User blog comment:LittlePeng9/Higher order set theory/@comment-10429372-20140926180316/@comment-1605058-20140926180619

First, if you want to throw these terms around, define w1DEF. Second, yes, there are OOST-undefinable ordinals between w1CK and w1. Let A be smallest of them. But then Rayo_A=Rayo_A+1=Rayo_A+w=..., so your SGH comparison actually fails (I knew someone is going to ask about that :P)

I believe it's possible, I wanted to make a section "Beyond?" later.