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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...

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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 $V=L$ .

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

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

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