Bitcoin Badger

This will enhance the ordinals expansion greatly as they are currenyly expanded based on multiplication and exponentiation only. Part I: Up Arrow & Ordinals for Sequential Functions The up arrow notation can be added to ordinals in the form of f α↑ k β (n), where f is FGH fundamental sequence function, where k and n are positive integers; α and β are ordinals. Similar to expansion of ordinals, α↑ k β is right associative (expanding from right to left). f α↑ k β+1 (n) = f ...(((α↑ k β)↑ k β)↑ k β)... (n) (with n copies of ↑ k 's). All other rules shall follow the rules of Up Arrow and the rules of ordinal expansion. Examples: f ω↑ω (2) = f ω↑2 (2) = f ω 2 (2) = ... f ω↑↑ω (2) = f ω↑↑2 (2) = f ω ω (2) = f ω 2 (2) = ... f ω↑↑(ω+1) (2) = f ((ω↑↑ω)↑↑ω) (2) = f ((ω↑↑ω)↑↑2) ((2) = ... f ε 0 ↑↑↑ε 0 (2) = f ε 0 ↑↑↑ ω ω (2) = f ε 0 ↑↑↑ ω 2 (2) = f ε 0 ↑↑↑ ω2 (2) = f ε 0 ↑↑↑ω+ω (2) = f ε 0 ↑↑↑ω+2 (2) = f ((ε 0 ↑↑↑ω+1)↑↑↑ω+1) (2) = f (((ε 0 ↑↑↑ω+1)↑↑↑ω)↑↑↑ω) (2) = ... f φ(ω,0)↑↑ω (2) =

This is the summary of six fastest growing functions I built, using existing fast growing functions, nested ordinals and notation ↥ . Enjoy! Part I: Nest Ordinals with Up Arrows 1.1- For ordinals ɑ, β, γ, δ and κ; positive integers a, b, c and d: The basic form is a↑ ɑ b and always expand from right to left. β↑γ# = β γ# (γ# denotes expressions of γ expansion in +, * and ^), e.g. β↑(γ+1) = β γ+1 .  β↑(γ 2 +γ) = β γ 2 +γ = β γ 2 *β γ . β↑ c+1 b = β↑ c β↑ c ... β↑ c β (with b copies of β). a↑ ɑ+1 b = a↑ ɑ a↑ ɑ ... a↑ ɑ a (with b copies of a's). β↑ ɑ+1 b = β↑ ɑ β↑ ɑ ... β↑ ɑ β (with b copies of β's). a↑ β↑ ɑ γ+1 b = a↑  ...(((β↑ ɑ γ)↑ ɑ γ)↑ ɑ γ)...  b (with b copies of ↑ ɑ γ 's). a↑ κ↑ δ β↑ ɑ γ+1 b = a↑ κ↑ δ ...(((β↑ ɑ γ)↑ ɑ γ)↑ ɑ γ)... b (with b copies of ↑ ɑ γ 's). All other rules shall follow the rules of Up Arrow  and the rules of ordinal collapsing. Define T(δ,d) = ...δ↑ δ↑ δ δ δ... (with d floors; both δ and d are user-defined). Examples: 2↑ ω 3