On a quasi-set theory
In modern physics there are apparent paradoxes concerning the identity or distinguishability of elementary particles. The author constructs a set theory \(S^*\) that takes this into account by allowing the presence of a certain kind of Urelement (called an \(m\)-atom) which does not obey the usual laws of identity. In addition to the usual apparatus of quantification theory, \(S^*\) has three unary predicate letters \({\mathfrak m}(x)\) (``\(x\) is an \(m\)-atom''), \({\mathfrak M}(x)\) (``\(x\) is an ordinary Urelement''), and \(\mathbb{Z}(x)\) (``\(x\) is a standard set, that is, the transitive closure of \(x\) contains no \(m\)-atoms''). There are two predicate letters, \(\in\) (membership) and, instead of \(=\), an indistinguishability relation \(\equiv\). Lastly, there is a unary function symbol qcard (quasicardinality). A quasiset is defined to be anything that is not an \(m\)-atom or an ordinary Urelement, and the axioms imply that anything that has a member is a quasiset and that all standard sets are quasisets. Suitable axioms are assumed so that the theory ZFU of sets with Urelements in interpretable in \(S^*\). (The replacement and choice axioms seem to be stated improperly, but the mistakes can be easily rectified.) For non-\(m\)-atoms, the relation \(\equiv\) satisfies the usual properties of an identity relation. The quasicardinal operation generalizes the usual cardinality operation to arbitrary quasisets; this is necessary because the usual connection between cardinality and one-one correspondences does not apply to quasisets.
scientific article; zbMATH DE number 120374