It is proved that--consistently--there can be no ccc closed P-sets in the remainder space ω*.

It is proved that--consistently--there can be no ccc closed P-sets in the remainder space $\omega^\ast$.

In this article, I make an attempt to define terms such as: tragedy, absurdity, and paradox as conceived of by the German essayist Walter Hilsbecher. This is a task which is both difficult and interesting, mainly because of the fact that concepts from within speculative philosophy are relatively rarely subject to scrupulous definitions. The reason for this state of affairs lies in the difficulty to capture the meaning of these concepts within a rigid framework. These problems also appear in Hilsbecher’s (...) 1972 work titled Tragizm, absurd, paradoks [Tragedy, Absurd, Paradox]. The philosophy of the absurd – the methodological separation of which I propose – has the absurd for its central concept, but it is a concept which eludes sharp definitions, too. This article aims to reconstruct Hilsbecher’s vocabulary of concepts crucial to the philosophy of absurdity in reference to the tradition of contemporary speculative philosophy. By locating the German writer’s thought in the context of other authors working in a similar thematic and stylistic context, I will outline this specific strand of philosophy. (shrink)

A set-theoretical proof of Gowers’ Dichotomy Theorem is presented together with its application to another dichotomy concerning asymptotic l 2 basic sequences.

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the σ-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of σ-ideals to include the completely (...) Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non(CRU 0); and a third concerning the size of add(CRU 0). We also study those classes of perfect sets which are bases for the class of always first category sets. (shrink)

We investigate several versions of a cardinal characteristic $ \frak f$ defined by Frankiewicz. Vojtáš showed ${\frak b} \leq{\frak f}$ , and Blass showed ${\frak f} \leq \min({\frak d},{\mbox{\rm unif}}({\bf K}))$ . We show that all the versions coincide and that ${\frak f}$ is greater than or equal to the splitting number. We prove the consistency of $\max({\frak b},{\frak s}) <{\frak f}$ and of ${\frak f} < \min({\frak d},{\mbox{\rm unif}}({\bf K}))$.

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the $\sigma$-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of $\sigma$-ideals to include the completely (...) Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non; and a third concerning the size of add$. We also study those classes of perfect sets which are bases for the class of always first category sets. (shrink)