Cardinal invariants of the continuum and combinatorics on uncountable cardinals
The author explores the connection between combinatorial principles on the reals, like stick and club, on the one hand, and the combinatorics of sets of reals on the other hand. He starts by showing the following fact: Assume that the null ideal \(\mathcal N\) is \(< \mathfrak{c}\)-additive. Then Martin's axiom holds for any \(\mathbb{C}_{\lambda}\), where \(\mathbb{C}\) denotes the corresponding Cohen algebra. Next, he deals with Gross spaces. It is an old result of Baur and Gross, that Gross spaces exist under CH. On the other hand it was shown by Shelah and Spinas that it is consistent that \(\mathfrak{c} = \aleph_2\) and there are no Gross spaces over any finite field, and it was shown by Shelah that it is consistent that \(\mathfrak{c} = \aleph_3\) and there are no Gross spaces over any at most countable field. Here, the author shows that under \(\mathfrak{c} = \aleph_2\) there is a Gross space over every countable field. It is shown that it is consistent that \(\mathfrak{c} = \aleph_2\), \(\clubsuit\) holds and the union of less than \(\mathfrak{c}\) many null sets does not cover the real line. There are results concerning Suslin trees and mad families. So it is shown: Assume that \(\clubsuit\) holds and the meagre ideal \(\mathcal M\) has a base of size \(\aleph_1\). Then there is a Suslin tree. Assume \(\clubsuit\) and there is a dominating family of size \(\aleph_1\). Then there is a mad family of size \(\aleph_1\). In two appendices the author solves problems of Fuchino, Shelah and Soukup, and of Kraszewski, respectively. So he shows that it is consistent that {cov}(\(\mathcal M\))\(=\aleph_2\) and \(\clubsuit_S\) holds for all stationary \(S \subseteq \omega_1\).
scientific article; zbMATH DE number 5121368
scientific article