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 GCH . Our main results then show that MOC corresponds to a weak version of global choice in models of the GCH : it can fail in models of the GCH without global choice, but also global choice can fail in models of MOC .