# Injection of the proper class of ordinals in every proper class

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 well-orderings of any rank-initial 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 set-sized family of well-orderings. Thus, we have a global selection of well-orderings of unboundedly many $V_\alpha$, and from this we can define a well-ordering 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 well-ordering 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 Gödel-Bernays set theory, in which I explain this answer and give several other related formulations and arguments.

• 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 ? Dec 2, 2014 at 19:19
• 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. Dec 2, 2014 at 19:25
• @JoelDavidHamkins Do you know of a reference for this argument? Dec 2, 2014 at 19:28
• @SamRoberts I don't know a reference; I just made it up...but probably it has been known before. Dec 2, 2014 at 19:30
• @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! Dec 2, 2014 at 19:34