On a Quasi-Set Theory

The main features of a theory that enables us to deal, in terms of a set theory, with collections of indistinguishable objects are presented. The fundamental idea is to restrict the concept of "identity" in the underlying logical apparatus. The basic entities of the theory are Urelemente of two sorts; to those called w-atoms, the usual concept of identity, in a precise sense, does not apply, but there exists a primitive equivalence relation called "the indistinguishability relation" that holds among them. The other sort of atoms (Matoms) are treated as Urelemente stricto sensu. The underlying logic is a kind of "nonreflexive logic" and reflects formally this situation. The intuitive motivation is twofold: seeking agreement with Schrόdinger's dictum that "identity" lacks sense with respect to the elementary particles of modern physics, and building WeyFs "effective aggregates" "directly", that is, dealing ab initio with indistinguishable objects; hence, their collection must not be considered a "set". Despite these motivations, in this paper quasi-set theory is delineated as a set theory, independently of its possible applications to other domains. 1 The intuitive idea of a quasi-set To understand intuitively what we mean by a quasi-set (qset for short), the reader may think of a classical set with atoms (in the sense of Zermelo-Fraenkel with Urelemente—ZFU). Suppose now that the atoms are of two sorts. In the first category we have the M-atoms, which can be thought of as the macroscopic objects of our environment. They will be treated as Urelemente of ZFU stricto sensu; hence, we will admit that classical logic is valid with respect to them in all its aspects. The atoms of the other kind (m-atoms) may be intuitively thought of as elementary particles of modern physics, and we will suppose, following Schrόdinger's ideas, that identity is meaningless with respect to them ([10], pp. 16-18). ι Then we will admit that the Traditional Theory of Identity (TTI) does not apply to the m-atoms. These facts enable us to hold, with regard to the m-atoms, that the concepts of indistinguishability and identity may not be equivalent. Therefore, roughly speaking we can say that a qset is a collection of objects (called elements) such that to the elements Received January 15, 1991; revised May 15, 1991 QUASI-SET THEORY 403 of one of the species (the m-atoms), the notion of identity (ascribed by classical logic and mathematics) lacks sense. This intuitive sketch is in a sense supported by certain philosophical views concerning the notions of identity, indistinguishability, and individuality in classical and quantum physics (see French [5], French and Redhead [6], and [10]). We will not discuss these philosophical questions here (but see Krause [8]); in this paper, we will study the quasi-set theory as a set-theory, independently of its philosophical motivations or its (possible) applications to other domains. In order to provide a motivation, we note only that in some domains of knowledge, such as quantum mechanics, chemistry, biology, or genetics (cf. Weyl [12], App. B), it is necessary to consider collections of entities that are capable of being in certain states, but such that it is impossible to say what elements belong to each particular state. Only the quantity of elements in each state may be known. Weyl called such collections effective aggregates of individuals ([12], p. 239). The idea is that it is not possible to distinguish among the elements that belong to the same state of an effective aggregate. It is important to note that such aggregates cannot be considered sets in the usual sense (ZFU, say), since in a set the elements are always distinguishable. This point was recently observed by Dalla Chiara and Toraldo di Francia [3]. In this paper, we present the main features of a theory of quasi-sets which intends to provide adequate mathematical tools for dealing with effective aggregates (in WeyPs sense) directly, that is, without using the subterfuge of distinguishing first (that is, considering their collection as a set), and then abstracting the distinction previously made (by the underlying mathematical apparatus), keeping only the quantity of the elements in each particular state. 2 The quasi-set theory We will denote the quasi-set theory by S*; the language of S* has the following primitive symbols: (i) connectives: -ι (not), v (or); the symbols Λ (and), => (implication), and (equivalence) are introduced as usual; (ii) Universal quantifier V (for all) and the existential quantifier 3 (there exists) are defined as usual; (iii) Three unary predicate symbols: m (m-object), cM (Λfobject), and Z (set), and two binary predicate symbols: E (membership) and = (indistinguishability); (iv) a unary functional symbol qcard (quasi-cardinality); (v) Parenthesis and comma; (vi) Individual variables: a denumerably infinite collection of variables. The concepts of term and formula, of bound variable, of closed formula, etc. are defined as usual. We observe that if x is a variable, m(x), CM(A:), and *Z(x) may be read "x is an m-object", "x is an M-object", and "Λ: is a set", respectively. Then, if x is a variable, the term qcard(x) means the quasi-cardinality of the qset x". We will use the following abbreviations: v α # ( . . . ) and 3DΛΓ( . . . ) for Vx(Π(x) =>(. . .)) and 3x(Π(x) A ( . . . )) respectively, where D stands for a predicate of the language or some of the ones defined below. The Postulates (axiom schemata and inference rules) for Propositional and Predicate levels are the standard ones, based on the primitive connectives we considered (as for instance, those of Hubert and Ackermann [7]). The only difference from the systems of classical first order logic is that instead of the Axioms of Identity, we have the Axioms of Indistinguishability: