Infinite Order Set Theory

Since we have first and second order set theory, naturally we should have third and higher order set theories, though I am not sure how they are defined. Anyway, let's assume they are well defined to produce this fundamental sequence:

Further Improved version:

For non-negative integer k and positive integer function R,

R₀(n) = Rayo's function (based on FOST or 1-st order set theory).

R_k-1(n) is defined as the function based on k-th order set theory.

Further growth shall be achieved through R_O(n) where O denotes any combination of countable ordinals.

The version below reaches only a growth rate of about $R_{\omega ^{\omega }}(n)$.

Improved version:

For non-negative integer k and positive integer function R,

R₀(n) = Rayo's function (based on FOST or 1-st order set theory); and

R_k-1(n) is defined as the function based on k-th order set theory.

R_1,0(n) = $R_{\omega }(n)$; R_1,1(n) = R_1,0 ⁿ (n); R_1,2(n) = R_1,1 ⁿ (n); and so on.

R_2,0(n) = $R_{1,\omega }(n)$; R_2,1(n) = R_2,0 ⁿ (n); R_2,2(n) = R_2,1 ⁿ (n); and so on.

R_1,0,0(n) = $R_{\omega ,\omega }(n)$; R_1,0,1(n) = R_1,0,0 ⁿ (n); R_1,0,2(n) = R_1,0,1 ⁿ (n); and so on.

R_1,1,0(n) = $R_{1,{0},\omega }(n)$; R_1,1,1(n) = R_1,1,0 ⁿ (n); R_1,1,2(n) = R_1,1,1 ⁿ (n); and so on.

R_2,0,0(n) = $R_{1,\omega ,\omega }(n)$; R_2,0,1(n) = R_2,0,0 ⁿ (n); R_2,0,2(n) = R_2,0,1 ⁿ (n); and so on.

Final form = $R_{n\blacktriangleleft {n}}(n)$, where $n\blacktriangleleft {n}$ = n,n, ... ,n,n (with n of n's)

With this improvement, the sequence is much stronger than original version which was strike-through for reference below.

For non-negative integer k and positive integer function R, define R₀(n) = Rayo's function (based on FOST or 1-st order set theory).

Then R_k+1(n) is defined as the function based on R_k(n)-th order set theory.

Example

R₀(984) = Rayo(984) > 65536.

R₁(984) is calculated based on R₀(984)-th or 66536^th order set theory.

R₂(984) is calculated based on R₁(984)-th order set theory.

Question

1. In this case, do you think$R_{\omega }(n)$, the diagonalization of the sequence, exist? If it does, I think it's not recursive, as R is uncomputable.
2. Do you think$R_{\omega }(n)$is at par with Large Number Garden Number for n > 10↑¹⁰10?
3. Do you think $R_{\omega }(n)$is at par with Bowers' Oblivion and Utter Oblivion for n > Rayo's number?