User blog:SuperSpruce/The T array function

Here is the T array function.

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, 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_n0=T_n[n].

5. Catastrophic Rule: Rules 1-4 don’t apply: [&,$a#]=[[[...[[&,$a-1#],$a-1#]...],$a-1#],$a-1#] with n (a-1)’s.

A few comparisons: T arrays vs. FGH ordinal

T_n[b]  b

T_n0 ω

T_n[0] ω2

T_n[0,1]  ω^2

T_n[0,[0]] ω^ω

T_n[0,0,1] ε_0

It would be appreciated if you would give me the growth rate of TT(n)=T_n[n,n,n,...,n,n] with n n’s. This function grows so fast that it is hard for me to analyze.