.” Concerning this point, it is worth clarifying how to understand the notion of possibility involved when speaking of possible acts of choice. We don’t assume any ontology of possible acts. If possible acts were understood as entities of a realm of possibilia, then the problem of arbitrary reference to the objects of the universe of discourse would be simply reduced to the even harder problem of arbitrary reference to possibilia. The force, if any, of our approach rests essentially on the view that acts of choice are no entities at all, neither actual nor possible. Acts do not exist, they are merely performable. The possibility to be performed is merely combinatorial. Besides it is non-epistemic: the possibility of a choice to the effect that a certain condition be satisfied by no means requires that the agents are aware of such an effect. This notion of possibility is perfectly compatible with the use of classical logic. For, one can recognize, by induction on the complexity of a formula, that the truth-value of a formula, relative to an assignment, is well-determined by the truth clauses. As an example, let us consider the case of an existential formula ∃XB. An instance of clause (vi) says that: ∃XB is true if it is performable a p.s.c. X* for X such that B turns out to be true. By the induction hypothesis, however a p.s.c. for X may be performed, it determines a truth value of B. The combinatorial possibilities concerning the performance of a p.s.c. are determined by the play rule governing plural choices, according to which every agent is allowed to choose an individual ad libitum. Hence it is well determined if the possibility is left that a p.s.c. be performed in such a way that B turns out to be true. Thus, ∃XB has a well-determined truth-value. Besides, our notion of combinatorial possibility justifies immediately, without any circularity, the comprehension principle of second-order logic: (CP) ∃X n ∀y1 . . . yn (X n y1 . . . yn ↔ A) (where A is any second-order formula without X free). For, in virtue of the arbitrarity of choices allowed by the choice rule, nothing can prevent the possibility of a p.r.c. (or a p.s.c.) such that the Author's personal copy M. Carrara, E. Martino chosen n-tuples are just the ones satisfying A. Possible occurrences in A of second-order quantifications cannot produce any circularity, since acts of choice, unlike properties, are all independent one of the other. Besides, our semantic of choices makes the following axiom of choice evident: (CA) ∀X 2 ∃Y 2 ∀x(∃yX 2 xy → ∃!z (Y 2 xy ∧ X 2 xz)) By using our semantics, we will reconstruct a structuralist set theory where the role of sets is played by our agents.6 By means of a p.r.c. a single individual can represent a plurality of individuals, precisely the plurality of its correlates. In this way we want to elaborate Black’s idea of a plural reference understood as a “plural pointing”.7 Call representation a binary p.r.c. such that no two distinct individuals represent the same plurality. The problem arises: could some possible representation satisfy the condition that, however a p.s.c. is performed, the chosen individuals have a representative? The answer is no, as we will see. However, we will show that, if ZFC is consistent, one can consistently assume the existence of a representation such that all small (in a sense to be specified) pluralities are representable. Relative to such a representation our agents form a secondorder model of ZFC. 3. Megethology As said in the introduction, Megethology is a powerful framework in which one can formulate strong assumptions about the size of the universe of individuals (corresponding to the existence of strongly inaccessible cardinals). Aim of this section is to reconstruct Megethology within the framework of our idealization and without mereology. Due to the fact that from a mathemat6 To better clarify our perspective, in To be is to be the object of a possible act of choice [9] we have compared our proposal with Kitcher’s work [11]. Kitcher has clearly realized the importance of an idealized notion of act vs. that of object. Unfortunately, he doesn’t explain what the ideal act of collecting consists in. Our approach can be seen as a proposal of a way to understand an act of collecting. For an alternative approach to plural quantification see [21]. See also [20] and [19]. 7 “The notion of ‘plural’ or simultaneous reference to several things at once is really not at all mysterious. Just as I can point to a single thing, I can point to two things at once using two hands, if necessary; pointing to two things at once need be no more perplexing than touching two things at once. Of course it would be a mistake to think that the rules for ‘multiple pointing’ follow automatically from the rules from pointing proper; but the requisite conventions are almost too obvious to need specification. The rules for ‘plural reference’ are not harder to elaborate”. ([1], p. 629). Author's personal copy Grounding Megethology on Plural Reference ical point of view the nature of individuals is irrelevant for a stucturalistic reconstruction, for the sake of ontological economy, we assume as individuals the same agents so that agents are both the choosers and the objects to be chosen. We introduce the basic notions and axioms of megethology. Let S be the universe of individuals. A plurality X is small ifdf there is no plural injection of S to X. Otherwise it is large. A plurality X is infinite ifdf there is an injection of X into a proper subplurality of X. Of course, the notion of injection is to be understood in terms of a p.r.c. We assume that a representation D is possible, such that the axioms below are satisfied. From now on the reference to such a D will be understood. The following axioms assure that the size of the universe S of individuals is suitably large for reconstructing set theory: (1) Axiom. Every small plurality has a representative. (2) Axiom. Some infinite plurality is small. (3) Axiom. The plurality of the representatives of all subpluralities of a small plurality is small. (4) Lemma. Some pluralities have no representetives. Proof. Suppose, by way of contradiction, that every plurality has a representative. We can reproduce the Russell paradox by taking the plurality X of all representatives that do not represent themselves and wondering whether the representative of the X s is or not one of the X s. (5) Corollary. Given any large plurality X, some subpluralities of X have no representatives. Proof. Let X be large. There is an injection f of the plurality S of all individuals into X. Suppose, by reduction, that every subplurality of X has a representative. We can define a new representation D’ by taking as the representative of any plurality Y the D-representative of the corresponding subplurality f (Y) of X. So every plurality has a D’ -representative, against lemma 4. (6) Theorem. Let X be a small plurality of representatives of small pluralities. The plurality Y, such that for all y, Yy iff y is represented by one of the X s, is small. Author's personal copy M. Carrara, E. Martino Proof. Suppose, by way of contradiction, that Y is large. We will show that it is possible to define a new representation D’, such that every subplurality of Y has a D’ -representative, against corollary 5. Let V be any subplurality of Y. V meets any plurality represented by some x of the Xs into a small plurality represented by some x V . The plurality Z of all x V , when x ranges over all the X s, is small. Let z be the representative of the Z s. As V is determined by Z, we can define D’ by taking z as the D’ -representative of V. Thus any subplurality of Y has a D’ -representative. 