


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
highertype 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 settheoretical
notations without the ONTOLOGY of highertype logic or the full power
of ZermeloFraenkel set theory. 


