User blog:Eightisthebestnumber/Function Transformer

It takes a function and output a faster function !

Definition
d∶ R→R

α is the smallest ordinal such that g_α eventually outgrows d ,where g represents the slow growing hierarchy.

T_d (n) = f_α (n) using the fast growing hierarchy.

Higher order T function
The n-th order T function is like the normal one except we use a modified version of the fast growing hierarchy where f_0(n) =  the limit of the (n-1)th order T function.

Comparisons
T_0 (n) = n + 1

T_1 (n) = 2n

T_2 (n) = 2^n*n

T_3 (n) ≈ tetration

T_4 (n) ≈ pentation

T_x (n) ≈ Ack(n)

T_(x+1) (n) ≈ g_n

T_(x^2 ) (n) ≈ TW(n) and MEa_n

T_(x^3 ) (n) ≈ a possible extension of TW(n) up to TW_n (n)

T_(x^x ) (n) ≈ a possible extension to TW_n (n) up to TW_(n_(n_(n_⋱ ) ) ) (n)

T_(x^(x^(10^100 ) ) ) (n) ≈ a googol-dimensional hypercube in BEAF

T_(x^(x^(x^3 ) ) ) (n) ≈ a super-super-super-super-dimensional hypercube in BEAF

T_(x↑↑x) (n) ∶ BEAF becomes ill-defined after this point,really powerful

T_(x↑↑↑x) (n) ≈ f_(ζ_0 ) (n),even more powerful

T_(T_x ) (n) ≈ f_φ(ω,0) (n)

T_(T_(x+1) ) (n) ≈ f_(Γ_0 ) (n)

T_(T_(x^x ) ) (n) ≈ f_SVO (n)

The lower bound of TREE(n) is between these two.

T_(T_(x^x + 1))  ≈ f_LVO (n)

T_(T_(x^^x) ) (n) ≈ f_BHO (n) = f_(ψ(Ω_2) ) (n)

T_(T_(T_(x + 1) ) ) (n) ≈ f_(ϑ(Γ_(Ω+1) ) )(n)

T_(T_(T_(x^^x) ) ) (n) ≈ f_(ψ(Ω_3) ) (n)

LIMIT :  T_(T_(T...(T_(x^^x) ) ) ) (n) ≈ f_(ψ(Ω_ω) ) (n)

T_LIMIT < f_(ψ(Ω_ω) ) (n+1)

Numbers
The First T number : T_(T_(T...(T_(x^^x) ) ) ) (8) with 8 Ts

$$f_{ψ(Ω_ω)} (8)$$

The Second T number : T_(T_(T...(T_(x^^x) ) ) ) (8) with 8 Ts using the 8th order T function.

$$f_{ψ(Ω_ω*ω)} (8)$$

The Third T number : T_(T_(T...(T_(x^^x) ) ) ) (8) with 8 Ts using the 8th super^8 order T function.

$$f_{ψ(Ω_ω*ω^ω} ) (8)$$

The Final T number : the limit of nested meta-orders_8.

$$f_{ψ(Ω_ω*\epsilon_0} ) (8)$$