Models of Self-Descriptive Set Theories

F. Honsell

Abstract: This paper is about Set Theory as it was originally intended: i.e. as a theory of the extensions of properties. We investigate, and prove relatively consistent to ZF, several restrictions to Frege's inconsistent Comprehension Principle in Set Theory different from Zermelo's ``Limitation of Size'' Principle.

More precisely we discuss models for highly self referential and self descriptive Set Theories. In these theories many interesting classes such as the membership relation or the class of all sets are themselves sets but sets are nonetheless closed under interesting operations (e.g. intersection, union and power set). We deal with models of non purely set theoretical theories for the foundations of Mathematics.

Our models carry naturally a peculiar topological structure. In fact any of them can be viewed as a x-compact x-metric space which coincides with the space of its closed subsets equipped with Hausdorff's x-metric. The comprehension properties of these models are a consequence of this topological structure.

Ideas and techniques in the theory of non-well-founded sets play a crucial role in this paper. Techniques similar to these have been widely used also in the theory of transition systems. In fact, the analogy of ``sets as processess'' establishes a correspondence between many concepts in these two areas, e.g. f-admissible relations and strong bisimulations, greatest f-admissible relation and strong observational congruence. We hope therefore that the paper might be of inspiration to the theory of process algebras, and many of the constructions developed in this paper might be fruitfully carried to that domain.

LFCS report ECS-LFCS-88-47

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