In this chapter, the basic theory of sets is developed axiomaticallyin a paraconsistent logic. The two main goals are (1) to establish atoolkit for elementary mathematics, and (2) to prove the mainantinomies of naive set theory. The two goals come together inproving the Burali-Forti paradox for the theory of ordinals. Alongthe way, results are proved about the universal set, various formsof “empty” sets, Russell’s set, the axioms ofZFC, fixed points, Cantor’s theorem, and the possibility of awell-ordering theorem. The Routley set is introduced and studied asa particularly inconsistent object.