# Injection of every proper class in the ordinal class

Is it possible in the set theory NBG (with local choice but without global choice) that every proper class injects in the proper class of ordinals ?

## 1 Answer

This is equivalent to global choice, since if $V$ itself injects into Ord, then there is a global well-ordering (defined by $x<y$ if $x$ maps to a smaller ordinal than $y$), and vice versa.

• It might be worth noting that global choice can indeed fail while the axiom of choice for sets holds. For example if we add a proper class of Cohen sets (to each regular cardinal) then we cannot linearly ordered the universe, and in that case we certainly cannot well-order it. So there is no injection into the ordinals. – Asaf Karagila Dec 2 '14 at 14:58