P Points with Countably Many Constellations

If the continuum hypothesis (CH) is true, then for any P point ultrafilter D (on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of) P points which lie above D, of order type N,. In particular, if D is an RK-minimal ultrafilter, then we have (CH) that there exist P-points with countably many constellations. 0. Introduction. Our main result is that in the presence of the continuum hypothesis (henceforth denoted CH), there exist P point ultrafilters on « with exactly S0 many constellations. Actually, we prove a somewhat stronger theorem about initial segments of the Rudin-Keisler (RK) ordering on the class of P points; in order to state this result, we begin with a few definitions. All ultrafilters here are nonprincipal ultrafilters on u = (0, 1, 2,...}. An ultrafilter D is a P point iff any function /: w —> w is either constant or finite-to-one on a set in D. P points have been studied extensively, and we shall assume basic results about them and their RK ordering; good references are [Bl and Pu]. If F) is a F point, let < P D denote the RK ordering on (equivalence classes of) P points which lie above D in RK. An initial segment of < PD means a downward closed subset, and the initial segment determined by E is {F: D < F < E) (we use < to denote the RK ordering). In his thesis [Ec], Eck showed (CH) that if D is any P point, then there exist P points E immediately above D in RK in the strong sense that any strict RK predecessor of F is a predecessor of D; we call such an F a strong immediate successor (s.i.s) of D. Iterating Eck's theorem w times yields the existence (CH), for any P point D, of initial segments of < P D of order type w. Our main theorem is the existence (CH) of initial segments of < P D of order type K1; the bulk of the article is devoted to its proof. In [B3], Blass proved the result just stated without the restriction to P points; that is, he showed (CH) that for any ultrafilter D, there exist initial segments of "RK above D" of order type X,. The proof involved reformulating the problem in model theoretic terms, and we shall take the same approach. Let N be the complete first order structure on w (i.e. the language for N contains names for every finitary function and relation on w). We use the term model to mean "nonstandard model of Th(N)", and we use */ to indicate the interpretation of the function /: u -* w in whichever model is under consideration. If D is an ultrafilter, then D-prod denotes Received by the editors July 16, 1984. The results in this paper were presented by the author at the winter meeting of the Association for Symbolic Logic held in Denver, Colorado, in January, 1983. 1980 Mathematics Subject Classification. Primary 04A20; Secondary 03H15. [ 1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page