There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

Blass, Andreas; Shelah, Saharon

doi:10.1016/0168-0072(87)90082-0

Cited by 122 publications

(123 citation statements)

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“…While the existence of semi-Q-points appears to be weaker than that of Q-points, in the models where it is known that Q-points do not exist (models obtained by iteratively adding Laver or Mathias reals, studied by Miller in [14], or Shelah's model for Blass's Near Coherence of Filters principle, described in [6]), semi-Q-points do not exist either. Moreover, when Kunen showed in [12] that selective ultrafilters do not exist in the random real model, he actually proved that semiselectives are not found there either.…”

Section: The Theoremsmentioning

confidence: 99%

On the Generic Existence of Special Ultrafilters

Canjar¹

1990

Proceedings of the American Mathematical Society

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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 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.

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Section: The Theoremsmentioning

confidence: 99%

On the Generic Existence of Special Ultrafilters

Canjar¹

1990

Proceedings of the American Mathematical Society

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“…To prove (3), let b be any branch of q. b is also a branch of p, so (2) shows that q∩split n (p) ⊇ b ∩ split n (p) = ∅. Proof of (4): Let b be a branch of p containing η.…”

Section: This Contradicts 21(d)mentioning

confidence: 99%

“…Since Unif (S) is equivalent to for every H : ω → ω, for every F ∈ [ n H(n)] <c , there exists f * ∈ ω ω such that for every f ∈ F there are infinitely many n satisfying f (n) = f * (n), 2.18 (2) shows that if we have c = ℵ 2 and Martin's Axiom for the forcing notions P T H (for all H), then we also have Unif (S). (In fact the "easy" implication "⇐" of this equivalence is sufficient.)…”

Section: Remarkmentioning

confidence: 99%

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Strong measure zero sets without Cohen reals

Goldstern¹,

Judah²,

Shelah

1993

J. symb. log.

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“…In [5] a model of set theory is constructed where c = ℵ 2 and there exist a simple P ℵ 1 point and a simple P ℵ 2 point. The simple P ℵ 1 point is generated by ℵ 1 many sets, thus u = ℵ 1 .…”

Section: Theorem 12mentioning

confidence: 99%

Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness

Shelah¹,

Tsaban²

2003

Journal of Applied Analysis

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Abstract. Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (τ -covers). We deal with two types of combinatorial questions which arise from this study.(1) Two new cardinals introduced in the topological study are expressed in terms of well known cardinals characteristics of the continuum. (2) We study the additivity numbers of the combinatorial notions corresponding to the topological diagonalization notions. This gives new insights on the structure of the eventual dominance ordering on the Baire space, the almost inclusion ordering on the Rothberger space, and the interactions between them.

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There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed

Cited by 122 publications

References 5 publications

On the Generic Existence of Special Ultrafilters

On the Generic Existence of Special Ultrafilters

Strong measure zero sets without Cohen reals

Critical Cardinalities and Additivity Properties of Combinatorial Notions of Smallness

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