Mathematical Logic Quarterly 46 (2):183-194 (2000)

Abstract
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
Keywords Class operator  Tree‐property  Gödel‐Bernay set theory  Fixed‐point  Kelley‐Morse set theory
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DOI 10.1002/(sici)1521-3870(200005)46:2
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