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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 ?

Gérard Lang
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1 Answers1

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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.

Joel David Hamkins
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  • 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 ? – Gérard Lang Dec 02 '14 at 19:19
  • I see; that's interesting. Perhaps one might begin by trying to understand the situation in the model at http://mathoverflow.net/a/110823/1946, which has NGB without global choice. – Joel David Hamkins Dec 02 '14 at 19:25
  • @JoelDavidHamkins Do you know of a reference for this argument? – Sam Roberts Dec 02 '14 at 19:28
  • @SamRoberts I don't know a reference; I just made it up...but probably it has been known before. – Joel David Hamkins Dec 02 '14 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! – Sam Roberts Dec 02 '14 at 19:34
  • Can I ask something about the class "W" which mentioned above? In my set theory's book it said there are two parallel universe.One is the universe of all the constructible sets which is class "V" shown above.And the other one is named the "true" universe(For example,if a set "A" has 2 elements,then "W" consists all the possible combination of all the different elements to form 2 elements.Which means there are unboundedly many of different sets which consisting 2 element) which is the "ultimate universe" which contain everything. (continued) – majin vegeta Jul 23 '16 at 09:48
  • I have two questions: 1. Is my understanding for class"W" correct? 2. Based on my question 1, if the global choice axiom holds, then "W" can be bijective with every proper class, and if the axiom of choice doesn't hold, then "V" should be injected to "W"? – majin vegeta Jul 23 '16 at 09:48