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I wrote descriptions of Fish numbers, large numbers that I created, in English.

Fish (2013) "Googology in Japan - exploring large numbers" http://gyafun.jp/ln/en.html

From abstract:

Fish, the author of this book, developed several versions of Fish numbers. With their complicated definitions using maps from functions to functions (functional), maps from "maps of functions to functions" to "maps of functions to functions", and so on, Fish number 3 (2003) reached the level of F[ωω+1 × 63], Fish number 5 (2003) reached the level of F[ε0], and Fish number 6 (2007) reached the level of F[φ2(0)], where φ is the Veblen function. Fish number 7 (2013) extended the definition of Rayo's function by introducing oracle formula in the micro-language of Rayo's definition to go beyond Rayo's number. He defined a hierarchy based on Rado's function, which is similar to fast-growing hierarchy but uncomputational. As the definitions of Fish numbers are available only in Japanese, the author translated them into English.

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