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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.

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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 '14 at 14:58

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