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Quasi-set theory is a formal mathematical theory for dealing with collections of indistinguishable objects, mainly motivated by the assumption that certain objects treated in quantum physics are indistinguishable.



The American Mathematical Society sponsored a 1974 meeting to evaluate the resolution and consequences of the 23 problems Hilbert proposed in 1900. An outcome of that meeting was a new list of mathematical problems, the first of which, due to Manin (1976, p. 36), questioned whether classical set theory was an adequate paradigm for treating collections of indistinguishable elementary particles in quantum mechanics. He suggested that such collections cannot be sets in the usual sense, and that the study of such collections required a "new language".

The use of the term quasi-set follows a suggestion in da Costa's 1980 monograph Ensaio sobre os Fundamentos da Lógica (see da Costa and Krause 1994), in which he explored possible semantics for what he called "Schrödinger Logics". In these logics, the concept of identity is restricted to some objects of the domain, and has motivation in Schrödinger's claim that the concept of identity does not make sense for elementary particles (Schrödinger 1952). Thus in order to provide a semantics that fits the logic, da Costa submitted that "a theory of quasi-sets should be developed", encompassing "standard sets" as particular cases, yet da Costa did not develop this theory in any concrete way. To the same end and independently of da Costa, Dalla Chiara and di Francia (1993) proposed a theory of quasets to enable a semantic treatment of the language of microphysics. The first quasi-set theory was proposed by D. Krause in his PhD thesis, in 1990 (see Krause 1992).

On the use of quasi-sets in philosophical discussions of quantum identity and individuality, see French (2006) and French and Krause (2006).On Schrödinger logics, see da Costa and Krause (1994, 1997), and French and Krause (2006).

Outline of the theory

We now exposit Krause's (1992) axiomatic theory \mathfrak{Q}, the first quasi-set theory; other formulations and improvements have since appeared. Krause builds on the set theory ZFU, consisting of Zermelo-Fraenkel set theory with an ontology extended to include two kinds of urelements:

Quasi-sets (q-sets) are collections resulting from applying axioms, very similar to those for ZFU, to a basic domain composed of m-atoms, M-atoms, and aggregates of these. The axioms of \mathfrak{Q} include equivalents of extensionality; axioms asserting the existence of the empty set, unordered pair, union set, and power set; Separation; the image of a q-set under a q-function is also a q-set; q-set equivalents of Infinity, Regularity, and Choice. Q-set theories based on other set-theoretical frameworks are, of course, possible.

\mathfrak{Q} has a primitive concept of quasi-cardinal, governed by eight additional axioms, intuitively standing for the quantity of objects in a collection. The quasi-cardinal of a quasi-set is not defined in the usual sense (by means of ordinals) because the m-atoms are assumed (absolutely) indistinguishable. Furthermore, it is possible to define a translation from the language of ZFU into the language of \mathfrak{Q} in such a way so that there is a 'copy' of ZFU in \mathfrak{Q}. In this copy, all the usual mathematical concepts can be defined, and the 'sets' (in reality, the '\mathfrak{Q}-sets') turn out to be those q-sets whose transitive closure contains no m-atoms.

In \mathfrak{Q} there may exist q-sets, called "pure" q-sets, whose elements are all m-atoms, and the axiomatics of \mathfrak{Q} provides the grounds for saying that nothing in \mathfrak{Q} distinguishes the elements of a pure q-set from one another, for certain pure q-sets. Within the theory, the idea that there is more than one entity in x is expressed by an axiom which states that the quasi-cardinal of the power quasi-set of x has quasi-cardinal 2qc(x), where qc(x) is the quasi-cardinal of x (which is a cardinal obtained in the 'copy' of ZFU just mentioned).

What exactly does this mean? Consider the level 2p of a sodium atom, in which there are six indiscernible electrons. Even so, physicists reason as if there are in fact six entities in that level, and not only one. In this way, by saying that the quasi-cardinal of the power quasi-set of x is 2qc(x) (suppose that qc(x) = 6 to follow the example), we are not excluding the hypothesis that there can exist six subquasi-sets of x which are 'singletons', although we cannot distinguish among them. Whether there are or not six elements in x is something which cannot be ascribed by the theory (although the notion is compatible with the theory). If the theory could answer this question, the elements of x would be individualized and hence counted, contradicting the basic assumption that they cannot be distinguished.

In other words, we may consistently (within the axiomatics of \mathfrak{Q}) reason as if there are six entities in x, but x must be regarded as a collection whose elements cannot be discerned as individuals. Using quasi-set theory, we can express some facts of quantum physics without introducing symmetry conditions (Krause et al. 1999, 2005). As is well known, in order to express indistinguishability, the particles are deemed to be individuals, say by attaching them to coordinates or to adequate functions/vectors like |ψ>. Thus, given two quantum systems labeled |ψ1> and |ψ2> at the outset, we need to consider a function like |ψ12> = |ψ1>|ψ2> ± |ψ2>|ψ1> (except for certain constants), which keep the quanta indistinguishable by permutations; the probability density of the joint system independs on which is quanta #1 and which is quanta #2. (Note that precision requires that we talk of "two" quanta without distinguishing them, which is impossible in conventional set theories.) In \mathfrak{Q}, we can dispense with this "identification" of the quanta; for details, see Krause et al. (1999, 2005) and French and Krause (2006).

