1. Online calculator for fast-growing hierarchy with Extended Buchholz Function

HomepageFor calculation of expressions of fast-growing hierarchy write below a natural number n>0 and a countable ordinal α written in the normal form for Extended Buchholz Function, i.e. α=0 or α=ψ_{0}(β_{1})+...+ψ_{0}(β_{k}), where k≥1 and ψ_{0}(β_{1})≥...≥ψ_{0}(β_{k}) and β_{i}∈C_{0}(β_{i}) for 1≤i≤k. Each β_{i} also should be written in the normal form, i.e. β_{i}=0 or β_{i}=ψ_{ν1}(γ_{1})+...+ψ_{νm}(γ_{m}). Write "_{" before each subscript and "}" after it, even if this subscript is just 0. Don't use spaces. Write "p" instead of "ψ". A non-zero finite ordinal μ in the line for α should be written as p_{0}(0)+...+p_{0}(0) where p_{0}(0) repeats μ times. For example, instead of ψ_{0}(ψ_{1}(0)+ψ_{1}(0)) write p_{0}(p_{p_{0}(0)}(0)+p_{p_{0}(0)}(0)). Some other examples of α you can find here. See also video with examples of work

n =

α =

2. Online calculator for Extended Arrows with Extended Buchholz Function

For calculation of expressions of Extended Arrows write below positive integers b, c and a countable ordinal α written in the normal form for Extended Buchholz Function. The requirements for filling in the line for α are the same as in case of the previous calculator.

b = ; c =

α =