Article révisé par les pairs
|In this paper, we study when a set-continuous operator has a fixed-point that is the intersection of a directed family. The framework of our study is the Kelley-Morse theory KMC- and the Gödel-Bernays theory GBC-, both theories including an Axiom of Choice and excluding the Axiom of Foundation. On the one hand, we prove a result concerning monotone operators in KMC- that cannot be proved in GBC-. On the other hand, we study conditions on directed superclasses in GBC- in order that their intersection is a fixed-point of a set-continuous operator. Finally, we illustrate our results with a solution to the liar paradox and a construction of maximal bisimulations.