We investigate , the minimum cardinality of a subset of that cannot be made convergent by multiplication with a single matrix taken from , for different sets of Toeplitz matrices, and show that for some sets it coincides with the splitting number. We show that there is no Galois-Tukey connection from the chaos relation on the diagonal matrices to the chaos relation on the Toeplitz matrices with the identity on as first component. With Suslin c.c.c. forcing we show that < is consistent...
We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [“Proper and Improper Forcing”, Ch. V]) and < ω₁-proper -bounding forcings adding reals. We show...
Let χ be the minimum cardinality of a subset of that cannot be made convergent by multiplication with a single Toeplitz matrix. By an application of a creature forcing we show that < χ is consistent. We thus answer a question by Vojtáš. We give two kinds of models for the strict inequality. The first is the combination of an ℵ₂-iteration of some proper forcing with adding ℵ₁ random reals. The second kind of models is obtained by adding δ random reals to a model of for some δ ∈ [ℵ₁,κ). It...
We show that in the -stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under the two algebras are isomorphic [15].
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