User blog:Sleepy Dragonfly/Uncomputable fast growing hierarchy

The fast growing hierarchy can only take recursive ordinals, i.e. f_(ω_1 CK) (n) is undefined because of a lack of fundamental sequence. By incorporating Rado’s sigma function, a relatively simple function with growth rate ~ω_1 CK, the fast growing hierarchy can be extended to include uncomputable functions.

u_0 (n)=Σ(n) u_(α+1) (n)=u_α^n (n) u_α (n)=u_(α[n]) (n) if α is a limit ordinal.

Examples u_1 (2)=Σ(4)=13 u_1 (3)=Σ^2 (6)≥Σ(3.514×〖10〗^18267 ) u_1 (4)=Σ^3 (13) u_2 (2)=u_1 (13)=Σ^13 (13) u_2 (3)≥u_1^2 (3.514×〖10〗^18267 )=u_1 (Σ^(3.514×〖10〗^18267 ) (3.514×〖10〗^18267 ))=Σ^((Σ^(3.514×〖10〗^18267 ) (3.514×〖10〗^18267 )) ) (Σ^(3.514×〖10〗^18267 ) (3.514×〖10〗^18267 ))

Questions - Is there an ordinal α that would let u_α (n) have growth rate ω_2 CK? ω_ω CK? Recursively inaccessible? Even higher? - Can this uncomputable fast growing hierarchy be used as a benchmark for uncomputable functions, which currently have no benchmark? - Can this uncomputable fast growing hierarchy be made stronger without being too complicated?