Dana Scott: Parametric Sets and Virtual Classes

Abstract

In an axiomatic development of geometry, there is much convenience to be found in treating various loci as sets. Thus, a line corresponds to the set of all points lying on the line; a circle, to the set of all points on the circumference.  Moreover, sets of sets are natural, say in considering pencils of lines or circles or conics. And families of pencils are used as well. Does geometry need a full set theory, therefore? In giving a negative answer, we shall consider higher-type sets introduced by parametric definitions with just finite lists of points as parameters. We will show how to formulate a simple axiomatization for such sets together with a notation for virtual classes. The objective is to have the USE of set-theoretical notations without the ONTOLOGY of higher-type logic or the full power of Zermelo-Fraenkel set theory.