Found Phys (2011) 41:1338–1354 DOI 10.1007/s10701-011-9552-5

A Discussion on Finite Quasi-cardinals in Quasi-set Theory Jonas Rafael Becker Arenhart

Received: 17 September 2010 / Accepted: 8 March 2011 / Published online: 23 March 2011 © Springer Science+Business Media, LLC 2011

Abstract Quasi-set theory Q is an alternative set-theory designed to deal mathematically with collections of indistinguishable objects. The intended interpretation for those objects is the indistinguishable particles of non-relativistic quantum mechanics, under one specific interpretation of that theory. The notion of cardinal of a collection in Q is treated by the concept of quasi-cardinal, which in the usual formulations of the theory is introduced as a primitive symbol, since the usual means of cardinal definition fail for collections of indistinguishable objects. In a recent work, Domenech and Holik have proposed a definition of quasi-cardinality in Q. They claimed their definition of quasi-cardinal not only avoids the introduction of that notion as a primitive one, but also that it may be seen as a first step in the search for a version of Q that allows for a greater representative power. According to them, some physical systems can not be represented in the usual formulations of the theory, when the quasi-cardinal is considered as primitive. In this paper, we discuss their proposal and aims, and also, it is presented a modification from their definition we believe is simpler and more general. Keywords Quasi-set theory · Quasi-cardinal · Quantum indistinguishability 1 Introduction One of the philosophical and foundational problems presented by quantum physics concerns the identity and individuality of indistinguishable particles. According to a widely accepted position on that issue (sometimes called Received View), indistinguishable particles are not individuals in quantum mechanics. This non-individuality of the particles should be understood in the sense that statements concerning their J.R. Becker Arenhart () Federal University of Santa Catarina, Campus Trindade, 88040-900, Florianópolis, SC, Brazil e-mail: [email protected]

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identity or difference do not make any sense (see [5] for references and further discussions). Historically, one of the main arguments for this position claims this understanding of the nature of the particles fits more naturally the so-called “Indistinguishability Postulate” adopted in quantum mechanics. According to this postulate, roughly speaking, given a permutation of quantum particles in some state, no measurement can distinguish between the original state (non-permuted) and the final state (permuted). In spite of its popularity, the Received View faces many challenges, two of which are of particular interest for us: (i) the first one is concerned on how we can make sense of the notion of non-individuality, a problem requiring the articulation of a “metaphysics of non-individuals”, (ii) the other is concerned on how to provide for an adequate formalism to deal with non-individuals, explaining in particular how we can develop a mathematical theory allowing entities for which identity statements do not make any sense. Now, both problems may be seen as tightly interrelated, the formal and metaphysical developments influencing each other. One instance of that relationship may be seen in the proposal to underpin the Received View by using the quasi-set theory Q framework. Like some authors suggest, the theory might be helpful to put some order in the categorial framework of the Received View (see [4, p. 102], [5]). Roughly speaking, Q is a set-theory meant to provide formal counterpart to the Received View, allowing not only objects for which the notion of identity is not defined (representing the non-individuals) and collections of those objects, but also all the usual collections existing in the classical set theory with urelemente, ZFU (Zermelo-Fraenkel with Urelemente). Now, there is a growing literature on the subject, and a topic that begins to be discussed inside the quasi-set theoretical framework is concerned with the relation between cardinals on the one hand, counting and ordinals on the other hand (see [2] and [1]). It is generally remarked that, from an experimental point of view, physicists know how many particles they are dealing with in practical situations. Hence, the collections of particles dealt with by physicists in the laboratory usually have a well defined cardinal in orthodox QM. That is granted even if we accept that particles in that theory may be perfectly indistinguishable, and more than that, even when they are considered to be non-individuals, in the sense mentioned before (namely, that identity does not make sense for them). On the other hand, since those objects are indistinguishable and non-identifiable, it seems we can not count them in the usual sense of orderly attributing a unique natural number to each element of the collection, for this kind of process presupposes we can identify the items that are being counted, something we can not do for quantum particles (according to the Received View, obviously). As a consequence, it seems we can not attribute an ordinal to collections of those objects. Now, the great problem stems from the fact that in general, the cardinal of a collection is defined through the notion of ordinal, we know how many items there are by counting them. Then, it seems that in handling non-individuals, we must keep both concepts, ordinality and cardinality, apart, and sense must be given to collections having a well defined cardinal, but no ordinal, attributed to them. Precisely that is done in Q, we keep the concepts of cardinal and ordinal separated, since, as we are discussing, we can not attribute an ordinal to every collection, although we know how many elements each of them have. To this end, the concept of cardinal is

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introduced as a primitive symbol in the language and called quasi-cardinal, because it generalizes the idea of cardinality beyond the realm of usual collections. Thus, even when we can not associate an ordinal to some collections, the cardinal is granted by the theory’s axioms (see [5, Chap. 7]). In a recent paper, Domenech and Holik (see [2]) claimed that both the picture of collections of particles always having a well defined cardinal and the procedure used to achieve its formal counterpart in Q are not completely satisfactory. According to them, besides having to introduce a further notion as primitive in the theory of quasisets, that way of granting cardinality for every collection limits the capacity of the theory to represent some interesting physical systems, for example, the ones whose particle number is not well defined. That is, they claim some physical systems, those not having a well defined cardinal, cannot be represented in Q when we assume the notion of quasi-cardinal as a primitive. In order to surpass those difficulties, they proposed that the notion of quasi-cardinal should be introduced by definition. Domenech and Holik claimed this move allows us to avoid the introduction of a primitive symbol in the language of the theory, and they also suggested that, with further investigations of their specifications, it may be used to pursue a formulation of Q allowing that some collections remain without a well defined cardinal. Our aim in this paper is to discuss both the way they pose the problems concerning cardinality in Q as it is originally formulated and their purported solution. We also propose an alternative definition of finite quasi-cardinal, related to the one given by Domenech and Holik. We claim our version is even more economic, easier to work with and it avoids some difficulties occurring in these authors’ definition. We are also less ambitious on proposing it. We show their results are derivable from ours, so, we argue their definition can be considered as a particular case of our definition. In the next section we sketch quasi-set theory, the framework of both our proposal and of Domenech and Holik’s definition.

