Notes on inconsistent set theory

Zach Weber

pp. 315-328

in: Koji Tanaka, Francesco Berto, Edwin D. Mares, Francesco Paoli (eds), Paraconsistency, Berlin, Springer, 2013

Abstract

The purpose of this paper is to highlight and discuss two ideas that play in to the axiomatic development of a paraconsistent naive set theory. We will focus on aspects of the theory that can be read right off the axioms, concerning intensional identity and unrestricted set existence. Both relate to inconsistency. To begin I lay out a relevant background logic, placing a strong emphasis on the restrictions such a logic must have in order to support an inconsistent set theory. The sections that follow proceed on the understanding that, while highly inconsistent, a good deal of control is being exerted on the theory through the weakened logic. The two features of a fully naive theory, identity and self-reference, dovetail throughout.