Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?
1 Answer
The answer is no. That principle is equivalent to global choice.
To see this, consider the class $W$ consisting of all wellorderings of any rankinitial segment $V_\alpha$, for any $\alpha$. If we had an injection of Ord into $W$, then there must be unboundedly many $\alpha$'s that are used, since each $V_\alpha$ has only a setsized family of wellorderings. Thus, we have a global selection of wellorderings of unboundedly many $V_\alpha$, and from this we can define a wellordering of the entire universe. Namely, $x<y$ if the rank of $x$ is lower than that of $y$, or if they have the same rank and $x<y$ in the first wellordering of some sufficiently large $V_\alpha$ to appear in the range of the injection of Ord into $W$.
Update. I made a blog post concerning these various equivalent formulations of The global choice principle in GödelBernays set theory, in which I explain this answer and give several other related formulations and arguments.

$\begingroup$ In fact, I hoped that the classes equinumerous with On could be considered as the minimal equivalence class of proper classes under injection, as the classes equinumerous with the universe class V form the maximal equivalence class of proper classes under injection. Now the question is, does such a minimal class exist ? $\endgroup$ Dec 2, 2014 at 19:19

$\begingroup$ I see; that's interesting. Perhaps one might begin by trying to understand the situation in the model at mathoverflow.net/a/110823/1946, which has NGB without global choice. $\endgroup$ Dec 2, 2014 at 19:25

$\begingroup$ @JoelDavidHamkins Do you know of a reference for this argument? $\endgroup$ Dec 2, 2014 at 19:28

$\begingroup$ @SamRoberts I don't know a reference; I just made it up...but probably it has been known before. $\endgroup$ Dec 2, 2014 at 19:30

$\begingroup$ @JoelDavidHamkins I think I made it up too at some point, as have a few others I've spoken to. I thought it might have been folklore, but perhaps not. Thanks! $\endgroup$ Dec 2, 2014 at 19:34