FANDOM


Go to part: 1

Here are the definitions. I will also explain AMAN in this analysis.

Also, you might ask: Why are you comparing AMAN with Lambda Notation and not standard ordinal notation? Well, starting from aepAMAN, AMAN gets so strong that standard ordinal notation gets too complicated at some point in the analysis. But Lambda Notation is simpler and stronger, so it is better for analyzing.

WIP!

sAMAN

When I say that array X = ordinal Y, it means that n(a|X) = fY(a).

Up to ω

First, increasing the first entry nests the base, which is the same as adding 1 to the ordinal. We start at the array 1, which is equal to the ordinal 0, since n(a|1) = n(a) = a+1 = f0(a).

AMAN array Lambda Notation
1 0
2 1
3 2
4 3
5 4

Up to ω^2

Next, increasing the second entry adds ω to the ordinal, since the second entry diagonalizes over the first entry, with the second entry decreased by 1 (Case 2).

AMAN array Lambda Notation
1,2 ω
2,2 ω+1
3,2 ω+2
1,3 ω2
2,3 ω2+1
1,4 ω3
1,5 ω4
1,6 ω5

Up to ω^ω

We can generalize this to arrays with any amount of entries. Increasing the n+1th entry adds ωn to the ordinal, since the n+1th entry diagonalizes over the nth entry, with the n+1th entry decreased by 1.

AMAN array Lambda Notation
1,1,2 ω2
2,1,2 ω2+1
3,1,2 ω2+2
1,2,2 ω2
1,3,2 ω2+ω2
1,1,3 ω22
1,2,3 ω22+ω
1,1,4 ω23
1,1,5 ω24
1,1,1,2 ω3
1,1,2,2 ω32
1,1,1,3 ω32
1,1,1,4 ω33
1,1,1,1,2 ω4
1,1,1,1,3 ω42
1,1,1,1,1,2 ω5
1,1,1,1,1,1,2 ω6
1,1,1,1,1,1,1,2 ω7

exAMAN

Up to ω^ω^2

Now, we introduce higher separators, starting with {2}. Entries with a {2} to the left diagonalize over the number of entries separated by commas, with the value of the entry decreased by 1 (Case 2.2). Adding an entry with a {2} to the left is the same as adding ω to the power of ω, and adding an entry with a comma to the left is the same as adding 1 to the power of ω.

AMAN array Lambda Notation
1{2}2 ωω
2{2}2 ωω+1
1,2{2}2 ωω
1,3{2}2 ωω+ω2
1,1,2{2}2 ωω2
1{2}3 ωω2
1,2{2}3 ωω2+ω
1{2}4 ωω3
1{2}1,2 ωω+1
1{2}2,2 ωω+1ω
1{2}1,3 ωω+12
1{2}1,1,2 ωω+2
1{2}1,1,1,2 ωω+3
1{2}1{2}2 ωω2
1{2}1,2{2}2 ωω2+ω+1
1{2}1{2}3 ωω22
1{2}1{2}1,2 ωω2+1
1{2}1{2}1,1,2 ωω2+2
1{2}1{2}1{2}2 ωω3
1{2}1{2}1{2}1,2 ωω3+1
1{2}1{2}1{2}1{2}2 ωω4
1{2}1{2}1{2}1{2}1{2}2 ωω5
1{2}1{2}1{2}1{2}1{2}1{2}2 ωω6

Up to ω^ω^ω

{2} diagonalizes over the number of entries separated with commas, {3} diagonalizes over the number of entries separated with {2}'s, and {x} diagonalizes over the number of entries separated with {x-1}'s (Case 2.2). Adding an entry with an {x+1} to the left adds ωx to the power of ω.