2 Quasi-set Theory Q Built as a first-order theory without identity, we can say, in a nutshell, that quasiset theory Q is a ZFU-like theory allowing two kinds of atoms, termed m-atoms (micro-atoms) and M-atoms (macro-atoms) (see [5, Chap. 7] and [6] for details). In the language of the theory, two unary predicate symbols m and M, for micro and macro atoms, respectively, distinguish between the atoms. The collections of Q, called quasi-sets (or q-sets), are defined as those objects that are not atoms, and a predicate symbol Q is introduced in the language in order to denote them. Q-sets are built by applying the usual set theoretical operations of union, intersection, power set, Cartesian product and difference, to mention the most well-known ones. Indistinguishability is denoted by a primitive binary relation symbol ≡. The axioms for it grant us that it is an equivalence relation coinciding with identity for every item of the theory’s domain, except for m-atoms and q-sets. That is, except if x and y are m-atoms or q-sets, we can be sure that x ≡ y if and only if x is identical to y; this grants us that identity and indistinguishability are not the same concept, since, for example, m-atoms may be indistinguishable without being identical. Identity enters the stage as a defined relation, such that it holds for M-atoms belonging to the

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same q-sets (M-atoms play the role of the usual atoms of ZFU), and for q-sets having the same elements. Though, for m-atoms identity is not defined, since they represent the non-individuals, that is, the non-individuality of m-atoms is granted by restricting the definition of identity, such that it does not hold for m-atoms. Identity relation is characterized by the usual properties of identity, viz. reflexivity and the substitution law. We define it through the following formula: Definition 1 x = y =def [(Q(x) ∧ Q(y) ∧ ∀z(z ∈ x ↔ z ∈ y)) ∨ (M(x) ∧ M(y) ∧ ∀z(x ∈ z ↔ y ∈ z))]. From now on, when we talk about identity in Q, we are mentioning that definition. In the quasi-set-theoretical hierarchy, built from the empty q-set and atoms, we can distinguish three kinds of q-sets: 1. q-sets whose only elements are m-atoms; 2. q-sets whose transitive closure contains no m-atoms; 3. q-sets whose elements are of both kinds. The q-sets pertaining to the second kind are called classical q-sets. They contain only elements for which identity is defined, and the same holds for their elements, their elements’ elements, and so on. We denote those q-sets by the predicate Z. Along with M-atoms, the q-sets satisfying Z are called the classical things of the theory. By using those things we can obtain inside Q all of the mathematics that can be obtained inside ZFU. In general, we denote a q-set satisfying a condition φ by [x : φ(x)]; if this q-set satisfies the predicate Z, that is, if it is a classical q-set, we shall denote it as usual by {x : φ(x)}. We can not formulate the definition of unordered pairs in the usual way, that is, by using the condition that its elements are those objects z identical to x or y, because identity is not defined for every element of the intended domain of discourse, i.e., using the classical definition we would not be able to form “pairs” of m-atoms. Then, we define the unordered pair of x and y as the q-set whose elements are indistinguishable from x or y, denoted by [x, y]. That restriction of the unordered pairs definition reflects itself also in the definition of ordered pairs of x and y, which is introduced by x, y =def [[x], [x, y]]. One can easily verify that if x and y are indistinguishable m-atoms, then x, y and y, x are indistinguishable, and so, we can not make any sense of order for indistinguishable m-atoms. A binary relation R is a q-set of ordered “pairs”. A quasi-function f (we call it simply function when there is no possibility of confusion) is a relation satisfying the following condition: if x, y ∈ f and w, z ∈ f and x ≡ w, then y ≡ z. That means indistinguishable things are mapped to indistinguishable things, and for classical things this notion coincides with the usual one. We can illustrate this definition with an example. Let’s suppose we are given a q-set A whose only element is one m-atom, and B has intuitively two indistinguishable m-atoms, then, the q-set whose elements are pairs composed by the element of A with elements of B in the second coordinate is a quasi-function, because it satisfies the definition, it maps indistinguishable things to indistinguishable things. But how are we going to make any sense on more specific kinds of functions like injections, surjections and bijections, which in their usual formulations draw on

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the concept of identity? Since identity is not always available to us, the alternative commonly proposed is the use of the concept of quasi-cardinal in order to remedy that situation. The quasi-cardinal concept enters the standard formulations of Q as a primitive function symbol qc, which attributes cardinality to q-sets by force of axiom, establishing that every q-set A has a quasi-cardinal given by qc(A). Then, returning to the case of the specific functions, a bijection, for example, is a quasi-function whose domain and counter domain have both the same quasi-cardinality. Thus, it follows that the notion of bijection, as it is usually introduced in Q, can not be of much help to define the notion of cardinality, since it assumes this very notion on its formulation! Therefore, a different route should be envisaged if we want to define cardinality within Q, and that is precisely what Domenech and Holik provided. Now, let us take a look at their proposal.

