User blog:SuperSpruce/The T array function: Part 2: Dimensional Arrays

Here is the T array function, which now includes dimensional arrays.

Attempt 1.

A string is any expression in the outer brackets that is a number, comma, or bracket.


 * 1) is any string (can be empty)

@ is any non-empty string.

$ is any number of opening brackets (can be 0 opening brackets)

& is any string involving just 0’s, commas, any dimensional separators, and opening brackets. It cannot be empty.

Rules:

1. Base Rule: T_n[0]=n+1

2. Recursion Rule: T_n[$a#]=T_(T_(...(T_(T_n[a-1])[a-1])...)[a-1])[a-1] with n T’s.

3. Tailing Rule: [@,0]=[@]

4. Fixed-Point Rule: [0] in an expression inside the outer brackets becomes n. For example: T_n[ [0] ]=T_n[n].

5. Comma Rule: {0} becomes a comma.

6. Dimensional Array Rule: T_n[0{$d#}$c#]=T_n[n{$d-1#}n{$d-1#}n...n{$d-1#}n{$d#}$c-1#] with n n’s.

7. Catastrophic Rule: Rules 1-6 don’t apply: [&,$a#]=[[[...[[&,$a-1#],$a-1#]...],$a-1#],$a-1#] with n (a-1)’s. This could be on any row on any plane on any realm (3D space) etc. The n is from the T_n.

A few comparisons: T arrays vs. FGH ordinal

T_n[b] b

T_n[ [0] ] ω

T_n[0] ω2

T_n[0,1] ω^2

T_n[0,[0]] ω^ω

T_n[0,0,1] ε_0

I’m just guessing from here on out, because this function grows so fast that it is hard for me to analyze.

T_n[0,0,0,1]   ζ_0

T_n[0{1}1]   φ(ω,0)

It would be appreciated if you would give me the growth rate of T_n[0{0,1}1]=T_n[n{[n{[n{[...{[n{[n]}n]}...]}n]}n]}1] with n [{’s. Γ_0 is my guess, but it could be wildly inaccurate.

I HATE the autocorrecting feature changing [ [0] ] with no spaces in between to 0.