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

In this article, I present a non-trivial but inconsistent set theory based on unrestricted comprehension. 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.