Strong, universal and provably non-trivial set theory by means of adaptive logic

Abstract

In this paper I present a non-trivial but inconsistent set theory based on the axioms of naive set theory. The theory is provably non-trivial and strong enough for most of the applications of regular mathematics. This is realized by distinguishing between strong and weak set membership and allowing for the derivation of strong membership from weak membership whenever this is not problematic (it does not lead to paradoxes). This idea of applying rules whenever unproblematic is formalized by means of an adaptive logic.

Against the iterative conception of set

“…The argument for adopting a paraconsistent version of naive set theory is articulated in Priest (2002). More recently, paraconsistent set theories have been developed and defended by Weber (2010Weber ( , 2012 and Verdee (2013).…”

Adaptive Fregean Set Theory

“…Verdée continued to work on the topic. This resulted in several lectures and papers [24,25] about some four Fregean set theories. For all straightforward choices of adaptive logics AL, Verdée found that the presence of Curry sets caused a major problem.…”

Paraconsistent Set Theory

“…Already in 1994, a naive set theory with paraconsistent basis was investigated in [15]; in 2010, Weber (see [16]), by revisiting Sylvan's ideas, introduced an axiomatic system for naive set theory (i.e., with a full comprehension principle) in a paraconsistent setting based on relevant logic. In [17] a non-trivial (but inconsistent) set theory based on unrestricted comprehension was also proposed, formalized by means of an adaptive logic.…”