We propose in this paper a family of algebraic models of ZFC based on the three-valued paraconsistent logic LPT0, a linguistic variant of da Costa and D’Ottaviano’s logic J3. The semantics is given by twist structures defined over complete Boolean agebras. The Boolean-valued models of ZFC are adapted to twist-valued models of an expansion of ZFC by adding a paraconsistent negation. This allows for inconsistent sets w satisfying ‘not (w = w)’, where ‘not’ stands for the paraconsistent negation. Finally, our framework is adapted to provide a class of twist-valued models generalizing Löwe and Tarafder’s model based on logic (PS 3,∗), showing that they are paraconsistent models of ZFC. The present approach offers more options for investigating independence results in paraconsistent set theory.