User blog comment:Mh314159/The psycholog of large numbers/@comment-35470197-20190708222539

Friedman, who created TREE and SCG, is the greatest googologist in the computable realm. Friedman's systems are surprisingly diagonalising all computable functions whose terminations can be provable under several theories. Namely, Friedman interprete provability under several theories into graph theory or game theory. Unlike other googologists, Friedman invented a way to create fast growing function without recursing along ordinals. That is why it is very difficult to understand why the systems are so strong.

For example, TREE growth faster than all computable functions whose terminations are provable under a system called \(\textrm{ACA}_0{+}\Pi_2^1{-}\textrm{BI}\). One of Friedman's greatest large numbers is the least transcendental integers, which is the output of a function which grows faster than all computable functions whose termination (restricted to standard natural numbers) are provable under (\textrm{ZFC]\) set theory, i.e. the usual mathemetics. This great number is of level 24 in my googological ruler, and hence larger than almost all computable large number ever defined. Once I understood the computation process of Friedman's transcendental integer system, I got very surprised that the system is outstandingly wonderful.