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 ?
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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.
answered Dec 2, 2014 at 14:37
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$\begingroup$ 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. $\endgroup$– Asaf Karagila ♦Dec 2, 2014 at 14:58