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 $x<_\alpha y$.

  • $\begingroup$ As far as I can tell, this question has been migrated from MathOverflow to MathOverflow. I wasn't previously aware that such vacuous migrations were possible. $\endgroup$ – Andreas Blass Aug 20 '13 at 19:08
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    $\begingroup$ I think it was migrated from meta? $\endgroup$ – Noah Schweber Aug 20 '13 at 19:09

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