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 (possibly 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 models, and to provide ``natural'' solutions for domain equations.

Philosophy of mathematics (msc 00A30), Modal logic (including the logic of norms) (msc 03B45), Nonclassical and second-order set theories (msc 03E70), Other hypotheses and axioms (msc 03E65), Models of arithmetic and set theory (msc 03C62), Other infinitary logic (msc 03C75), Foundations, relations to logic and deductive systems (msc 18A15), Hyperspaces (msc 54B20)
Software Engineering [SEN]

Baltag, A. (1998). STS : a structural theory of sets. Software Engineering [SEN]. CWI.