# On models of ZFC, does there exist a bijection between Von Neumann universe and the ordinal class?

Von Neumann universe $$V:=\bigcup_{\alpha \in Ord}V_\alpha$$, in which $$V_{\alpha+1}=\wp(V_\alpha)$$ and $$V_\alpha=\bigcup_{\xi<\alpha}V_\alpha$$ for limit ordinals.

My question: Is $V$ equipotent to the ordinal class Ord? Note that they are both proper class, hence seems do not have cardinality, but it seems bijections(seems proper class too) can still be built...

This is the axiom of global choice. Suppose that there were a formula $\varphi(x,y)$ defining a class bijection from $\mathbf{ON}$ to $V$; then $V$ would have a global well-ordering and would satisfy a strong form of the axiom of choice. Thus, such a formula is not provable in $\mathsf{ZF}$, or even in $\mathsf{ZFC}$.

The axiom of global choice does follow from $V=L$ .

• V=L seems so strong, many axioms(like GCH, also CH) follow it. – Popopo Mar 21 '13 at 16:08
• @Popopo: Yes, it’s a very strong axiom. – Brian M. Scott Mar 21 '13 at 16:13

The assertion "$V$ is in bijection with $\sf Ord$" is equivalent to the axiom of global choice, namely there is a choice function on every class of non-empty sets.

This is consistent with ZFC, for example in a model of the axiom $V=L$ (the constructible universe) this axiom is true, but it is consistent that the axiom fails. The construction (that I have in mind) of a counterexample can be a bit technical, as it involves class forcing.

But the point is that this is consistent with ZFC, but it is not provable from ZFC.

• Asaf just out of curiosity, how is the PhD coming along? You must be nearing completion by now. – goblin Mar 28 '14 at 9:17
• Near completion of my research proposal, but there is still a lot if work in that direction. – Asaf Karagila Mar 28 '14 at 11:23
• Good luck. $\!\;$ – goblin Mar 28 '14 at 11:27