User blog comment:P進大好きbot/Ordinal Notation Associated to a Proper Class of Ordinals/@comment-27513631-20180626200006/@comment-27513631-20180629230133

"Nope. It neglects the great effort of Rathjen and others to interpret the ordinal notations with large cardinals to ones well-defined in ZFC..."

Your point being...? The large cardinals aren't relevant to the recursive definitions, they're just used to prove they terminate. You can use the consistency (Σ-1 soundness?) of ZFC+LCAs for that. As far as googology is concerned, it's motivation, nothing more.