User blog comment:LittlePeng9/Higher order set theory/@comment-5150073-20140926142538/@comment-5529393-20140926144906

No, LittlePeng9 defines Rayo_w(n) differently than that.

It certainly seems to me that wOST can define Rayo(m,n), since given m we can quantify over x(m+1), which we know allows us to define Rayo_m(n). We have to make sure the definition can be uniform over m, but I don't see why it wouldn't be. So we can define Rayo_n (n), Rayo_{Rayo_n(n)} (n), and so on. so Rayo_w(n) will grow faster than any such construction.