Quasi-set theory is a way to operationalize Heinz Post's (1963) claim that quanta should be deemed indistinguishable "right from the start."

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  \mathfrak{Q} 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, ,\mathfrak{Q} 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 `\equiv ', 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  \mathfrak{Q} , 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 \mathfrak{Q} 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  \mathfrak{Q} . 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  \mathfrak{Q} 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  \mathfrak{Q} . 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  \mathfrak{Q} that corresponds to the "Urelemente" of ZFU are identified with the "M"-atoms of  \mathfrak{Q} ). The `sets' in  \mathfrak{Q} will be called "q"-sets, or just "sets" for short. To make the distinction, the language of  \mathfrak{Q} 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  \mathfrak{Q} , so that the translations of the postulates of ZFU are theorems of  \mathfrak{Q} ; thus, there is a `copy' of ZFU in  \mathfrak{Q} , and we refer to it as the `classical' part of  \mathfrak{Q} . 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  \mathfrak{Q} 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)  \{x : \varphi(x)\} for the former and  [x : \varphi(x)] for the latter. In  \mathfrak{Q} , 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,  x \not= y , are not well formed formulas if either "x" or "y" stand for "m"-objects). Notwithstanding, the primitive relation  \equiv 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 x = Ey (read ' "x" and "y" are extensionally identical') iff they are both qsets having the same elements (that is, \forall z (z \in x \Rightarrow z \in y) ) or they are both "M"-atoms and belong to the same qsets (that is,  \forall z (x \in z \Rightarrow y \in z) ). From now on, we shall not bother to always write = E, 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 α such that for no β < α there does not exist a bijection from β to α. 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 qc(x), 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 w \in x , then of course (x - \{w \}) \cup \{z\} = x iff z = w. 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  \mathfrak{Q} we can prove the following theorem, where [[z]] (and similarly [[w]]) 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 z \in x . If w \equiv z and  w \notin x , then there exists [[w]] such that (x - [[z]]) \cup [[w]] \equiv x

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 (x - [[z]]) \cup [[w]]), 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 (x - [[z]]) \cup [[w]]. So, within  \mathfrak{Q} , 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.

See also


  • Newton da Costa (1980) Ensaio sobre os Fundamentos da Lógica. São Paulo: Hucitec.
  • da Costa, N. C. A. and Krause, D. (1994) "Schrödinger logics," Studia Logica 53: 533-550.
  • ------ (1997) "An Intensional Schrödinger Logic," Notre Dame Journal of Formal Logic 38: 179-94.
  • Dalla Chiara, M. L. and Toraldo di Francia, G. (1993) "Individuals, kinds and names in physics" in Corsi, G. et al., eds., Bridging the gap: philosophy, mathematics, physics. Kluwer: 261-83.
  • Domenech, G. and Holik, F. (2007), `A Discussion on Particle Number and Quantum Indistinguishability', "Foundations of Physics" vol. 37, no. 6, pp 855-878.
  • Domenech, G., Holik, F. and Krause, D., “Q-spaces and the foundations of quantum mechanics”, Foundations of Physics 38 (11) Nov. 2008, 969-994.
  • Falkenburg, B.: 2007, "Particle Metaphysics: A Critical Account of Subatomic Reality", Springer.
  • French, Steven (2006) "Identity and Individuality in Quantum Theory," The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.).
  • French, S. and Krause, D. (2006) Identity in Physics: A Historical, Philosophical, and Formal Analysis. Oxford Univ. Press.
  • French, S. and Rickles, D. P. (2003), `Understanding Permutation Symmetry', in K. Brading and E. Castellani, "Symmetries in Physics: New Reflectio, Cambridge University Press, pp. 212-238.
  • Krause, Decio (1992) "On a quasi-set theory," Notre Dame Journal of Formal Logic 33: 402-11.
  • Krause, D., Sant'Anna, A. S. and Volkov, A. G. (1999) "Quasi-set theory for bosons and fermions: quantum distributions," Foundations of Physics Letters 12: 51-66.
  • Krause, D., Sant'Anna, A. S., and Sartorelli, A. (2005) "On the concept of identity in Zermelo-Fraenkel-like axioms and its relationship with quantum statistics," Logique et Analyse: 189-192, 231-260.
  • Manin, Yuri (1976) "Problems in Present Day Mathematics: Foundations," in Felix Browder, ed., Proceedings of Symposia in Pure Mathematics, Vol. XXVIII. Providence RI: American Mathematical Society.
  • Post, Heinz (1963) "Individuality in physics," The Listener, 10 October 1963: 534-537. Reprinted in (1973) Vedanta for East and West: 14-22.
  • Erwin Schrödinger (1952) Science and Humanism. Cambridge Un. Press.


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