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By R. Michael Canjar and Thomas J. Jech


ABSTRACT. Assume that there is no dominating family of reals of cardinality < c. We show that there then exists an ultrafilter on the set of natural numbers such that its associated Mathias forcing does not adjoin any real which dominates all ground model reals. Such ultrafilters are necessarily P-points with no Q-points below them in the Rudin-Keisler order. Notation and terminology, ta is the set of natural numbers. The symbols i, j, k, l, m, n will be used exclusively to denote its elements. All our filters (and ultrafilters) are proper, nonprincipal and on ta. We use the term "real " exclusively for elements of "ta. We say that a real / dominates a real g iff 3m Vn> mf{n)> g{n). Following [13] we let d denote the minimum cardinality of a dominating family of reals and b denote the minimum of an unbounded family; i.e., a family B such that there is no real g which dominates every real from B. c is the cardinality of the continuum. With respect to forcing, we adhere to the convention that "p < q " indicates that p extends q. Following [1] we will write Next(X, n) for min{a; G X\x> n} when X is an infinite subset of ta. P is the set of finite

Year: 2013
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