AMAN Array Lambda Notation
1{3}2 ωω2
2{3}2 ωω2+1
1,2{3}2 ωω2
1,1,2{3}2 ωω22
1{2}2{3}2 ωω2ω
1{2}1,2{3}2 ωω2ω+1
1{2}1{2}2{3}2 ωω2ω2
1{3}3 ωω22
1{2}2{3}3 ωω22+ωω
1{3}4 ωω23
1{3}1,2 ωω2+1
1{3}1,1,2 ωω2+2
1{3}1{2}2 ωω2
1{3}1{2}1,2 ωω2+ω+1
1{3}1{2}1{2}2 ωω2+ω2
1{3}1{2}1{2}1{2}2 ωω2+ω3
1{3}1{3}2 ωω22
1{3}1{3}1,2 ωω22+1
1{3}1{3}1{2}2 ωω22+ω
1{3}1{3}1{3}2 ωω23
1{3}1{3}1{3}1{3}2 ωω24
1{4}2 ωω3
1{4}1,2 ωω3+1
1{4}1{2}2 ωω3
1{4}1{3}2 ωω32
1{4}1{4}2 ωω32
1{4}1{4}1{4}2 ωω33
1{5}2 ωω4
1{5}1{5}2 ωω42
1{6}2 ωω5
1{7}2 ωω6
1{8}2 ωω7

ntAMAN

Up to Ψ(0)

You might have noticed something... {1} = ω0 in the power of ω, {2} = ω1 in the power of ω, and {x+1} = ωx in the power of ω. Does this seem familiar to you?

To get to Ψ(0), we can just let the x be an array, so we can extend the power tower as much as we like. (we use Cases 2.2 and 2.3)

AMAN array Lambda Notation
1{1,2}2 ωωω
2{1,2}2 ωωω+1
1,2{1,2}2 ωωω
1{2}2{1,2}2 ωωωω
1{3}2{1,2}2 ωωωω2
1{1,2}3 ωωω2
1{1,2}1,2 ωωω+1
1{1,2}1{2}2 ωωω
1{1,2}1{1,2}2 ωωω2
1{1,2}1{1,2}1{1,2}2 ωωω3
1{2,2}2 ωωω+1
1{2,2}1{2,2}2 ωωω+12
1{3,2}2 ωωω+2
1{1,3}2 ωωω2
1{2,3}2 ωωω2+1
1{1,4}2 ωωω3
1{1,1,2}2 ωωω2
1{1,1,1,2}2 ωωω3
1{1{2}2}2 ωωωω
1{1{2}1,2}2 ωωωω+1
1{1{2}1{2}2}2 ωωωω2
1{1{3}2}2 ωωωω2
1{1{3}1{3}2}2 ωωωω22
1{1{4}2}2 ωωωω3
1{1{1,2}2}2 ωωωωω
1{1{2,2}2}2 ωωωωω+1
1{1{1,3}2}2 ωωωωω2
1{1{1,1,2}2}2 ωωωωω2
1{1{1{2}2}2}2 ωωωωωω
1{1{1{2}1{2}2}2}2 ωωωωωω2
1{1{1{3}2}2}2 ωωωωωω2
1{1{1{1,2}2}2}2 ωωωωωωω
1{1{1{1,1,2}2}2}2 ωωωωωωω2
1{1{1{1{2}2}2}2}2 ωωωωωωωω
1{1{1{1{1,2}2}2}2}2 ωωωωωωωωω
1{1{1{1{1{2}2}2}2}2}2 ωωωωωωωωωω

epAMAN

Up to Ψ(1)

Now, we introduce a new separator, called /. 1/2 expands to 1{1{1{...}2}2}2, but 1/3 does not expand to 1{1{1{...}2}2}2/2. Instead, it expands to 1{1{1{.../2}2/2}2/2}2/2. The / separator is called an expanding separator, because it takes the array and expands by nesting it inside of {}.

We have 1{1/2}2/2 = Ψ(0)2 and 1{1/2}A/2 = Ψ(0)B, where A is an array and B is the ordinal that is equal to A. Next, 1{1/2}1{1/2}2/2 = Ψ(0)2, 1{2/2}2/2 = Ψ(0)ω, and 1{A/2}2/2 = Ψ(0)ωB. Next, 1{1{1/2}2/2}2/2 = Ψ(0)Ψ(0), 1{A{1/2}2/2}2/2 = Ψ(0)Ψ(0)ωB, and 1{1{1/2}A/2}2/2 = Ψ(0)Ψ(0)B. This gets us to Ψ(1).

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