Quasi-set Theory - Some Further Details

Some Further Details

Intuitively, a quasi-set is a collection of objects such that some of them may be indistinguishable without turning out to be identical. Of course this is not a strict `definition' of a quasi-set, but act more or less as Cantor's `definition' of a set as ``any collection into a whole "M" of definite and separate, that is, distinguishable objects "m" of our intuition or our thought" serving just to provide an intuitive account of the concept. For detail we recommend the discussion in (French and Krause 2006).

The quasi-set theory, that have been denoted by has in its main motivations some considerations taken from quantum physics, mainly in considering Schr\"odinger's idea that the concept of identity do not make sense when applied to elementary particles (Schr\"odinger 1952, pp.17-18). In his words, he considered just non-relativistic quantum mechanics. Another motivation is, in our opinion, the need, stemming from philosophical worries, of dealing with collections of absolutely indistinguishable items that not need be the same ones. (This is of course a way of speech.) Of course, viewed from a formal point of view, can also be developed independently of any intended interpretation, but here we shall always keep in mind this `quantum' motivation since, after all, it is the intended interpretation that has originated the problem of the development of the theory.

The first point is to guarantee that identity and indistinguishability (or indiscernibility) will not collapse into one another when the theory is formally developed. We assume that identity, that will be symbolized by `=', is not a primitive relation, but the theory has a weaker concept of indistinguishability, symbolized by `', instead. This is just an equivalence relation and holds among all objects of the considered domain. If the domain is divided up into objects of two kinds, the "m"-objects, that standing for `micro-objects', and "M"-objects, for `macro-objects', and quasi-sets of them (probably having other quasi-sets as elements as well), then the identity (defined with all the properties of standard identity of ZF) can be defined for "M"-objects and quasi-sets having no "m"-objects in their transitive closure. Thus, if we take just the part of theory obtained by ruling out the "m"-objects and collections (quasi-sets) whose have "m"-objects in their transitive closure, we obtain a copy of ZFU (ZF with Urelemente); if we further eliminate the "M"-objects, we get just a copy of the `pure' ZF.

Technically, expressions like "x = y" are not always well formed, because they are not formulas when either "x" or "y" denote "m"-objects. We express that by saying that the concept of identity does not make sense for all objects. One time else, it should be understood that this is just a way of speech. The $m$-objects to which the defined concept of identity does not apply are termed non individual by historical reasons (French and Krause, 2006). As a result from the axioms of the theory, we can form collections of "m"-objects which have no identity—in this sense; these collections may have a cardinal (termed its `quasi-cardinal') but not an associated ordinal. Thus, the concepts of ordinal and cardinal are independent, as in some formulations of ZF proper. So, informally speaking, a quasi-set of $m$-objects is such that its elements cannot be identified by names, counted, ordered, although there is a sense in saying that these collections have a cardinal which cannot be defined from ordinals.

It is important to remark that, when is used in connection with quantum physics, the "m"-objects are thought of as representing quantum entities (henceforth q-objects), but they are not necessarily `particles' in the standard sense. Generally speaking, whatever `objects' sharing the property of being indistinguishable can also be values of the variables of . For a survey of the various different meanings that the word `particle' has acquired in connection with quantum physics see (Falkenburg 2007).

Another important feature of is that standard mathematics can be developed using its resources, because the theory is conceived in such a way that ZFU (and hence also ZF, perhaps with the axiom of choice, ZFC) is a subtheory of . In other words, the theory is constructed so that it extends standard Zermelo-Fraenkel with "Urelemente" (ZFU); thus standard sets of ZFU must be viewed as particular qsets, that is, there are qsets that have all the properties of the sets of ZFU, and the objects of that corresponds to the "Urelemente" of ZFU are identified with the "M"-atoms of ). The `sets' in will be called "q"-sets, or just "sets" for short. To make the distinction, the language of encompasses a unary predicate "Z" such that "Z(x)" says that "x" is a set. It is also possible to show that there is a translation from the language of ZFU into the language of, so that the translations of the postulates of ZFU are theorems of ; thus, there is a `copy' of ZFU in, and we refer to it as the `classical' part of . In this copy, all the usual mathematical concepts can be stated, as for instance, the concept of ordinal (for the "q"-sets). This `classical part' of plays an important role in the formal developments of the next sections.

