Ultrafilters and types on models of arithmetic (Q1058509)

The author studies partition properties of models \(M=<I,R>\) (where \(R\subseteq {\mathcal P}(I))\) of fragments of second order arithmetic in terms of the existence of certain special ultrafilters on R. The notion of a definable ultrafilter is introduced, it is an ultrafilter U such that the family of subsets of I which are codable in the appropriate ultrapower of M is R itself. Several characterizations of partition properties by means of the existence of appropriate ultrafilters are obtained. Especially the properties \(I\to (I)^ 2_ 2\) and \(I\to (I)^ 3_ 2\) are characterized in this way. This throws some new light on the famous problem: does \(I\to (I)^ 2_ 2\) imply \(I\to (I)^ 3_ 2?\) One of the theorems proved in the paper gives a partial answer to this problem. Mainly, if M is a model of arithmetic comprehension, then \(I\to (U)^ 3_ 2\) iff \(I\to (U)^ 2_ 2\) for some kinds of ultrafilters U (where \(I\to (U)^ m_ n\) should be read as usual with the additional remark that the required homogeneous set is in U). Some of the characterization theorems concern types of M instead of ultrafilters U. In analogy with various kinds of ultrafilters on N, various kinds of types are considered, e.g. p-points and selective types.