Suppose that T is a set theory with global choice (for example ZF+ global choice). Question: Does T prove the existence of a well-order on the whole universe V whose restriction to the class On of ordinals is the natural well-order on ON ? Gérard Lang
Yes, use global choice to choose, for each ordinal $\alpha$, a well-ordering $<_\alpha$ of the sets of rank $\alpha$ in the cumulative hierarchy. Then well-order all the sets by putting $x$ before $y$ if either $x$ has lower rank than $y$ or they have the same rank $\alpha$ and $x<_\alpha y$.