A critical study on the concept of identity in Zermelo-Fraenkel-like axioms

Abstract

According to Cantor, a set is a collection into a whole of defined and separate (we shall say distinct) objects. So, a natural question is ``How to treat as `sets' collections of indistinguishable objects?". This is the aim of quasi-set theory, and this problem was posed as the first of present day mathematics, in the list resulting from the Congress on the Hilbert Problems in 1974. Despite this pure mathematical motivation, quasi-sets have also a strong commitment to the way quantum physics copes with elementary particles. In this paper, we discuss the axiomatics of quasi-set theory and sketch some of its applications in physics. We also show that quasi-set theory allows us a better and deeper understanding of the role of the concept of equality in mathematics.