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Almost disjoint families and “never” cardinal invariants

Charles Morgan, Samuel Gomes da Silva (2009)

Commentationes Mathematicae Universitatis Carolinae

We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to ω 1 under the effective weak diamond principle ♦ ( ω , ω , < ) , answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property ( a ) in spaces from almost disjoint families,...

Almost disjoint families and property (a)

Paul Szeptycki, Jerry Vaughan (1998)

Fundamenta Mathematicae

We consider the question: when does a Ψ-space satisfy property (a)? We show that if | A | < p then the Ψ-space Ψ(A) satisfies property (a), but in some Cohen models the negation of CH holds and every uncountable Ψ-space fails to satisfy property (a). We also show that in a model of Fleissner and Miller there exists a Ψ-space of cardinality p which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).

Another ⋄-like principle

Michael Hrušák (2001)

Fundamenta Mathematicae

A new ⋄-like principle ⋄ consistent with the negation of the Continuum Hypothesis is introduced and studied. It is shown that ¬ ⋄ is consistent with CH and that in many models of = ω₁ the principle ⋄ holds. As ⋄ implies that there is a MAD family of size ℵ₁ this provides a partial answer to a question of J. Roitman who asked whether = ω₁ implies = ω₁. It is proved that ⋄ holds in any model obtained by adding a single Laver real, answering a question of J. Brendle who asked whether = ω₁ in such models....

Another proof of a result of Jech and Shelah

Péter Komjáth (2013)

Czechoslovak Mathematical Journal

Shelah’s pcf theory describes a certain structure which must exist if ℵ ω is strong limit and 2 ℵ ω > ℵ ω 1 holds. Jech and Shelah proved the surprising result that this structure exists in ZFC. They first give a forcing extension in which the structure exists then argue that by some absoluteness results it must exist anyway. We reformulate the statement to the existence of a certain partially ordered set, and then we show by a straightforward, elementary (i.e., non-metamathematical) argument that such partially...

Around splitting and reaping

Jörg Brendle (1998)

Commentationes Mathematicae Universitatis Carolinae

We prove several results on some cardinal invariants of the continuum which are closely related to either the splitting number 𝔰 or its dual, the reaping number 𝔯 .

Axial permutations of ω²

Paweł Klinga (2016)

Colloquium Mathematicae

We prove that every permutation of ω² is a composition of a finite number of axial permutations, where each axial permutation moves only a finite number of elements on each axis.

Borel sets with large squares

Saharon Shelah (1999)

Fundamenta Mathematicae

 For a cardinal μ we give a sufficient condition ⊕ μ (involving ranks measuring existence of independent sets) for: ⊗ μ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| =μ) then it contains a 2 ℵ 0 -square and even a perfect square, and also for ⊗ μ ' if ψ ∈ L ω 1 , ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming M A + 2 ℵ 0 > μ for transparency, those three conditions ( ⊕ μ , ⊗ μ and ⊗ μ ' ) are equivalent, and from this we deduce that...

Cardinal invariants of the lattice of partitions

Barbara Majcher-Iwanow (2000)

Commentationes Mathematicae Universitatis Carolinae

We study cardinal coefficients related to combinatorial properties of partitions of ω with respect to the order of almost containedness.

Cardinal invariants of ultraproducts of Boolean algebras

Andrzej Rosłanowski, Saharon Shelah (1998)

Fundamenta Mathematicae

We deal with some problems posed by Monk [Mo 1], [Mo 3] and related to cardinal invariants of ultraproducts of Boolean algebras. We also introduce and investigate several new cardinal invariants.

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