On a quasi-set theory
type: Item
instance of: scholarly article
title: On a quasi-set theory
zbMATH Open document ID: 0774.03032
author: Décio Krause
published in: Notre Dame Journal of Formal Logic
publication date: 1993-02-22T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 81P99
zbMATH DE Number: 120374
zbMATH Keywords: set theory for modern particle physics
zbMATH Keywords: quasicardinality
zbMATH Keywords: Urelement
zbMATH Keywords: \(m\)- atom
zbMATH Keywords: quantification theory
zbMATH Keywords: indistinguishability relation
zbMATH Keywords: quasiset
reviewed by: Elliott Mendelson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1305/ndjfl/1093634404
OpenAlex ID: W2018421420
DOI: 10.1305/NDJFL/1093634404
description: scientific article; zbMATH DE number 120374