ABSTRACT
A natural counterpart to the investigations is to look at models that are in some sense as ‘small’ as possible. A nonstandard universe formed using a selective ultrafilter is definitely as small as an ω1-saturated nonstandard universe can get: no nonstandard universe can be properly embedded in it. This chapter outlines some unique features of nonstandard universes formed using selective ultrafilters and ultrafilters satisfying various weakenings of selectivity. It discusses distinguishing features of selective ultrafilters with respect to topology and measure theory, respectively. The chapter examines nonstandard topology in a selective universe, focusing on the S-topology and also discusses some properties of the Loeb measure in selective universes. It aims to identify the properties that distinguish nonstandard topology in a selective universe from topology in run-of-the-mill nonstandard universes. A major goal in foundational research in nonstandard analysis has been to pinpoint properties of nonstandard models beyond transfer and ω1-saturation that are useful in applications.