3 Domenech and Holik on Quasi-cardinals In their A discussion on particle number and quantum indistinguishability (see [2]),1 G. Domenech and F. Holik proposed a definition of quasi-cardinals in quasi-set theory that deals only with finite q-sets. We list, as follows, the ones we consider their three main theses in that work: 1. their definition shows that, at least when we deal with finite q-sets, it is not necessary to introduce the notion of quasi-cardinality by axioms, as it has been done in the usual presentations of the theory (see, for example, [5, Chap. 7]). That is, there is an alternative solution to the problem concerning the introduction of quasicardinals in Q which is more economical than the usual one; 2. they suggest that the definition presented by them may be seen as a first step in the formulation of a version of the theory allowing that some q-sets remain without an associated quasi-cardinal. Through these q-sets, it would be possible to represent some physical systems that appear in relativistic quantum physics, viz. the ones without a well defined particle number. According to the authors, that move may help us to overcome an inadequacy of the theory Q as it is standardly formulated, since the axioms employed for quasi-cardinals grant that every q-set has a quasicardinal, allowing no place for such systems; 3. their definition should capture a specific sense from the notion of counting, taken as an effective physical process which can be performed in a laboratory, for example. We shall leave the discussion of thesis 3 to the end, because it deals with interesting metaphysical questions we intend to explore with more details elsewhere. Let us begin with thesis 1, which is really a major novelty in the work of Domenech and Holik. As we mentioned before, in the usual formulations of Q, in order to guarantee that every q-set has an associated quasi-cardinal, it is common to employ a primitive unary function symbol qc for which specific axioms are provided (see [5, Chap. 7]). 1 When we mention Domenech and Holik’s work, unless otherwise stated, we are always referring to [2].

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Perhaps, that is the simplest way of solving the problem concerning how to associate cardinals to q-sets having m-atoms as elements. As we have already seen, the quasi-cardinal concept is usually taken as primitive, for in Q we can not employ the usual definitions of cardinals, like for example, the standard one à la von Neumann, which defines the cardinal of a collection as the least initial ordinal equipollent to it (see [3]). In order to apply that definition to a q-set of indistinguishable m-atoms, we would have to well-order and put it in a one-to-one correspondence with an ordinal. The first difficulty in doing that, as we have already commented, is that we can not well-order a q-set whose elements are indistinguishable m-atoms, because the very definition of a well-order relation would require that the concept of identity be defined for the elements of the collection being well-ordered, something that is not possible for m-atoms. Also, we can not put that q-set in one-to-one correspondence with an ordinal, because the notion of a one-to-one correspondence, besides involving the identity relation in its definition, can only be reasonably introduced in the usual formulation of Q with some help from the concept qc. Thence, since we are trying to define precisely the notion of quasi-cardinality, it is not available for us yet. As we can see, there is no straightforward way to define quasi-cardinality and so, the claim that the notion of quasi-cardinal can be defined, even in the face of so many difficulties, should be seen as a great achievement. In fact, Domenech and Holik suggest that this move may deliver us even more than the solution of a technical difficulty. According to them, the introduction of quasicardinality as a primitive notion in Q has a very serious shortcoming: there are, in quantum theory, some aggregates of particles that can not be represented in the theory of quasi-sets when it is formulated like that (see [2, pp. 858, 859]). More precisely, they claim the introduction of quasi-cardinals as primitive endows every q-set with a quasi-cardinal, and it is precisely that feature which limits the representative power of Q, because it prevents us from representing systems of particles whose notion of particle number is not well defined, for the sole reason that q-sets without a cardinal are impossible within that framework. The paradigmatical examples in which that kind of situation arises are the cases of quantum systems that are not in the eigenstate of a particle number operator, a common situation in relativistic quantum physics. So, according to Domenech and Holik, granting that every q-set has a quasi-cardinal is to grant too much. After that discussion, it follows that great weight is endowed to thesis 2, and we can see clearly its relation to thesis 1: since we want some qsets to remain without an associated quasi-cardinal, and since that seems impossible when we adopt the notion of quasi-cardinal as a primitive notion of quasi-set theory, one possible way out consists in trying to define the notion of quasi-cardinal, and we should try to do it in such a way that some q-sets fail to have a well defined quantity of elements. Basically, that, together with thesis 3, is the program developed in [2]. Before considering the proposal made by the authors to surpass those (as they see it) undesired features of Q, it is interesting to mention here that the main motivation for the first development of quasi-set theory was non-relativistic quantum theory. It is true French and Krause gave the first steps for the treatment of quantum field theories inside quasi-set theory in [5, Chap. 9]. But, in spite of that, the non-relativistic theory has provided the main ideas for its development. Then, even if we may try to accommodate some features of quantum field theory in Q, quasi-set theory was

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first designed to deal with non-relativistic quantum physics, and certainly Domenech and Holik are right when they claim that some modifications are in order if we want to deal properly with the relativistic theory. However, we believe they are not completely right on claiming quasi-set theory, as it stands, is defective, because it simply was not meant to deal with relativistic systems. Domenech and Holik propose the introduction of the quasi-cardinal concept as a defined notion, no longer considering it as primitive. Now, let us see, informally, how they attempt to do it. The core of the definition suggested by them is based on the idea that one can furnish a procedure according to which it is possible to take out the elements of a collection one by one, until the collection become empty if it is finite. That is, given a non-empty q-set A, we can, using the resources of Q (plus two extra axioms introduced by them), take out from A what we would intuitively count as a unique element of it, and verify whether the q-set thus obtained, call it A , is empty or not. If A is empty, we know A had only one element. If A is not empty, on the other hand, we repeat the procedure taking out another element from it. Once again, we verify whether the obtained q-set, call it A

