There may be no nowhere dense ultrafilter
نویسنده
چکیده
We show the consistency of ZFC + ”there is no NWD-ultrafilter on ω”, which means: for every non-principal ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for every nowhere dense set A of reals, {n : f(n) ∈ A} / ∈ D. This answers a question of van Douwen, which was put in more general context by Baumgartner. ∗ The research partially supported by “Basic Research Foundation” of the Israel Academy of Sciences and Humanities. Publication 594.
منابع مشابه
. L O ] 1 5 N ov 1 99 6 There may be no nowhere dense ultrafilter
We show the consistency of ZFC + ”there is no NWD-ultrafilter on ω”, which means: for every non-principal ultrafilter D on the set of natural numbers, there is a function f from the set of natural numbers to the reals, such that for every nowhere dense set A of reals, {n : f(n) ∈ A} / ∈ D. This answers a question of van Douwen, which was put in more general context by Baumgartner. ∗ The researc...
متن کامل1 5 D ec 1 99 7 REGULAR SUBALGEBRAS OF COMPLETE BOOLEAN ALGEBRAS
There is shown that there exists a complete, atomless, σ-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is undecidable in ZFC. A subalgebra B of a Boolean algebra A is called regular whenever for every X ⊆ B, supB X = 1 implies supA X = 1; see e.g. Heindorf and Sha...
متن کاملOn 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...
متن کاملRegular Subalgebras of Complete Boolean Algebras
It is proved that the following conditions are equivalent: (a) there exists a complete, atomless, σ–centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra, (b) there exists a nowhere dense ultrafilter on ω. Therefore the existence of such algebras is undecidable in ZFC. In ”forcing language” condition (a) says that there exists a non– trivial σ–centered for...
متن کاملApplications of the Covering Property Axiom
Applications of the Covering Property Axiom Andrés Millán Millán The purpose of this work is two-fold. First, we present some consequences of the Covering Property Axiom CPA of Ciesielski and Pawlikowski which captures the combinatorial core of the Sacks’ model of the set theory. Second, we discuss the assumptions in the formulation of different versions of CPA. As our first application of CPA ...
متن کامل