Ultrafilters and
Ultraprod ucts
Appendix A
Since we need them throughout this book, here's a brief account of ultrafil-
ters and ultraproducts.Ultrafilters. An ultrafilter on a set labels every subset as "big" or "small"
in such a way that a finite intersection of big _subsets is big and a finite union
of small sets is small.
Definition A.1. Let I be a set. A filter on the set I is a nonempty family
U of subsets of I with the following properties:
· (1) (nontriviality) 0 ~ U;
(2) (finite intersection property) if Io, Ii EU, then there exists J EU
such that Jc Ion Ii;
(3) (directedness) if Io EU and Io c Ii c I, then Ii EU.
A nonempty family U' of subsets of I with properties (1) and (2) is called a
filter base. Such a family can always be enlarged to a filter by including all
subsets which contain a member of U'. A filter U is called an ultrafilter if it
satisfies
(4) (maximality) for any subset Io CI, either Io EU or I\ Io EU.
It is easy to check that an ultrafilter U has the property
(5) if LJ~=l h EU, then at least one of h's is in U.
Example A.2. Let I be a set. The principal ultrafilter generated by io EI
is the family of all subsets which contain io.- 445