Furthermore, it should be recalled that the theory is constructed so that the relation of indiscernibility, when applied to "M"-atoms or "M"-sets, collapses into standard identity of ZFU. The "q"-sets are qsets whose transitive closure, as usually defined, does not contain "m"-atoms or, in other words, they are constructed in the "classical" part of the theory.

In order to distinguish between "Z"-sets and qsets that may have "m"-atoms in their transitive closure, we write (in the metalanguage) for the former and for the latter. In, we term `pure' those qsets that have only "m"-objects as elements (although these elements may be not always indistinguishable from one another, that is, the theory is consistent with the assumption of the existence of different kinds of "m"-atoms—that is, not all of them must be indiscernible from one another), and to them it is assumed that the usual notion of identity cannot be applied (that is, let us recall, "x = y", as well as its negation, are not well formed formulas if either "x" or "y" stand for "m"-objects). Notwithstanding, the primitive relation applies to them, and it has the properties of an equivalence relation.

The concept of ' extensional identity', as said above, is a defined notion, and it has the properties of standard identity of ZFU. More precisely, we write (read ' "x" and "y" are extensionally identical') iff they are both qsets having the same elements (that is, ) or they are both "M"-atoms and belong to the same qsets (that is, ). From now on, we shall not bother to always write, using simply the symbol "= for the extensional equality, as we have done above.

Since "m"-atoms are to stand for entities which cannot be labeled, for they do not enter in the relation of identity, it is not possible in general to attribute an ordinal to collections whose elements are denoted by "m"-atoms. As a consequence, for these collections it is not possible to define the notion of cardinal number in the usual way, that is, through ordinals. (We just recall that an ordinal is a transitive set which is well-ordered by the membership relation, and that a cardinal is an ordinal such that for no there does not exist a bijection from to . In the version of the theory we shall be considering, to remedy this situation, we admit also a primitive concept of quasi-cardinal which intuitively stands for the `quantity' of objects in a collection.(The notion of quasi-cardinal can be defined for finite quasi-sets; see Domenech and Holik 2007.) The axioms for this notion grant that certain quasi-sets "x" (in particular, those whose elements are "m"-objects) may have a quasi-cardinal, written, even when it is not possible to attribute an ordinal to them.

To link the relation of indistinguishability with qsets, the theory also encompasses an `axiom of weak extensionality', which states (informally speaking) that those quasi-sets that have the same quantity (expressed by means of quasi-cardinals) of elements of the same sort (in the sense that they belong to the same equivalence class of indistinguishable objects) are indistinguishable by their own. One of the interesting consequences of this axiom is related to the quasi-set version of the non observability of permutations, which is one of the most basic facts regarding indistinguishable quanta (for a discussion on this point, see French and Rickles 2003). In brief, remember that in standard set theories, if, then of course iff . That is, we can 'exchange' (without modifying the original arrangement) two elements iff they are \textit{the same} elements, by force of the axiom of extensionality. In contrast, in we can prove the following theorem, where (and similarly ) stand for a quasi-set with quasi-cardinal 1 whose only element is indistinguishable from "z" (respectively, from "w") --the reader shouldn't think that this element "is identical to either" "z" or "w", for the relation of equality doesn't apply to these items; the set theoretical operations can be understood according to their usual definitions):

Theorem: (Unobservability of Permutations) Let "x" be a finite quasi-set such that "x" does not contain all indistinguishable from "z", where "z" is an "m"-atom such that . If and, then there exists such that

The theorem works to the effect that, supposing that "x" has "n" elements, then if we `exchange' their elements "z" by corresponding indistinguishable elements "w" (set theoretically, this means performing the operation ), then the resulting quasi-set remains \textit{indistinguishable} from the one we started with. In a certain sense, it does not matter whether we are dealing with "x" or with . So, within, we can express that `permutations are not observable', without necessarily introducing symmetry postulates, and in particular to derive `in a natural way' the quantum statistics (see French and Krause 2006, chap.7). Further applications to the foundations of quantum mechanics can be seen in Domenech et al. 2008.