## 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*

Previous |

Index |

Next