On the Generic Existence of Special Ultrafilters

We introduce the concept of the generic existence of P-point, Qpoint, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically iff semiselectives do iff mc = c, and we show that ß-point ultrafilters exist generically iff semiß-points do iff mc — d , where d is the minimal cardinality of a dominating family of functions and m is the minimal cardinality of a cover of the real line by nowhere-dense sets. These results complement a result of Ketonen, that P-points exist generically iff c = d , and one of P. Nyikos and D. H. Fremlin, that saturated ultrafilters exist generically iff mc = c = 2 g(n). Following [21] we let d denote the minimum cardinality of a dominating family of functions. We let c be the cardinality a'o) and mc be the minimum cardinality of a cover of the real line consisting of nowhere-dense sets. The notation mc comes from "Martin's Axiom for countable partial orders"; mc is the maximum cardinal X such that the following holds: given any countable partial order P and fewer than X dense subsets of P, there exists a generic G C P which meets each of the dense subsets. Also, mc may be characterized as the maximum cardinal X such that, given any family of functions H of size < X , we may find a function g such that V/z e H, 3n e co g(n) = h(n). Some of the basic properties of this cardinal, including proofs of the equivalence of these characterizations, may be found in [1], [15], and [22]. It is clear from the above discussion that mc < d < c. In this paper we will consider the question of the existence of various special sorts of ultrafilters, whose definitions we now mention. An ultrafilter U is called a P-point if every function is either finite-to-one or bounded on a set in U . An ultrafilter U is called a Q-point or rare ultrafilter if every finite-to-one function is one-to-one on a set in U . An ultrafilter U is said to be a semi£?-point (or Received by the editors July 2, 1988 and, in revised form. May 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary' 03E05; Secondary 03E35, 54H05. ©1990 American Mathematical Society 0002-9939/90 $1.00 + 5.25 per page