On the combinatorial principle P(c)

Bell, Murray G.

doi:10.4064/fm-114-2-149-157

Cited by 116 publications

(72 citation statements)

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“…In [3] it was shown that if there is such a Boolean algebra and this Boolean algebra is dense in P(ω), then there is a space which is Mazur but not Gelfand-Phillips 1 . Briefly speaking, the minimal generation implies that every measure on A is in the sequential closure of measures of finite support.…”

Section: Problem 37mentioning

confidence: 99%

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Cardinal coefficients associated to certain orders on ideals

Borodulin–Nadzieja

Farkas

2011

Arch. Math. Logic

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We study cardinal invariants connected to certain classical orderings on the family of ideals on ω. We give topological and analytic characterizations of these invariants using the idealized version of Fréchet-Urysohn property and, in a special case, using sequential properties of the space of finitely-supported probability measures with the weak * topology. We investigate consistency of some inequalities between these invariants and classical ones, and other related combinatorial questions. At last, we discuss maximality properties of almost disjoint families related to certain ordering on ideals.

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Section: Problem 37mentioning

confidence: 99%

“…Proof Let 2 ω = 2 ω 1 be arbitrarily large. We will construct two filters F and G generated by ⊆ * -descending sequences {Ẋ α : α < ω 1 } and {Ẏ α : α < ω 1 } inductively in a model obtained by an ω 1 stage finite-support iteration of σ -centered forcing notions (P α ,Q β ) α≤ω 1 ,β<ω 1 .…”

Section: Theorem 43 Assume Gch Then In Vmentioning

confidence: 99%

Cardinal coefficients associated to certain orders on ideals

Borodulin–Nadzieja

Farkas

2011

Arch. Math. Logic

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“…Thus, the entire proof needs MA only for a-centered notions of forcing. It is a theorem of Bell [2] that this special case of MA is equivalent to the assertion, known as P(c), that, if J^ is a family of fewer than c subsets of co and if every finite subfamily has infinite intersection, then there is an infinite A ç co such that A -B is finite for all B e&. Summarizing, we have the following result is a P-point, since this is always true.…”

Section: It Remains To Prove That For Every Finite Q ç a F\j(a -Q) mentioning

confidence: 83%

“…Set Aa = B. Induction hypotheses (2) and (3) are clearly preserved, and (1) is trivial (with Q empty) for ß = y. The first preliminary observation (right after (3)) and the induction hypothesis (1) with y in place of a then yield (1) for all ß < a.…”

Section: It Remains To Prove That For Every Finite Q ç a F\j(a -Q) mentioning

confidence: 98%

On Strongly Summable Ultrafilters and Union Ultrafilters

Blass¹,

Hindman²

1987

Transactions of the American Mathematical Society

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Abstract.We prove that union ultrafilters are essentially the same as strongly summable ultrafilters but ordered-union ultrafilters are not. We also prove that the existence of ultrafilters of these sorts implies the existence of P-points and therefore cannot be established in ZFC.I. Introduction. The purpose of this paper is to clarify the connections between the strongly summable ultrafilters introduced in [9] and the union ultrafilters and ordered-union ultrafilters introduced in [3]. We show that strongly summable ultrafilters and union ultrafilters are essentially the same in that every ultrafilter of either sort is isomorphic to one of the other sort via an isomorphism that (almost everywhere) respects the operations of summation and union. We also show that the existence of such ultrafilters, deduced in [9, 3] from the continuum hypothesis (CH) or Martin's axiom (MA), cannot be deduced in ZFC alone, for it implies the existence of P-point ultrafilters. Finally, we show that union ultrafilters and ordered-union ultrafilters are not essentially equivalent; we construct, using CH or MA, a union ultrafilter that is not isomorphic to any ordered-union ultrafilter via an isomorphism that respects unions.For any set A of natural numbers, let FS(^4) be the set of all sums of nonempty finite subsets of A. An ultrafilter on the set w of natural numbers is strongly summable if it has a base consisting of sets of the form FS(^), with A infinite.We write F for the set of all finite nonempty subsets of to. For A QF, let FU(^) be the set of all unions of nonempty finite subsets of A. An ultrafilter on F is a

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“…Theorem 3.1 ( [3]). The cardinal p is the least cardinal such that there is a σ-centerd partially ordered set ( 3 ) P and a collection D of p dense subsets of P for which there is no centered subset G ⊆ P intersecting each member of D.…”

Section: Casementioning

confidence: 99%