User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-27513631-20180626200006/@comment-27513631-20180626233124

Yeah, but NBG is a conservative extension, and the actual theory used is kinda irrelevant if you stay recursive, as picking pretty much anything strong enough (think subsystems of second order arithmetic) gives you the desired results.