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.

Q-pointness, P-pointness and feebleness of ideals

“…(£) otherwise. Thus g a (8) > fr,+p{<t) for all £ < e E (5). Hence g a for a < p. is a ^-<*-tower in case K = co, and a /*-<*-tower otherwise.…”

Pseudo P-points and splitting number

“…Furthermore ℘(N, P ) is equal to the union of the increasing sequence {℘(N, P aα κ ) : α < λ}. It then follows immediately from condition (7) that s is forced to be λ. By Proposition 2.8, P forces that b ≤ κ.…”

“…κ ) that is forced by P a δ κ to be a free ultrafilter on N, then E δ ∩ A a δ i is not empty for all i < κ, (4) a δ+1 is the Cohen ω1 -extension of a δ and ζ δ+1 = ζ δ , (5) if α = δ + 1 then ζ α+1 = ζ α and a α+1 is the Cohen θα -extension of a α where θ α = |P aα κ | ℵ0 (6) if α = δ + 2, then ζ α+1 = ζ α and a α+1 is an L(D)-extension of a α , (7) if α ∈ (δ + 2, δ + ω), then ζ α+1 is the minimal value strictly above ζ α such thatQ α+1 = h(ζ α+1 − 1) has cardinality less than κ and is a P aα κ -name of a poset that is forced to be ccc, and a α+1 = a α * Q α+1 as in Definition 2.12.…”

“…Choose any γ < λ so thatQ * Fn((γ, γ + ω), 2) is not in the list {h(ζ) : ζ ≤ ζ α } and let h(ζ ′ ) =Q * Fn((γ, γ + ω), 2). It follows from inductive conditions (1) and (7), that the sequence {ζ α : α ∈ λ} is unbounded in λ. So we may choose limit δ < λ maximal so that ζ δ ≤ ζ ′ .…”

Generic Existence of Mad Families

“…Extensions of AD families to maximal ones have been previously investigated by Leathrum in [19] and by Fuchino, Geschke, and Soukup in [8]. Generic existence of ultrafilters has been introduced by Canjar in [7] and was recently investigated by Brendle and Flašková in [5].…”