STS: a structural theory of sets

Abstract

We explore a non-classical, universal set theory, based on a purely 'structural' conception of sets. A set is a transfinite process of unfolding of an arbitrary (possible large) binary structure, with identity of sets given by the observational equivalence between such processes. We formalize these notions using infinitary modal logic, which provides partial descriptions for set structures up to observational equivalence. We describe the comprehension and topological properties of the resulting set-theory, and we use it to give non-classical solutions to classical paradoxes, to prove fixed-point theorems that relate recursion and corecursion, to formalize 'super-large', reflexive categories and 'super-large' circular modes, and to provide 'natural' solutions for domain equations. Note: a preliminary version of this paper was awarded 'the best paper award' at the 1998 conference on Advances in Modal Logic (AiML '98 - Uppsala) and will appear in the proceedings of AiML.