On the refinement and countable refinement numbers
In his paper ``More on set-theoretic characteristics of summability of sequences by regular (Toeplitz) matrices'' [Commentat. Math. Univ. Carolinae 29, 97-102 (1988; Zbl 0653.40002)], \textit{P. Vojtaš} defined \(\mathfrak t\) and \(\mathfrak t_\sigma\), two cardinal invariants of the continuum called the refinement number and countable refinement number, respectively. From their definition it is immediate that \(\mathfrak t\leq\mathfrak t_\sigma\). Vojtaš asked whether they are actually equal. NEWLINENEWLINENEWLINEThe authors of this paper investigate the possibility that \(\mathfrak t <\mathfrak t_\sigma\) is consistent with ZFC. The smallest size of an ultrafilter base on the natural numbers and the dominating number play a role in their investigation. A typical result is: \(\mathfrak t <\mathfrak t_\sigma\) cannot be obtained by a countable support iteration of length \(\omega_2\) of proper posets of continuum size over a model of CH. The authors conclude that current forcing machinery may be insufficient to the establish the consistency of \(\mathfrak t <\mathfrak t_\sigma\). Of course, as they point out, it may be the case that \(\mathfrak t =\mathfrak t_\sigma\) is a theorem of ZFC.
scientific article