, is empty or not. If it is empty, we stop there, if not, we go on taking one more element out of it. The procedure goes on that way. In some cases, for some q-sets, the procedure will eventually reach the empty q-set in a finite number of repetitions of those steps, but in other cases, the procedure may be repeated an indefinite number of times and we never reach the empty q-set. Two possibilities result from that: the q-set may be “emptied” in a finite number of steps, and we say that it is finite by definition, or we can not get it emptied through such a procedure, and we call it infinite. Let us deal with the two cases separately. In the first possibility, we apply to a given q-set A a procedure which eventually reaches the empty q-set, giving us, according to the authors’ terminology, a descending chain of the following type: ∅ ⊆ · · · ⊆ A

⊆ A ⊆ A, where A denotes the q-set which results from A, when we take out of it “only one” element (A is called a direct descendant of A in Domenech and Holik’s terminology), and the same holds for A

in relation to A , etc. In the second possibility, the descending chains will be such that it does not matter how many elements of A we pick out from the q-sets forming each member of the chain, we never reach the empty q-set, and the chains will have the following form: · · · ⊆ A

⊆ A ⊆ A. In these cases, the q-set will not have a quasi-cardinal associated with it. For the chains of the first type, those reaching the empty q-set in a finite number of steps, the authors show how we can “count” the number of steps necessary to perform the procedure. The idea is that we can associate a natural number to each element of the descending chain, and that association will satisfy the condition that some natural number n will be attributed to A, number n − 1 to A , the direct descendant of A, and so on, until we attribute number 0 to the empty q-set. It can be demonstrated that this natural number, in case there is one, is unique. Also, we note the definition of finite q-set presented by these authors rests essentially on this particular notion of

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numbering: the finite q-sets are those for which a natural number exists which can be used to count the steps of a descending chain from this q-set which reaches the empty q-set. As we shall see, since that number is unique, it will also be, by definition, the quasi-cardinal of the mentioned q-set. As a first consequence of that definition, we have that every finite q-set (in the sense specified above) will have a finite quasi-cardinal associated with it, and that some q-sets remain without an associated quasi-cardinal. Could the ones falling in the last category be candidates to represent systems without a well defined particle number? Well, to consider this question, there is one important feature of Domenech and Holik’s definition that deserves to be mentioned: the q-sets that remain without an associated quasi-cardinal, according to their definition, are precisely those that are not finite. It must be noticed that Domenech and Holik recognize this fact, and do not claim that the present definition provides a satisfactory solution to the problem posed in thesis 2.2 In fact, there is a good reason to follow Domenech and Holik here and not accept such an easy way out: if we consider the q-sets that do not have an associated quasi-cardinal as representing physical systems for which the concept of particle number is not well defined, we end up knowing that all such collections are in fact infinite. Then, intuitively, we know their cardinality, even though we are not allowed to talk about it, should be at least ℵ0 . Hence, it is not the case that the quasicardinal of those q-sets is not really well-defined, it is simply left unspecified. That situation may be easily understood by the use of a classical analogous: when working in ZFC, nothing prevents us from defining cardinality for finite sets only. It would not follow, under that circumstances, that for infinite collections the cardinal is not well defined, it is simply unspecified. It seems to us that the same holds in the case of Domenech and Holik’s definition in Q. Then, as those authors suggest, further studies should be done in order to both apply the definition also to infinite q-sets as well as to encompass in Q systems without a well-defined cardinal. Now, considering the main idea underlying Domenech and Holik’s definition, one of its greatest virtues concerns the fact that, at least for the particular process of emptying a q-set they describe, it does not matter which specific element is taken out in each particular step, because ideally, the number of elements of the q-set does not depend on the order in which each element is taken and neither on the identification of which element is taken at a specific step. Thus, the impossibility of identification of some elements will not forbid the counting of q-sets whose elements are indistinguishable. Obviously, that idea mirrors some kind of laboratory practice, where we can employ a specific procedure to count the elements of a collection without being able to identify them. In order to give an example of such procedure, we consider the case in which we wish to count how many electrons an Helium atom has. We can put the atom in a cloud chamber and ionize it with radiation; the result will be one track of an ion and one of an electron. Repeating the procedure, we would once again see the track of one ion and one electron. Since after this stage we can not extract more electrons, the process ends in two steps, and we know the atom has two electrons. 2 We thank an anonymous referee for pointing this to us. Domenech and Holik are cautious enough to

suggest that some kind of improvement should be made in quasi-set theory, but they do not claim to have provided for such an improvement with their definition.

