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Here are the definitions.
WIP!
 (1) Nil = n(24) = 25
 (2) UnaryNil = n(1) = 2
 (3) BinaryNil = n(2) = 3
 (4) TernaryNil = n(3) = 4
 (5) OctalNil = n(8) = 9
 (6) DecimalNil = n(10) = 11
 (7) DuodecimalNil = n(12) = 13
 (8) HexadecimalNil = n(16) = 17

Here are the rules:
 0 < a (a â‰ 0)
 N(a,0) > N(b,0) (a > b)
 N(a,b) > N(a,c) (b > c)
 Î©_{x} > N(a,b) (b

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!
When I say that array X = ordinal Y, it means that n(aX) = f_{Y}(a).
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(a1) = n(a) = a+1 = f_{0}(a).
AMAN array Lambda Notation
1 0
2 1
3 2
4 3
5 4
Next, increasing the second entry adds Ï‰ to the ordinal, sâ€¦
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I'm surprised that no one has tried to define this before. This is easy to define, so where's the huge flaw?
The language of First Order Theory Theory (FOTT) consists of statements, theories, and operators.
In FOTT, a variable is a set. A set is a collection of things, like {1,2,4,8,6,327,pi,Canada,Swiss Cheese,{NO U, THAT IT, CLEAR IT, STOP IT, @person TRIGGERED},...}.
A statement is an operator applied to variables, statements and/or theories (you'll see what theories are later). Here are the basic operators in FOTT:
 a=b is true iff variable a is the same as variable b.
 aâˆˆb is true iff variable a contains variable b.
 Â¬(a) is true iff statement a is not true
 (a)âˆ§(b) is true iff statements a and b are true
 âˆƒa(b) is true iff there doesn't exist a vâ€¦
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Since Aarex made the array notation here, I will make one too.
It is very similar to AAN at the beginning, but both notations eventually split paths.
You can find the analysis for this notation here.
WIP!
a is called the base.
 Let # is the rest of the expression.
 n(a) = a+1
 n(a1) = n(a)
 n(a# 1) = n(a#)
 n(ab #) = n(ab #)[a]
 n(ab #)[1] (b > 1) = n(ab1 #)
 n(ab #)[c] (b,c > 1) = n(n(ab #)[c1]b1 #)
 Else, follow the process, starting from the second entry of the array.
Growth Rate: Ï‰^{Ï‰}
Here, arrays are of the form a,b,c,..., where a,b,c,... are positive integers.
Note that Case 2 is terminal but Case 1 is not.
Let the entry is the entry that you're on.
 Case 1: If the value of the entry is 1, jump to the next entry.
 Case 2: If the value of the entry isâ€¦
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