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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 under the effective weak diamond principle , answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property in spaces from almost disjoint families,...
We consider the question: when does a Ψ-space satisfy property (a)? We show that if 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 which has property (a). We extend a theorem of Matveev relating the existence of certain closed discrete subsets with the failure of property (a).
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....
Shelah’s pcf theory describes a certain structure which must exist if is strong limit and 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...
We prove several results on some cardinal invariants of the continuum which are closely related to either the splitting number or its dual, the reaping number .
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.
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