The cofinality of the symmetric group and the cofinality of ultrapowers (Q2040186)

In what follows, we assume the reader is familiar with some well-known cardinal invariants of the continuum (like \(\mathfrak{b}\), \(\mathfrak{g}\) and \(\mathfrak{d}\)). In the paper under review, the authors compare the cardinal \(\mathfrak{mcf}\), which is the minimal cofinality of the ultrapower \((\omega,<)\) by a non-principal ultrafilter on \(\omega\), and the cofinality of the symmetric group on \(\omega\), \(\textrm{cf}(\textrm{Sym}(\omega))\). Notice that both cardinals are regular in \(\mathbf{ZFC}\). By previous known results (due to Blass, Mildenberger, Brendle, Losada and Thomas), in \(\mathbf{ZFC}\) both cardinals have value in the interval \([\mathfrak{g},\mathfrak{d}]\). The authors present two forcing constructions. The first forcing shows the relative consistency of \(\aleph_1 = \mathfrak{b} = \mathfrak{mcf} < \aleph_2 = \textrm{cf}(\textrm{Sym}(\omega))\). The second forcing separates the two cardinals in the second direction above \(\mathfrak{b}\), meaning that the consistency of \(\aleph_1 = \mathfrak{b} = \textrm{cf}(\textrm{Sym}(\omega)) < \mathfrak{mcf} = \aleph_2\) is established.