4. Interpretation of ZFC Set Theory Let us reconstruct the theory ZFC of pure sets (i.e. without ur-elements). We define the plurality U of sets as the individuals belonging to every plurality X satisfying the following conditions: (i) Xo, where o is the representative of the empty plurality, (ii) if x is the representative of a small subplurality of X, then Xx. It follows that every set is the representative of a small plurality of sets. We use α, β, γ. . . as variables for sets. Define membership as the inverse of representation D: α ∈ β ifdf α is one of the individuals represented by β. So o is the empty set and will be indicated by the usual symbol ∅. Let us verify the axioms of ZFC. From axioms 1 and 3 it follows immediately the (7) Axiom of Extensionality. Two sets are identical iff they have the same members. Since two sets form a small plurality, the following axiom follows: (8) Axiom of Pair. For all α, β there is a set {α, β} whose members are exactly α and β. (9) Axiom of Union. For every nonempty set α, there is a set β whose members are all members of members of α. Proof. From theorem 6 it follows that the plurality X of members of members of α is small, so it forms a set. From axiom 3 it follows immediately that: Author's personal copy Grounding Megethology on Plural Reference (10) Axiom of Powerset. For every set α there is a set β whose members are the subsets of α. Any subplurality of the members of a set is small, so it forms a set. In particular: (11) Axiom of Separation. The members of a set satisfying any formula form a set. Since any plural function maps small pluralities into small pluralities, we get in particular: (12) Axiom of Replacement. If φ is a functional formula, i.e., such that ∀x∃!yφ, the φ-image of a set is a set. From axiom 4 it follows directly the axiom of choice: (13) Axiom of choice. If α is a set of nonempty pairwise disjoint sets, there is a set β that shares a unique member with every member of α. Proof. A plural choice function (see axiom 4) maps the members of α into a small plurality of sets, which are the members of the required set β. Define the plurality N of Zermelo natural numbers as the least plurality X of sets satisfying the following clauses: (i) ∅ ∈ X; (ii) if x ∈ X then {x } ∈ X. One can verify that N is Dedekind infinite according to the plural function of ordered pairs (x, {x }). One can also verify that there is an embedding of N into any infinite plurality, in particular (by axiom 6) into an infinite small plurality. So the plurality N is small and therefore determines a set. Consequently, we get the axiom: (14) Axiom of Infinite. Some set is infinite. (19) Axiom of Regularity. If α = ∅, there is a β ∈ α, such that β ∩ α = ∅. Proof. The plurality of ∅ and the sets α, for which there is a β ∈ α, such that β ∩ α = ∅, satisfies clauses (i), (ii) of the definition of U. Thus X = U. 5. Categoricity Given the universe S of individuals, the universe U of sets depends on the representation relation D. But one can easily recognize that if D* is a new Author's personal copy M. Carrara, E. Martino representation satisfying the same axioms for D, the universe U* relative to D* is isomorphic to U. In fact, let R be the least plural relation such that (i) R (o, o* ); (ii) if H is a small subplurality of R, x the D-representative of the first components, x* the D* -representative of the second components of the ordered pairs of H, then R (x, x* ). It is easily seen that R is an isomorphism. 6. Concluding Remarks Our approach to megethology starts from the observation (developed in [9]) that the possibility, in principle, of referring to any individual of the universe of discourse of a mathematical theory is essentially, although implicitly, presupposed in mathematical reasoning. In order to make such presupposition explicit and to clarify the sense of the locution “in principle,” we have taken as individuals a team of ideal agents. By means of this device, we have obtained an idealized version of Kripke’s notion of direct reference based on the crucial distinction between acts and things. Our framework provides a new perspective for understanding the ontological innocence of plural reference and plural quantification. Speaking of a plurality of individuals is nothing but a linguistic device for referring to the individuals selected by an act of choice. Moreover, our team of agents provides an intuitive means for simulating relations, without introducing the latter as a particular sort of entities. One could perhaps object that our approach to pluralities is far from being ontological innocent just because of our commitment to a so large team of infinitely many agents. Of course we cannot deny that agents are entities. In this respect our approach is not in agreement with Lewis’ claim that the transition from singular to plural reference is free of any further ontological commitment: such a transition requires a surplus of idealization. However, it would be a mistake to conclude that our commitment to agents is as strong as the traditional commitment to classes. First, as observed in [9], since all that we know about an arbitrary class of individuals, settheoretically understood, is that it is built up by its members, singular reference to a class must be obtained through plural reference to its members; and our team of agents serves just the purpose of making the idealization of plural reference explicit. Second, while classes are abstract entities, agents are concrete, though imaginary, entities. Our approach shows how, adopting a suitable structuralist view of mathematics, one can avoid the assumption of abstract entities: infinitely many concrete entities are sufficient for modeling any mathematical theory, provided that their infinity is suitably large. Author's personal copy Grounding Megethology on Plural Reference References [1] Black, M., The elusiveness of sets, Review of Metaphysic 24:615–636, 1971. [2] Boolos, G., To Be Is to Be a Value of a Variable. 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Martino FISPPA Department Section of Philosophy University of Padua Piazza Capitaniato 3 35139 Padova, Italy massimiliano.carrara@unipd.it