An ordinal-connection axiom as a weak form of global choice under the GCH

The minimal ordinal-connection axiom \(MOC\) was introduced by the first author in R. Freire. (South Am. J. Log. 2:347–359, 2016). We observe that \(MOC\) is equivalent to a number of statements on the existence of certain hierarchies on the universe, and that under global choice, \(MOC\) is in fact equivalent to the \({{\,\mathrm{GCH}\,}}\). Our main results then show that \(MOC\) corresponds to a weak version of global choice in models of the \({{\,\mathrm{GCH}\,}}\): it can fail in models of the \({{\,\mathrm{GCH}\,}}\) without global choice, but also global choice can fail in models of \(MOC\).