Is ZFC with existence of Grothendieck universe (variant: Grothendieck universe containing every given set) provable in ZFC to be equiconsistent with ZFC?
If not, what else it may be equiconsistent with? (And in which formalistic it may be proved?)
This matter was much discussed on MathOverflow. The links below and the Wikipedia page should answer the question and the one that appears in the comments.