|Time:|| 12 - 1:20 p.m.
Dougherty Hall 1209
|Speaker:|| Dana Scott
Hillman University Professor
Departments of Computer Science, Philosophy and Mathematical Sciences
Carnegie Mellon University
Parametric sets and virtual classes
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 Zermelo-
Fraenkel set theory.