User blog comment:LittlePeng9/The largest number ever/@comment-2033667-20140914215715

We can generalize this with f(n) = the value of N in nth-order Robinson arithmetic. Amazingly, f(n) is provably recursive in ZFC + "for all k, there exists a k-ultracardinal," but not in any consistent fragment of ZFC + for all k, "there exists a k-ultracardinal"!