User blog comment:Ubersketch/Lexicographic ordinal notation up to the Bachmann-Howard ordinal/@comment-30869823-20190913225449/@comment-35470197-20190917004441

@Ubersketch

I think that training of analysis migh help you to understand OCFs more clearly. Of course, I mean analysis precisely based on the definition instead of intuition.

First, please check ψ(ε_0) = ψ(ε_0+1), which is one of the most confusing computation for beginners with computation based on intuition. Be careful that people often state "ψ(ε_0+1) = ψ(ε_0)×ω because it also holds for UNOCF". I am not sure who began to set this weird equality as an agreed-upon rule of UNOCF, though. (Here, "agreed-upon" just means "widely accepted by analysts based on UNOCF".)

Next, please check ψ(ε_0) = ψ(Ω_1), which is not so difficult after understanding ψ(ε_0) = ψ(ε_0+1). Finally, please check ψ(Ω×ω) = ψ(Ω×ω+α) for any ordinal α. After then, you can understand why the growth stucks at Ω×ω, and why solutions suggested by Fejfo and Plain'N'Simple actually work.