2. The ultrapower construction
The study of ultrapowers originates in model theory, although it has found applications both in algebra and in analysis. However, it is accurate to say that it is mainly exploited in set theory. Here I present the basic idea, showing its close connection to the study of measurable cardinals, defined last lecture.
Suppose first that is an ultrafilter over a set We want to define the ultrapower of the universe of sets by The basic idea is to consider the product of many copies of the structure We want to “amalgamate” them somehow into a new structure For this, we look for the “typical” properties of the elements of each “thread” and add an element to whose properties in are precisely these typical properties. We use to make this precise, by saying that a property is typical of the range of iff This leads us to the following definition, due to Dana Scott, that adapts the ultrapower construction to the context of proper classes:
Definition 1 Let be an ultrafilter over a nonempty set We define the ultrapower of by as follows:
For say that
This is easily seen to be an equivalence relation. We would like to make the elements of to be the equivalence classes of this relation. Unfortunately, these are all proper classes except for the trivial case when is a singleton, so we cannot within the context of our formal theory form the collection of all equivalence classes.
Scott’s trick solves this problem by replacing the class of with
Here, as usual, All the “classes” are now sets, and we set
We define by saying that for we have
(It is easy to see that is indeed well defined, i.e., if and then iff )
(The ultrapower construction is more general than as just defined; what I have presented is the particular case of interest to us.) The remarkable observation, due to is that this definition indeed captures the typical properties of each thread in the sense described above: