Inconsistent set theory lives on. My paper "Transfinite Cardinals in Paraconsistent Set Theory," has just been accepted by the Review of Symbolic Logic. Here's the abstract:
"This paper develops a (non-trivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem."