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Quasi-set theory: a formal approach to a quantum ontology of properties

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Abstract

In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for dealing with aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.

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Notes

  1. We are grateful to one of the anonymous reviewers for suggesting the need of further clarification of this point.

  2. This is similar to name \(\mathcal {R}\) the collection \(\mathcal {R} =\{x:x\notin x\}\) (Russell’s set), which can be expressed in the language of ZFC but is not a set of this theory, supposed consistent.

  3. For instance, in a Lithium atom \(1s^{2}2s^{1}\), it suffices to say that there is one electron in the outer shell; it does not matter which one (really, it makes no sense to identify a particular electron there).

  4. It is interesting here that we would be in trouble to teach quasi-set theory to children. For instance, take a qset x with cardinal 2 so that its elements (call them y and z) are indiscernible. Now try to write the qset \(\mathcal {P}(x)\). It cannot be done in a meaningful way. Actually, the two subsets with quasi-cardinal 1 are indiscernible (by the Weak Extensionality Axiom, see below), so something like \(\mathcal {P} (x)=[\emptyset ,[y],[z],x]\) has no clear meaning. Nevertheless, as we will see from axiom (\(qc_{7}\)), the quasi-cardinal of \(\mathcal {P}(x)\) is 4.

  5. As shown by Domenech and Holik (2007), we can define quasi-cardinals for finite qsets in \(\mathfrak {Q}\), without resulting that the qset will have an associated ordinal in the usual sense.

  6. When quasi-set theory is applied to elemental particles, this allows us to say that the two electrons in an Helium atom in its fundamental state have different values of spin in a given direction.

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Authors and Affiliations

  1. Instituto de Física La Plata - CONICET, La Plata, Argentina

    Federico Holik

  2. University of Buenos Aires and ANPCyT, Buenos Aires, Argentina

    Juan Pablo Jorge

  3. Federal University of Santa Catarina, Florianópolis, Brazil

    Décio Krause

  4. Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Décio Krause

  5. University of Buenos Aires and CONICET, Buenos Aires, Argentina

    Olimpia Lombardi

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  1. Federico Holik
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  2. Juan Pablo Jorge
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  3. Décio Krause
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  4. Olimpia Lombardi
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Correspondence to Federico Holik.

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Holik, F., Jorge, J.P., Krause, D. et al. Quasi-set theory: a formal approach to a quantum ontology of properties. Synthese 200, 401 (2022). https://doi.org/10.1007/s11229-022-03884-8

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