Quasi-set theory is mainly motivated by the assumption that certain objects treated in quantum physics are indistinguishable and don't have individuality.
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.
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.
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.
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.
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 that are 'singletons', although we cannot distinguish among them.
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.
Using quasi-set theory, we can express some facts of quantum physics without introducing symmetry conditions (Krause et al. 1999, 2005).
Quasi-set theory is a way to operationalize Heinz Post's (1963) claim that quanta should be deemed indistinguishable "right from the start."