Sets as singularities in the intensional universe

Abstract

This paper is motivated by the search for a natural and deductively powerful extension of classical set theory. A theory of properties U is developed, based on a system of relevant logic related to RQ. In U the set {a, b, c,...} is identified with the property [x: x=a ∨ x=b ∨ x=c...]. The universe of all sets V, is identified with the property of being a hereditary set. The main result is that relevant implication → collapses to material implication ⊃ for sentences with quantifiers restricted to V. This demonstrates the naturalness of the system. However, an aparent lack of deductive power leads to the conclusion that the best extension of classical set theory is to be found in intensional theories with the unrestricted comprehension schema based on weak relevant logics. The author has obtained similar collapses of → to ⊃ for these systems.