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Notice that it does not matter which electron was the first and which one was the second, we can not identify them, although we can determine how many of them there are (see [2, p. 867]). Despite the ingenious idea lying behind this technical apparatus, in order to have all that in our hands, we need to adopt two specific additional postulates. Then, although the proposed definition is economic in terms of primitive notions, because it treats quasi-cardinality as a defined concept, the specific approach of Domenech and Holik still needs the introduction of two additional axioms, because without them, the desired results can not be demonstrated and the desired concept of quasi-cardinal can not be adequately introduced. As we see it, the mentioned fact imposes a weakening on thesis 1. We shall postpone to a later section a detailed formulation of the axioms introduced by them, mentioning those axioms now only to show they are of a different nature from the other axioms of quasi-set theory. The first axiom grants to us the existence of descending chains, at least one for each non-empty q-set (see [2, p. 868]), whether it is finite or infinite. In fact, it is not easy to see how that can be proven in Q for q-sets in general without the axioms for quasi-cardinal. As we can see, this is an existential statement of non-constructive character, granting us that some q-set exists (the descending chain) without specifying how we can obtain it. There is also a second axiom added to Q to grant that, given finite q-sets x and y such that x ≡ y, if x ⊆ y, then x is equal to y, but if y ⊆ x, then x is equal to y (see [2, p. 869]). This last axiom works to the effect that as we proceed in the elimination of elements of a q-set A, we can always distinguish a qset from its direct descendent. Later on, we shall present the formal versions of both axioms, written in the language of Q. For this moment, we have only presented them informally, in order to motivate Domenech and Holik’s definition of quasi-cardinal and to discuss their role in the foundations of quasi-set theory. Still concerning the last point, it is interesting to consider the motivations to adopt those axioms. That is, the reasons we should have to justify the addition of those statements as axioms in our theory of quasi-sets. The first reason we may propose, obviously, is that without them we can not proceed to state the definition of finite quasi-cardinal according to Domenech and Holik. But, from a foundational point of view, the introduction of new axioms in a theory, and mainly the ones postulating nonconstructively the existence of some collection, must be motivated somehow, and we must show the adoption of the extra axioms is a reasonable and fruitful addition to the ones already accepted (see [8] for the motivation of the axioms for ZFC). In the case of quasi-set theory, we would like to know, for example, whether the new axioms proposed by Domenech and Holik would help us to understand the universe of quasisets intuitively described by the already accepted axioms, or whether they would help us to settle some important problems which cannot be dealt with adequately only with the help of the later. We may, then, ask for the reasons to adopt as an axiom the statement granting that for every non-empty q-set there exists at least one descending chain (Domenech and Holik’s first axiom), and how the adoption of that statement may help us to understand, for example, the notion of quasi-cardinal. The crucial question, then, seems to be: is there something that can be said in favour of Domenech and Holik’s axioms? How could we justify the acceptance of those two additional postulates as definitive postulates of Q? Domenech and Holik

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give us no hints on that topic. As we have said before, the first reason that comes to our mind is that the new postulates should be accepted because they allow us to define the notion of quasi-cardinal. However, we might give a counter-argument, then, by stating that using the same idea we may also justify the axioms for quasi-cardinals, the ones employed in the theory when it is formulated with that concept as primitive, since they even allow us to deduce the facts about descending chains postulated by Domenech and Holik. Moreover, one may argue that to assume quasi-cardinality as a primitive notion has an advantage, because in this case we do not need to make a détour through notions like descending chains and direct descendants in order to operate with quasi-cardinals, we can go straight to the point. As a second reason favoring Domenech and Holik’s axioms, we could argue on their plausibility; they make perfect sense in these authors’ construction context, and seem innocuous to the other well established parts of the theory. But, looking at it from the other side, if plausibility is all that matters, we must concede the axioms which were originally formulated for the notion of quasi-cardinal are also plausible. They generalize well-known properties of cardinals, thus, we also have good reasons to adopt them. Therefore, it seems much more should be said on this topic, the study of the foundations of quasi-set theory must be more deeply investigated if we wish to have a better understanding of the impact of those new axioms in the theory. In Sect. 5 we deal with some of these questions when restricted to finite q-sets, and show that the statements used by Domenech and Holik as axioms can be derived as theorems in Q in this particular case. We leave the discussion of the general case, though, for another occasion. Now, let us see an alternative definition of quasi-cardinals.

4 Finite Quasi-cardinals In this section, we present an alternative definition of finite quasi-cardinals in Q. A much simpler definition than the one presented by Domenech and Holik, it needs no additional axioms. For simplicity, throughout this section, we shall use the word “cardinal” instead of “quasi-cardinal”. Our presentation is rigorous, although we do not give all the proofs. The motivation behind the definition we present now is the same as Domenech and Holik’s: we can make a finite q-set empty, by taking out its elements one by one. So, we may count the number of steps required to complete that process and say it is the number of elements from the q-set, i.e., its cardinal. In order to do that, we do not need to identify the elements being taken out at each step, and so, this process is suitable to determine the cardinal of collections of m-atoms. First of all, we state the axiom of choice in the formulation we will use: Axiom 1 (AC) If A is a q-set whose elements are non-empty q-sets, then, there is a q-function f such that for every B ∈ A, f (B) ∈ B. Since we want to restrict our approach to finite q-sets, we must give an account on what we mean by “finite”. We should remark first that the usual definition of finite collections, stating that a collection is finite if and only if there is a one-toone correspondence between that collection and a natural number is not available in

