Ultrafilters and ultraproducts

Abstract

The topics of this thesis are properties that distinguish between the 22Xo isomorphism-classes (called types) of non-principal ultrafilters on o. In particular we investigate various orders on ultrafilters.

The Rudin-Frolik order is a topologically invariant order on types; it had been shown that there are types with 2X o predecessors in this order, and that, assuming the C.H., for every there are types with n predecessors. We shew that, assuming the C.H., there is a type with X_o predecessors.

The next two main results can be phrased in terms of the minimal elements of these orders. Both assume the C.H. We find an ultrafilter that is a p-point (minimal in M.E.Rudin's "essentially greater than" order) that is not above any Ramsey ultrafilter (minimal in the Rudin-Keisler order). We also find an ultrafilter minimal in Blass' "initial segment" order that is not a p-point. These ultrafilters generate ultrapowers with interesting model-theoretic properties.

We then investigate the classification of ultrafilters when the C.H. is no longer assumed. We find various properties of ultrafilters, sometimes by assuming some substitute for the C.H. such as Martin's Axiom, and sometimes without assuming any additional axiom of set-theory at all. Finally we relate the structure of ultrapowers to the existence of special sorts of ultrafilters.