On the generic existence of special ultrafilters

Abstract

Abstract.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 semi-ß-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<c.All our filters (and ultrafilters) are proper, nonprincipal, and on co. By the character of a filter we mean the minimum cardinality of a generating set for the filter. We use the term function exclusively for elements of mco. We say that a function / dominates a function g iff Vzz e co f(n) > 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.

The covering number for category and partition relations on Pω(λ)

“…The converse was proved by Canjar [6] and by Bartoszyński and Judah [1]. The equivalence can be generalized as follows.…”

“…Canjar [6] showed that cov(M ) = d if and only if every ideal on ω generated by less than d sets can be extended to a Q-point. We will now generalize his result.…”

The least cardinal for which the Baire category theorem fails

“…There is a natural duality between the theory of filters on ω and the theory of ω-covers of a set S. We make this explicit since it can be used to obtain some of our lemmas also directly from some of Canjar's results in [2].…”

“…Then f is finite-to-one. By [2], Lemma 7, choose an infinite set A ⊂ ω such that {A} ∪ {A α : α < κ} has the finite intersection property, and f is one-to-one on A. Put V = {U n : n ∈ A}.…”

“…Define a function f : ω → ω such that for each n we have f (n) = m if U n ∈ U m \U m+1 . Then select, by [2], Lemma 7, an infinite subset A of ω such that {A} ∪ {A n α : n < ω, α < κ} has the finite intersection property and on A either f is bounded, or else f is one-to-one.…”