User blog:SuperSpruce/The T Array Function Part 4: Double Comma T Function (DCT)

Here is the T array function, which now includes Double Comma arrays. This should reach ψ(ε_((Ω_2)+1)) after 5 failed attempts to reach this ordinal.

Attempt 1 at definition

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

An entry is a integer greater than or equal to 0.

@ is any non-empty string.
 * 1) and % are any strings (can be empty)

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

& is everything inside the outer brackets.

&& is everything inside the outer brackets, but a is decreased by 1.

Entry a is at Layer x, where x is the amount of curly braces an entry is enclosed by. The base layer is x=0.

Rules: [APPLY IF POSSIBLE BEFORE STARTING SCANNING PROCESS]

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]=[@] There cannot be anything after the 0. This rule still applies when the brackets are braces (denoting dimensional separators)

4. Preceding rule: [0[ = [[ (delete the 0)

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

6. Meta-Fixed-Point Rule: [0]2 becomes n{n{n{...{n{n}n}...}n}n}n with n sets of curly braces.

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

8. Double Comma Rule: {0,,} becomes a double comma.

However, the other rules will require a scanning process to check when they will be applied.

Now comes the scanning process. Start at the first entry in the brackets.

A. If the entry is 0, jump to the next entry at that layer. Then check that entry (return to the beginning of step A).

B. If the entry is not 0, then check what is immediately before it.

Ba. If a comma is immediately before it, then apply the catastrophic rule. The process ends.

Bb. If a double comma is immediately before it, apply the meta-catastrophic rule. The process ends.

Bc. If a [ is immediately before it, check what is immediately before that.

Bd. If that is a comma, return to step Ba. If that is a double comma, return to step Bb. If that is a [, return to step Bc. If that is a }, go to step C.

C. If a } is immediately before it, follow these steps.

Ca. Jump to the first entry of that dimensional separator.

Cb. If it is not a 0, apply the dimensional array rule. The process ends.

Cc. If it is a 0, jump to step A.

Dimensional Array Rule and Catastrophic Rule:

9. 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.

10. Catastrophic Rule: Follow these steps. One starts with [%0,$a#].

10a. Take the n from T_n and label it as s. The T_n still exists; we’re basically just copying n and calling it s. s will change later, but n will not change with it.

10b. Now find the innermost pair of brackets that surrounds the 0 in the [%0,$a#]. It can have any subscript. That pair of brackets is the brackets shown in the [%0,$a#].

10c. Let M_1 = [%0,$a-1#] and M_(r+1) = [%(M_r),$a-1#].

10d. Now do this with M_s.

10e. The pair of brackets the surrounds M_s must have the exact same subscript as the brackets shown in the [%0,$a#] initially.

11. Meta-Catastrophic Rule: Follow these steps. One starts with [%0,,$a#].

11a. Take the n from T_n and label it as s. The T_n still exists; we’re basically just copying n and calling it s. s will change later, but n will not change with it.

11b. Now find the innermost pair of brackets that surrounds the 0 in the [%0,,$a#]. It can have any subscript. That pair of brackets is the brackets shown in the [%0,$a#].

11c. Let M_1 = [%0,,$a-1#]2 and M_(r+1) = [%(M_r),,$a-1#]2.

11d. Now do this with M_s.

11e. The pair of brackets the surrounds M_s must have the exact same subscript as the brackets shown in the [%0,,$a#] initially.

A few comparisons: T arrays vs. FGH ordinal. The ordinal notations used here are Cantor’s normal form, binary Veblen function, and Madore’s ψ function. Previous comparisons can be seen here.

[0]2 ψ(εΩ+1)

[0,,1]2 ψ(ωψ1(1)+1)

[0,,0,1]2 ψ(ωψ1(1)+Ω)

[0,,0,,1]2 ψ(ωω ψ1(1)+1 )

[0{1,,}1]2 ψ(ψ1(2))

[0{1,,}0,,1]2 ψ(Ω2)

[0{1,,}{1,,}1]2 ψ(Ω2ω)

[0{0,1,,}1]2 ψ(Ω2Ω)

[0{0,,1,,}1]2 ψ(Ω2Ω2)

[0{0,,0,,1,,}1]2 ψ(Ω2Ω2 2 )

[0{0{1,,}1,,}1]2 ψ(Ω2Ω2 ω )

[0{0{0,,1,,}1,,}1]2 ψ(Ω2Ω2 Ω2 )

The limit of this notation is ψ(εΩ 2+1 ).

This analysis could be wrong, but I think it is correct.

Feel free to correct me on this complicated notation if I’m wrong!