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quasi-set theory. The reason, as we have discussed before, is that the notion of one-toone correspondence is formulated in Q using the quasi-cardinal concept, and that is no longer part of the theory’s vocabulary being used as our framework now (since it is the concept we wish to define). For this reason, we employ the Tarskian definition of finiteness, but, before stating it, we need the notion of a ⊂-minimal element: Definition 2 Given a q-set B whose elements are q-sets, we call an element A of B ⊂-minimal if ∀C(C ∈ B → ¬(C ⊂ A)). Now, the Tarski finite q-sets are defined as follows: Definition 3 A q-set A is finite in the sense of Tarski if every non-empty collection of subsets of A have a ⊂-minimal element. In the following, when we talk about finite q-sets, we are talking about Tarskifinite q-sets, unless it is stated otherwise. The next step is the definition of the strong singleton from an item A, which we will denote by A , as in [5, p. 293]. Definition 4 1. If A ∈ B, we call SA the q-set [S ∈ P(B) : A ∈ S] (here, P denotes the power set operation);  2. A =def t∈SA t. Informally speaking, the strong singleton A is a q-set containing what we would intuitively take as only one element indistinguishable from A (if A is m-atom, we can not prove in Q that the element in question is A itself, proving that would require identity). Domenech and Holik have also argued that notion precisely encapsulates the intuitive idea of a q-set with only one element [2, p. 865]. Next and speaking intuitively once again, we define a function which takes elements out of any q-set, and it performs that task by taking “only one at each time”. We call it subtraction function. Given a finite q-set A, by (AC), there is a choice function g for P(A)\{∅}, where \ denotes the difference operation between q-sets. Thus, we define the subtraction function h from P(A) to P(A) as: Definition 5 1. If B is non-empty, then h(B) = B\g(B) ; 2. If B is empty, then h(B) = ∅. So, h “extracts” an element from each subset B of A, resulting in another subset of A that has, intuitively, one less element than B. Applying the function to the empty q-set, it results in the empty q-set again. We must remember that, when we say we extract “only one element” at a time, we are speaking only intuitively, as a kind of heuristic in the metamathematics, because to give a formal proof of that fact inside Q we would need the concept of cardinal already defined. Going ahead, let us remember that P(A) is a q-set whose elements are also q-sets, thus, the concept of identity is defined for it (since identity is defined for q-sets, see Definition 1). Under those conditions, the recursion theorem, which is also a theorem of Q, can be applied. We define by recursion a q-function f from ω to P(A):

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Definition 6 1. f (0) = A; 2. f (n + 1) = h(f (n)). The function f defined that way counts the number of times we repeat the operation of taking one element out of a finite q-set A. In other words, we start with the original q-set A, when 0 elements are taken out, and map (with f ) 0 to A. Each time we take one element out from A we put the convenient number in the argument of the function; then, for example, when we take out the first element from A (that is, when we apply the procedure for the first time), we map 1 (by f ) to the q-set obtained that way from A. In the next step, when we are taking a second element from A, we map 2 to the subset obtained from A in the previous step, and so on. Since A is a Tarskifinite q-set, we can be sure the sequence so defined will always arrive at the empty q-set for some n ∈ ω. Really, if it were otherwise it would violate the definition of Tarski-finiteness. Thence, any finite q-set will have only a finite number of stages in the process of “eliminating” its elements and becoming empty. Given this, we define the cardinal of a finite q-set as the least number in which we have reached the empty q-set while performing the process here described. Definition 7 The cardinal of A, denoted qc(A), is the least natural number n such that f (n) = ∅. With a little more work we may prove the cardinal of a q-set is unique, and so the introduction of the symbol qc is justified. We call counting function the q-function f given in the definition of cardinal above when it is restricted to the least n ∈ ω such that f (n) = 0. So, in order to establish the cardinal of a finite q-set, we must give it a counting function. Notice we did not need any further axiom to introduce the notion of cardinal. One might now go on and prove various results, for example, the theorem stating that for objects satisfying Z, the cardinal as defined above coincides with the usual cardinal, which was defined through ordinals. We shall not develop the theory any further here, because all we need is already at our disposal. Let us see how that definition is related to Domenech and Holik’s one.

5 The Relation Between both Definitions of Quasi-cardinal Now, let us see how Domenech and Holik’s definition may be translated into our terminology, and how the formulas they had used as axioms can be shown to be theorems, when they are restricted to finite q-sets. Here, we also present rigorously the ideas developed by those authors which we previously discussed only in an informal level. Differently from the mentioned authors, we will restrict ourselves to finite q-sets in the sense of Tarski. Therefore, even though one of the axioms introduced by them is also valid for infinite q-sets (the one granting the existence of descending chains for non-empty q-sets), we will derive it, here, from our approach restricting it to finite q-sets. This can be done without damaging the work of the mentioned authors, because that axiom is never applied to infinite q-sets on their way to define finite quasi-cardinal. Besides, we shall see every Tarski-finite q-set is also finite in Domenech and Holik’s sense.

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The first important notion present in Domenech and Holik’s work not present in the usual formulations of quasi-set theory is the idea that Y is a direct descendant from a q-set X, denoted by DDX (Y ), with the proviso that X is non-empty. Formally, we define the symbol DDX (Y ) as follows: Definition 8 DDX (Y ) ↔ ∃z(z ∈ X ∧ Y = X\z ). We could generalize that definition to include the empty q-set with the following condition: if X is empty, it is its own direct descendant. Taking into account our definition of quasi-cardinals given in the previous section, how can we understand the direct descendants in that framework? It follows immediately that, for the empty q-set, in our approach, given a counting function f for X, f (0) = X, and as we have just stipulated, X is its own direct descendant. For a finite non-empty q-set in general, given a counting function f , if qc(X) = n, then, for each f (k) in the counting process of the quasi-cardinal definition, with k ≤ n − 1, we will have f (k + 1) as the direct descendant from f (k), because according to the definition of f , f (k + 1) = h(f (k)) = f (k)\g(f (k)) , and so, f (k + 1) is the right candidate, it satisfies the definition to be a direct descendant from f (k). The next important concept is the descending chain of X. This notion is introduced by Domenech and Holik for q-sets in general, finite and infinite. But here, as we have already said before, we shall deal only with the finite case. It is important to mention that we have not introduced, yet, the definition of finite and infinite q-sets as proposed by those authors; instead, here we are considering finite in the sense of Tarski. The definition for the symbol CDX (γ ), which reads “γ is a descending chain from X”, is the following: Definition 9 CDX (γ ) ↔ (γ ∈ ((X)) ∧ X ∈ γ ∧ ∀z∀y(z ∈ γ ∧ y ∈ γ ∧ z = y → (z ⊆ y ∨ y ⊆ z)) ∧ ∀z(z ∈ γ ∧ z = ∅ → ∃Y (Y ∈ γ ∧ DDZ (Y ) ∧ ∀w(w ∈ γ ∧ DDZ (w) → w = Y )))). In order to translate that definition in the terms of our approach, it is enough to mention that according to our definition, given a q-set X with a counting function f and such that qc(X) = n, we might consider the q-set γ = [f (k) : 0 ≤ k ≤ n] as a descending chain from X. If we consider each of the conjunctions in the Definition 9 of a descending chain as a condition imposed on γ , we will easily see the q-set, which was proposed by us, satisfies each of the conditions. Notice the first clause is trivially satisfied, and, since by definition X = f (0), then, f (0) ∈ γ , and the second condition is also immediately satisfied. Moreover, we must notice that for f (k) and f (m) elements of γ representing different steps in the counting of X, if k < m, we have immediately that f (k) ⊆ f (m), satisfying the third clause. As to the last condition, we have that for any f (k) such that k = n, as we saw before, we will have f (k + 1) as its direct descendant, and its uniqueness can be easily verified. With those notions translated into our scheme, we may show the first formula used as an axiom by Domenech and Holik is a theorem in quasi-set theory, when it is restricted to finite q-sets. The first axiom proposed by them is the Descending chains axiom, and as we commented before, it postulates non-constructively the existence of a particular kind of q-sets:

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Axiom 2 ∀Q X(X = ∅ → ∃γ (CDX (γ ))). Theorem 1 The sentence employed by Domenech and Holik as their first axiom, when restricted to finite q-sets, is a theorem of Q. Proof First note that for any non-empty finite q-set X, there is always a choice function g, and so, its subtraction function h is well defined. The counting function for X is established by the recursion theorem as we presented before. As we have argued, this counting function always comes to an end for some n, and we will have by definition that qc(X) = n. As we have indicated before, with this we guarantee the existence of a descending chain for X.  It is important to emphasize, again, that we are talking about finite q-sets only, because in our approach infinite q-sets play no prominent role, and in Domenech and Holik’s work infinite q-sets play no role in the definition of finite quasi-cardinal. As in the case of direct descendants, for a greater generality, it would be reasonable to concede that the empty q-set has a descending chain, whose only element is the empty q-set itself, and as we have argued above, it is not difficult to notice our definition would immediately grant a descending chain to that case. It would be interesting to consider our definition when not restricted to the finite q-sets. In that case, when it is applied to an infinite q-set, a counting function would not reach the empty q-set at any finite number of steps, and the result would match exactly the role proposed by Domenech and Holik to their descending chains, when they are related to infinite q-sets. That could help us to prove, indirectly, that infinite q-sets have a denumerable subset. We restrict ourselves, here, to the finite case, and as we have mentioned before, we are not particularly concerned here with an attempt to provide an answer to the problem raised by thesis 2. Our next step is to consider the authors’ definition of finite q-set. The finite q-sets, in our approach, will also be finite in theirs. Regarding that notion, we can see a difference in both approaches by noting we first define finite q-sets, and then, we go on defining quasi-cardinality for them. On the other hand, the mentioned authors define quasi-cardinality for some q-sets, and then call them finite. In order to avoid any confusion, each time we mention that a q-set is finite, without further qualification, we mean finite in the sense of Tarski. For finiteness in the sense of Domenech and Holik we will write Fin(X). The definition of Fin(X) is the following: Definition 10 If X is a non-empty q-set Fin(X) ↔ ∃n(n ∈ ω ∧ ∀γ (CDX (γ ) → ∃F (F ⊆ γ × n+ ∧ qf (F ) ∧ n, X ∈ F ∧ ∀z(z ∈ γ → ∃j (j ∈ n+ ∧ j, z ∈ F )) ∧ ∀j (j ∈ n+ ∧ j = 0 → DDF (j ) (F (j − 1)))))). On that definition, qf (F ) means that F is a quasi-function and n+ is the successor of n, which is defined as usual by n+ =def n ∪ {n}. We could, again, for the sake of generality, allow the empty q-set to be finite according to their sense, and that could be easily achieved by doing the generalizations of direct descendant and descending chain that have been suggested above. In their presentation, Domenech and Holik derive that statement as a theorem, which relies on one additional axiom we will consider soon. The q-function F , in the Definition 10, “labels” the steps in the counting

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procedure, allowing us to know how many times the process of taking out an element from a q-set has been repeated. Now, we show that Tarski finiteness implies Fin. Theorem 2 If X is finite in the sense of Tarski, then Fin(X). Proof If X is finite in the sense of Tarski, then, according to our definition, there is a counting function f for X, such that qc(X) = n, for some n. As we have shown before, [f (k) : 0 ≤ k ≤ n] is a descending chain. Surely, the quasi-function F from n+ in [f (k) : 0 ≤ k ≤ n] defined as F (k) = f (n − k) satisfies the conditions above, and the theorem is proved.  The next step, before going to the definition of quasi-cardinal according to Domenech and Holik, consists in presenting their second axiom. That one is used to grant, for example, that for any finite q-set (in their sense), the quasi-function F , whose existence is granted in their definition of finiteness, is such that F (0) = ∅. Axiom 3 ∀Q X∀Q Y (Fin(X) ∧ Fin(Y ) ∧ X ≡ Y → ((X ⊆ Y → X = Y ) ∧ (Y ⊆ X → X = Y ))). In order to show that statement is a theorem in our system, we will employ an axiom of Q, the Axiom of Weak Extensionality, which can be formulated after our definition of quasi-cardinal is already available (and so, the axiom is restricted here to finite q-sets). Intuitively, the axiom grants that q-sets X and Y , having the same quantity of indistinguishable elements, are indistinguishable and reciprocally. More precisely, if for each equivalence class of X by the indistinguishability relation there is an equivalence class of Y by this same relation such that both have the same quasicardinality, and reciprocally, then, X and Y are indistinguishable, and the converse also holds (For more discussions on that axiom, see [5, pp. 290–291]. Theorem 3 The statement used by Domenech and Holik as their second axiom is a theorem of Q when it is restricted to finite q-sets. Proof Let X and Y be finite q-sets satisfying the hypothesis of axiom 2. Let us suppose that X ⊆ Y , but that X = Y . Then, for at least one z, z ∈ Y \X. In this case, either there is an element in Y from which z is indistinguishable, or there is no such element. In the second case, from the axiom of weak extensionality, since not every element of Y have a kind of “correspondent” in X, since there is no element in X indistinguishable from z and since every element of X is in Y , we have that it is not the case that X ≡ Y , contradicting our hypothesis. In the first case, if z is indistinguishable from some element of X, it will be the element of one of the equivalence classe of Y by the indistinguishability relation. It follows that, since X ⊆ Y , one of the classes from X will be a proper subclass of the corresponding class of elements from Y , and will have a smaller quasi-cardinal; thus, it follows from the weak extensionality axiom that it is not the case that X ≡ Y , contradicting our hypothesis again. A totally analogous argument deals with the case in which Y ⊆ X. 

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One should keep in mind that statement was derived with the help of our definition of quasi-cardinal. With that axiom in our hands and following the approach of Domenech and Holik, some propositions can be derived to grant that the n in the definition of finiteness is unique, and that allows them to define quasi-cardinal (see [2, p. 870]). We shall use the symbol qcard(X) to distinguish it from our qc(X). The definition is the following: Definition 11 If X is a non-empty q-set and Fin(X), then, qcard(X) = n, where n is the natural number that appears in the definition of Fin(X). Also, by definition, qcard(∅) = 0. Now, let us show how this definition relates to ours: Theorem 4 For any finite q-set X, if qc(X) = n, then, also qcard(X) = n. Proof Let X be a finite q-set in the sense of Tarski. As we saw above, it follows that also Fin(X). Given a counting function f for X, let us suppose that qc(X) = n. Then, as we argued above, γ = [f (k) : 0 ≤ k ≤ n] is a descending chain for X, and the q-function F from n+ in γ such that F (k) = f (n − k) satisfies the definition of Fin(X), and in particular F (n) = X. It then follows immediately that qcard(X) = n.  Now, using the definition of qcard one can follow the work of Domenech and Holik and derive, for finite q-sets, the usual formulas used as axioms for quasicardinality when this is considered as a primitive in the theory (see [5, Chap. 7]). We will not pursue that work here, indicating only that it can be done by following the presentation in [2, pp. 871–873]. Since we can do with less everything done by Domenech and Holik, because we do not need any further axioms, our claim that our approach generalizes theirs seems to be justified.

6 Concluding Remarks We have argued Domenech and Holik are not fully justified in their criticism of Q. In fact, we believe it would be a nice research program to look for modifications of Q in which the “missing” q-sets might be represented adequately. But the theory, as it stands, was simply not designed for that. As we mentioned before, some suggestions to deal with quantum field theory were made in [5, Chap. 9]. But, deeper changes will have to be provided for in a version of the theory of quasi-sets willing to deal with the relativistic theory, in order to overcome the problems Domenech and Holik point out to the notion of quasi-cardinal. Also, the definition they proposed could be refined and improved. We hope we have convinced the reader that the concept of quasi-cardinal, when restricted to the finite case, may be adequately defined in Q, without further axioms. Furthermore, this is achieved in such a way that a kind of counting process is defined for finite qsets, even the ones whose elements are indistinguishable m-atoms. Therefore, we can

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make sense on the concept of counting even in the absence of identity for the items being counted. One of the great merits of Domenech and Holik’s paper is pointing in that direction, and here, we want to comment briefly on the weight of thesis 3, presented before. The possibility of counting without identifying the items being counted raises new interesting metaphysical questions: what is the relation between “counting” and identity? Since it seems possible to count indistinguishable things for which identity does not make any sense (through a non-standard notion of counting, as we have seen), the identity concept is not necessarily linked to the notion of counting. Then, one can coherently envision that identity may be absent from a conceptual framework dealing with quantum mechanical particles, as it was argued in [5]. Thus, as research progresses, more and more conceptual tools can be added to the Received View bag and it cannot be charged with incoherence or conceptual fuzziness. Therefore, besides being interesting for discussions on foundations of quantum mechanics, an interesting link is made between a metaphysical question on the possibility of counting without identity, and some effective processes carried out in laboratory. Those connections come to give plausibility not only to the Received View, as we have mentioned before, but also, to the more general metaphysical thesis that there might be items for which counting can be separated from identity (see [7, Chap. 3]). Furnishing alternative notions of counting that encode some process performed in the laboratory, like Domeneh and Holik’s definition does, makes the metaphysical thesis look more respectable to those who tend to see it with diffidence. Interesting as they are, though, we leave those further discussions for a future work.

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